Chapter 4.pdf

4. Context-rich Problems
Introduction
What Are the Characteristics of a Good Group Problem? Twenty-one Characteristics That Can Make a Problem Difficult How to Judge If a Problem is a Good Group Problem Linear Kinematics Problems
One-dimensional Motion at a Constant Velocity One-dimensional Motion at a Constant Acceleration One-dimensional Motion, Both Constant Velocity and Two-dimensional Motion, Constant Acceleration (Projectile) Two-dimensional Motion, Both Constant Velocity and Force Problems
No Acceleration (a = 0), No Force Components No Acceleration (a = 0), Force Components Force and Linear Kinematics Problems
Force and Circular Motion at a Constant Speed
Conservation of Energy and Conservation of Momentum
Conservation of Energy (Mechanical, Gravitational) Conservation of Energy (Mechanical) and Force Conservation of Energy (Mechanical) and Momentum Rotational Kinematics and Dynamics Problems
Center of Mass, Moment of Inertia, and Rotational Kinematics Conservation of Energy and Heat Problems
Oscillations and Waves Problems
10. Electricity and Magnetism Problems
What Are Context Rich Problems?
Context-rich problems are designed to encourage students to use an organized, logical problem-solvingstrategy instead of their novice, formula-driven, "plug-and-chug" strategy. Specifically, context rich problemsare designed to encourage students to (a) consider physics concepts in the context of real objects in the realworld; (b) view problem-solving as a series of decisions; and (c) use the fundamental concepts of physics toqualitatively analyze a problem before the mathematical manipulation of formulas.
Consequently, all context-rich problems have the following characteristics:
• Each problem is a short story in which the major character is the student. That is, each problem statement uses the personal pronoun "you." • The problem statement includes a plausible motivation or reason for "you" to calculate something.
• The objects in the problems are real (or can be imagined) -- the idealization process occurs explicitly.
• No pictures or diagrams are given with the problems. Students must visualize the situation by using • The problem cannot be solved in one step by plugging numbers into a formula.
These characteristics emphasize the need for students to make decisions by using their physics knowledge.
They encourage students to view physics problem-solving as something that they can do successfully andimagine doing in their future careers. They discourage the view that problem solving in physics is a purelymathematical exercise with no real-world applications for the average person.
What Are The Characteristics of a Good Group Problem?
Group problems should be more difficult to solve than easy problems typically given on an individual test.
But the increased difficulty should be primarily conceptual, not mathematical. Difficult mathematics is bestaccomplished by individuals, not by groups. So problems that involve long, tedious mathematics but littlephysics, or problems that require the use of a shortcut or "trick" that only experts would be likely to know donot make good group problems. In fact, the best group problems involve the straight-forward application ofthe fundamental principles (e.g., the definition of velocity and acceleration, the independence of motion in thevertical and horizontal directions) rather than the repeated use of derived formulas (e.g., v 2 Twenty-one Characteristics That Can Make a Problem More Difficult
There are twenty-one characteristics of a problem that can make it more difficult to solve than a standardtextbook exercise: Approach
1 Cues Lacking
A. No explicit target variable. The unknown variable of the problem is not explicitly stated.
Unfamiliar context. The context of the problem is very unfamiliar to the students (e.g., cosmology,molecules).
2 Agility with Principles
A. Choice of useful principles. The problem has more than one possible set of useful concepts that could be applied for a correct solution.
Two general principles. The correct solution requires students to use two major principles (e.g.,torque and linear kinematics).
C. Very abstract principles. The central concept in the problem is an abstraction of another abstract concept. (e.g., potential, magnetic flux).
3 Non-standard Application
A. Atypical situation. The setting, constraints, or complexity is unusual compared with textbook Unusual target variable. The problem involves an atypical target variable when compared withhomework problems.
Analysis of Problem
4 Excess or Missing Information
A. Excess numerical data. The problem statement includes more data than is needed to solve the Numbers must be supplied. The problem requires students to either remember or estimate anumber for an unknown variable.
C. Simplifying assumptions. The problem requires students to generate a simplifying assumption to 5 Seemingly Missing Information
A. Vague statement. The problem statement introduces a vague, new mathematical statement.
Special conditions or constraints. The problem requires students to generate information from theiranalysis of the conditions or constraints.
C. Diagrams. The problem requires students to extract information from a spatial diagram.
6 Additional Complexity
A. More than two subparts. The problem solution requires students decompose the problem into Five or more terms per equation. The problem involves five or more terms in a principle equation(e.g., three or more forces acting along one axes on a single object).
C. Two directions (vector components). The problem requires students to treat principles (e.g., Mathematical Solution
7 Algebra Required
A. No numbers. The problem statement does not use any numbers.
Unknown(s) cancel. Problems in which an unknown variable, such as a mass, ultimately factorsout of the final solution.
C. Simultaneous equations. A problem that requires simultaneous equations for a solution.
8 Targets Math Difficulties
A. Calculus or vector algebra. The solution requires the students to sophisticated vector algebra, such Lengthy or Detailed Algebra. A successful solution to the problem is not possible without workingthrough lengthy or detailed algebra (e.g., a messy quadratic equation).
BEWARE! Good group problems are difficult to construct because they can easily be made too complex
and difficult to solve. A good group problem does not have all of the above difficulty characteristics, but
usually only 2- 5 of these characteristics.
How to Create Context-rich Group Problems
One way to invent context-rich problems is to start with a textbook exercise or problem, then modify theproblem. You may find the following steps helpful: If necessary, determine a context (real objects with real motions or interactions) for the textbook
exercise or problem. You may want to use an unfamiliar context for a very difficult group problem.
Decide on a motivation -- Why would anyone want to calculate something in this context? Determine if you need to change the target variable to (a) make the problem more than a one-step exercise, or (b) make the target variable fit your motivation.
Determine if you need to change the given information (or target variable) to make the problem anapplication of fundamental principles (e.g., the definition of velocity or acceleration) rather than aproblem needing the application of many derived formulas.
Write the problem like a short story.
Decide how many "difficulty" characteristics (characteristics that make the problem more difficult)you want to include, then do some of the following: (a) think of an unfamiliar context; or use an atypical setting or target variable; (b) think of different information that could be given, so two approaches (e.g., kinematics and forces) would be needed to solve the problem instead of one approach (e.g., forces), or sothat more than one approach could be taken (c) write the problem so the target variable is not explicitly stated; (d) determine extra information that someone in the situation would be likely to have; or leave out common-knowledge information (e.g., the boiling temperature of water); (e) depending on the context, leave out the explicit statement of some of the problem idealizations (e.g., change "massless rope" to "very light rope"); or remove some information that studentscould extract from an analysis of the situation; take the numbers out of the problem and use variable names only; (g) think of different information that could be given, so the problem solution requires the use of vector components, geometry/trigonometry to eliminate an unknown, or calculus.
Check the problem to make sure it is solvable, the physics is straight-forward, and the mathematicsis reasonable. After you have written the problem, solve it yourself and use the judging strategy(next section) to determine its difficulty.
• physical work (pushing, pulling, lifting objects vertically, horizontally, or up ramps) • sports situations (falling, jumping, running, throwing, etc. while diving, bowling, playing golf, tennis, • situations involving the motion of bicycles, cars, boats, trucks, planes, etc.
• astronomical situations (motion of satellites, planets) • heating and cooling of objects (cooking, freezing, burning, etc.) Sometimes it is difficult to think of a motivation. We have used the following motivations: • You are . . . . (in some everyday situation) and need to figure out . . . .
• You are watching . . . . (an everyday situation) and wonder . . . .
• You are on vacation and observe/notice . . . . and wonder . . . .
• You are watching TV or reading an article about . . . . and wonder . . .
• Because of your knowledge of physics, your friend asks you to help him/her . . . .
• You are writing a science-fiction or adventure story for your English class about . . . . and need to • Because of your interest in the environment and your knowledge of physics, you are a member of a Citizen's Committee (or Concern Group) investigating . . . .
• You have a summer job with a company that . . . . Because of your knowledge of physics, your • You have been hired by a College research group that is investigating . . . . Your job is to determine • You have been hired as a technical advisor for a TV (or movie) production to make sure the science is correct. In the script . . . ., but is this correct? • When really desperate, you can use the motivation of an artist friend designing a kinetic sculpture! Decision Strategy for Judging Problems
Outlined below is a decision strategy to help you decide whether a context-rich problem is a good individualtest problem, group practice problem, or group test problem.
1. Read the problem statement. Draw the diagrams and determine the equations needed to solve the problem (through plan-a-solution step).
• the problem can be solved in one step, • the problem involves long, tedious mathematics, but little physics; or • the problem can only be solved easily using a "trick" or shortcut that only experts would be likely to know. (In other words, the problem should be a straight-forward application of fundamentalconcepts and principles.) 3. Check for the twenty-one characteristics that make a problem more difficult: 4. Decide if the problem would be a good group practice problem (20 - 25 minutes), a good group test problem (45 - 50 minutes), or a good (easy, medium, difficult) individual test problem, depending onthree factors: (a) the complexity of mathematics, (b) the timing (when problem is to be given to students),and (c) the number of difficulty characteristics of the problem: Type of Problem
Diff. Ch.
Group Practice Problems should be
Group Test Problems can be more
Individual Problems can be easy,
medium-difficult, or difficult:
There is considerable overlap in the criteria, so most problems can be judged to be both a good group
practice or test problem and a good easy, medium-difficult, or difficult individual problem.
Problems in This Booklet
Most of the context-rich problems in this booklet were group and individual test problems given in thealgebra-based introductory physics courses and the calculus-based courses at the University of Minnesota.
The problems vary greatly in length and difficulty. The more difficult problems were usually given ascooperative group problems. The problems also vary in quality. Feel free to edit, revise, and improve them! To discourage memorization and focus students' attention on the fundamental concepts necessary to solvethe problems, the tests include all equations and constants necessary to solve the problems. No otherequations are allowed to appear in the students solutions unless explicitly derived from the given equations.
These equations represent the fundamental concepts taught in the courses. A few new equations are addedfor each successive test, so the information available is the accumulation from the beginning of the course.
The next two pages contain the mathematical and physics accumulated at the end of the algebra-basedcourse and the calculus based course. All of the problems in this section can be solved with the equations onthese sheets.
The context-rich problems in this booklet are grouped according to the fundamental concepts andprinciple(s) required for a solution (instead of the typical textbook chapter or topic organization): linearkinematics problems, force problems, force with linear kinematics, force and circular motion, conservationproblems.
Equations: One-Semester Algebra-based Course
This is a closed book, closed notes exam. Calculators are permitted. The only formulas and constantswhich may be used in this exam are those given below. You may, of course, derive any expressions youneed from those that are given. If in doubt, ask. Define all symbols and justify all mathematical expressionsused. Make sure to state all of the assumptions used to solve a problem. Each problem is worth 25 points.
Useful Mathematical Relationships:
Fundamental Concepts:
Under Certain Conditions:
Useful constants: 1 mile = 5280 ft, 1 ft = 0.305 m, g = 9.8 m/s2 = 32 ft/s2, 1 lb = 4.45 N,
G = 6.7 x 10-11 N m2/kg2, ke = 9.0 x 109 N m2 / C2, e = 1.6 x 10-19 C
Equations: Two-Semester Calculus-based Course
Useful Mathematical Relationships:
Fundamental Concepts and Principles:
Under Certain Conditions:
Useful constants: 1 mile = 5280 ft, 1km = 5/8 mile, g = 9.8 m/s2 = 32 ft/s2 , 1 cal = 4.2 J,
RE = 4x103 miles, G = 6.7x10-11 Nm2/kg2, ke = 9.0 x 109 Nm2/C2, e = 1.6 x 10-19 C,
µo = 4p x 10-7 T m/A
Linear Kinematics Problems
The problems in this section can be solved with the application of the kinematics relationships. The problemsare divided into five groups according to the type of motion of the object(s) in the problem: (1) one-dimensional motion at a constant velocity; (2) one-dimensional motion at a constant acceleration; (3) one-dimensional motion, both constant velocity and constant acceleration, (4) two-dimensional (projectile)motion, and (5) two-dimensional motion, both constant velocity and constant acceleration.
One-dimensional, Constant Velocity
You are writing a short adventure story for your English class. In your story, two submarines on asecret mission need to arrive at a place in the middle of the Atlantic ocean at the same time. They startout at the same time from positions equally distant from the rendezvous point. They travel at differentvelocities but both go in a straight line. The first submarine travels at an average velocity of 20 km/hrfor the first 500 km, 40 km/hr for the next 500 km, 30 km/hr for the next 500 km and 50 km/hr for thefinal 500 km. In the plot, the second submarine is required to travel at a constant velocity, so thecaptain needs to determine the magnitude of that velocity.
It is a beautiful weekend day and, since winter will soon be here, you and four of your friends decideto spend it outdoors. Two of your friends just want to relax while the other two want some exercise.
You need some quiet time to study. To satisfy everyone, the group decides to spend the day on theriver. Two people will put a canoe in the river and just drift downstream with the 1.5 mile per hourcurrent. The second pair will begin at the same time as the first from 10 miles downstream. They willpaddle upstream until the two canoes meet. Since you have been canoeing with these people before,you know that they will have an average velocity of 2.5 miles per hour relative to the shore when theygo against this river current. When the two canoes meet, they will come to shore and you should bethere to meet them with your van. You decide to go to that spot ahead of time so you can study whileyou wait for your friends. Where will you wait? It's a sunny Sunday afternoon, about 65 °F, and you are walking around Lake Calhoun enjoying thelast of the autumn color. The sidewalk is crowded with runners and walkers. You notice a runnerapproaching you wearing a tee-shirt with writing on it. You read the first two lines, but are unable toread the third and final line before he passes. You wonder, "Hmm, if he continues around the lake, Ibet I'll see him again, but I should anticipate the time when we'll pass again." You look at your watchand it is 3:07 p.m. You recall the lake is 3.4 miles in circumference. You estimate your walking speedat 3 miles per hour and the runner's speed to be about 7 miles per hour.
You have joined the University team racing a solar powered car. The optimal average speed for thecar depends on the amount of sun hitting its solar panels. Your job is to determine strategy byprogramming a computer to calculate the car’s average speed for a day consisting of different raceconditions. To do this you need to determine the equation for the day’s average speed based on thecar’s average speed for each part of the trip. As practice you imagine that the day’s race consists ofsome distance under bright sun, the same distance with partly cloudy conditions, and twice thatdistance under cloudy conditions. 5.
Because of your technical background, you have been given a job as a student assistant in a Universityresearch laboratory that has been investigating possible accident avoidance systems for oil tankers.
Your group is concerned about oil spills in the North Atlantic caused by a super tanker running into an iceberg. The group has been developing a new type of down-looking radar which can detect largeicebergs. They are concerned about its rathershort range of 2 miles. Your research director has told you that the radar signal travels at the speed oflight which is 186,000 miles per second but once the signal arrives back at the ship it takes thecomputer 5 minutes to process the signal. Unfortunately, the super tankers are such huge ships that ittakes a long time to turn them. Your job is to determine how much time would be available to turn thetanker to avoid a collision once the tanker detects an iceberg. A typical sailing speed for super tankersduring the winter on the North Atlantic is about 15 miles per hour. Assume that the tanker is headingdirectly at an iceberg that is drifting at 5 miles per hour in the same direction that the tanker is going.
The following three problems are mathematically equivalent, with different contexts.
You and your friend run outdoors at least 10 miles every day no matter what the weather (wellalmost). Today the temperature is at a brisk 0 oF with a -20 oF wind chill. Your friend, a real runningfanatic, insists that it is OK to run. You agree to this madness as long as you both begin at your houseand end the run at her nice warm house in a way that neither of you has to wait in the cold. You knowthat she runs at a very consistent pace with an average speed of 3.0 m/s, while your average speed is aconsistent 4.0 m/s. Your friend finishes warming up first so she can get a head start. The plan is thatshe will arrive at her house first so that she can unlock the door before you arrive. Five minutes later,you notice that she dropped her keys. If she finishes her run first she will have to stand around in thecold and will not be happy. How far from your house will you be when you catch up to her if youleave immediately, run at your usual pace, and don't forget to take her keys? Because of your technical background, you have been given a job as a student assistant in a Universityresearch laboratory that has been investigating possible accident avoidance systems for automobiles.
You have just begun a study of how bats avoid obstacles. In your study, a bat is fitted with atransceiver that broadcasts the bats velocity to your instruments. Your research director has told youthat the signal travels at the speed of light which is 1.0 ft/nanosecond (1 nanosecond is 10-9 seconds).
You know that the bat detects obstacles by emitting a forward going sound pulse (sonar) which travelsat 1100 ft/s through the air. The bat detects the obstacle when the sound pulse reflect from theobstacle and that reflected pulse is heard by the bat. You are told to determine the maximum amountof time that a bat has after it detects the existence of an obstacle to change its flight path to avoid theobstacle. In the experiment your instruments tell you that a bat is flying straight toward a wall at aconstant velocity of 20.0 ft/s and emits a sound pulse when it is 10.0 ft from the wall.
You have been hired to work in a University research laboratory assisting in experiments to determinethe mechanism by which chemicals such as aspirin relieve pain. Your task is to calibrate your detectionequipment using the properties of a radioactive isotope (an atom with an unstable nucleus) which willlater be used to track the chemical through the body. You have been told that your isotope decays byfirst emitting an electron and then, some time later, it emits a photon which you know is a particle oflight. You set up your equipment to determine the time between the electron emission and the photonemission. Your apparatus detects both electrons and photons. You determine that the electron andphoton from a decay arrive at your detector at the same time when it is 2.0 feet from your radioactivesample. A previous experiment has shown that the electron from this decay travels at one half thespeed of light. You know that the photon travels at the speed of light which is 1.0 foot pernanosecond. A nanosecond is 10-9 seconds.
One Dimensional, Constant Acceleration
You are part of a citizen's group evaluating the safety of a high school athletic program. To help judgethe diving program you would like to know how fast a diver hits the water in the most complicateddive. The coach has his best diver perform for your group. The diver, after jumping from the highboard, moves through the air with a constant acceleration of 9.8 m/s2. Later in the dive, she passesnear a lower diving board which is 3.0 m above the water. With your trusty stop watch, youdetermine that it took 0.20 seconds to enter the water from the time the diver passed the lower board.
How fast was she going when she hit the water? As you are driving to school one day, you pass a construction site for a new building and stop to watchfor a few minutes. A crane is lifting a batch of bricks on a pallet to an upper floor of the building.
Suddenly a brick falls off the rising pallet. You clock the time it takes for the brick to hit the ground at2.5 seconds. The crane, fortunately, has height markings and you see the brick fell off the pallet at aheight of 22 meters above the ground. A falling brick can be dangerous, and you wonder how fast thebrick was going when it hit the ground. Since you are taking physics, you quickly calculate the answer.
Because of your knowledge of physics, and because your best friend is the third cousin of the director,you have been hired as the assistant technical advisor for the associate stunt coordinator on a newaction movie being shot on location in Minnesota. In this exciting scene, the hero pursues the villain upto the top of a bungee jumping apparatus. The villain appears trapped but to create a diversion shedrops a bottle filled with a deadly nerve gas on the crowd below. The script calls for the hero toquickly strap the bungee cord to his leg and dive straight down to grab the bottle while it is still in theair. Your job is to determine the length of the unstretched bungee cord needed to make the stuntwork. The hero is supposed to grab the bottle before the bungee cord begins to stretch so that thestretching of the bungee cord will stop him gently. You estimate that the hero can jump off the bungeetower with a maximum velocity of 10 ft/sec. straight down by pushing off with his feet and can react tothe villain's dropping the bottle by strapping on the bungee cord and jumping in 2 seconds.
You are helping a friend devise some challenging tricks for the upcoming Twin Cities FreestyleSkateboard Competition. To plan a series of moves, he needs to know the rate that the skateboard,with him on board, slows down as it coasts up the competition ramp which is at 30° to the horizontal.
Assuming that this rate is constant, you decide to have him conduct an experiment. When he istraveling as fast as possible on his competition skateboard, he stops pushing and coasts up thecompetition ramp. You measure that he typically goes about 95 feet in 6 seconds. Your friend weighs170 lbs. wearing all of his safety gear and the skateboard weighs 6 lbs.
You have a summer job working for a University research group investigating the causes of the ozonedepletion in the atmosphere. The plan is to collect data on the chemical composition of the atmosphereas a function of the distance from the ground using a mass spectrometer located in the nose cone of arocket fired vertically. To make sure the delicate instruments survive the launch, your task is todetermine the acceleration of the rocket before it uses up its fuel. The rocket is launched straight upwith a constant acceleration until the fuel is gone 30 seconds later. To collect enough data, the totalflight time must be 5.0 minutes before the rocket crashes into the ground.
One Dimensional, Constant Velocity and Constant Acceleration
You have landed a summer job as the technical assistant to the director of an adventure movie shothere in Minnesota. The script calls for a large package to be dropped onto the bed of a fast movingpick-up truck from a helicopter that is hovering above the road, out of view of the camera. Thehelicopter is 235 feet above the road, and the bed of the truck is 3 feet above the road. The truck istraveling down the road at 40 miles/hour. You must determine when to cue the assistant in thehelicopter to drop the package so it lands in the truck. The director is paying $20,000 per hour for thechopper, so he wants you to do this successfully in one take.
Just for the fun of it, you and a friend decide to enter the famous Tour de Minnesota bicycle race fromRochester to Duluth and then to St. Paul. You are riding along at a comfortable speed of 20 mphwhen you see in your mirror that your friend is going to pass you at what you estimate to be a constant30 mph. You will, of course, take up the challenge and accelerate just as she passes you until youpass her. If you accelerate at a constant 0.25 miles per hour each second until you pass her, how longwill she be ahead of you? In your new job, you are the technical advisor for the writers of a gangster movie about Bonnie andClyde. In one scene Bonnie and Clyde try to flee from one state to another. (If they got across thestate line, they could evade capture, at least for a while until they became Federal fugitives.) In thescript, Bonnie is driving down the highway at 108 km/hour, and passes a concealed police car that is 1kilometer from the state line. The instant Bonnie and Clyde pass the patrol car, the cop pulls onto thehighway and accelerates at a constant rate of 2 m/s2. The writers want to know if they make it acrossthe state line before the pursuing cop catches up with them.
The University Skydiving Club has asked you to plan a stunt for an air show. In this stunt, twoskydivers will step out of opposite sides of a stationary hot air balloon 5,000 feet above the ground.
The second skydiver will leave the balloon 20 seconds after the first skydiver but you want them bothto land on the ground at the same time. The show is planned for a day with no wind so assume that allmotion is vertical. To get a rough idea of the situation, assume that a skydiver will fall with a constantacceleration of 32 ft/sec2 before the parachute opens. As soon as the parachute is opened, theskydiver falls with a constant velocity of 10 ft/sec. If the first skydiver waits 3 seconds after steppingout of the balloon before opening her parachute, how long must the second skydiver wait after leavingthe balloon before opening his parachute? Because parents are concerned that children are learning "wrong" science from TV, you have beenasked to be a technical advisor for a science fiction cartoon show on Saturday morning. In the plot, avicious criminal (Natasha Nogood) escapes from a space station prison. The prison is locatedbetween galaxies far away from any stars. Natasha steals a small space ship and blasts off to meet herpartners somewhere in deep space. The stolen ship accelerates in a straight line at its maximumpossible acceleration of 30 m/sec2. After 10 minutes all of the fuel is burned up and the ship coasts ata constant velocity. Meanwhile, the hero (Captain Starr) learns of the escape while dining in the prisonwith the warden's daughter (Virginia Lovely). Of course he immediately (as soon as he finishesdessert) rushes off the recapture Natasha. He gives chase in an identical ship, which has an identicalmaximum acceleration, going in an identical direction. Unfortunately, Natasha has a 30 minute headstart. Luckily, Natasha's ship did not start with a full load of fuel. With his full load of fuel, Captain Starr can maintain maximum acceleration for 15 minutes. How long will it take Captain Starr's ship tocatch up to Natasha's? Because parents are concerned that children are learning "wrong" science from TV, you have beenasked to be a technical advisor for a new science fiction show. The show takes place on a spacestation at rest in deep space far away from any stars. In the plot, a vicious criminal (Alicia Badax)escapes from the space station prison. Alicia steals a small space ship and blasts off to meet herpartners somewhere in deep space. If she is to just barely escape, how long do her partners have totransport her off her ship before she is destroyed by a photon torpedo from the space station? In thestory, the stolen ship accelerates in a straight line at its maximum possible acceleration of 30 m/sec2.
After 10 minutes (600 seconds) all of the fuel is burned and the ship coasts at a constant velocity.
Meanwhile, the hero of this episode (Major Starr) learns of the escape while dining with the station'scommander. Of course she immediately rushes off to fire photon torpedoes at Alicia. Once fired, aphoton torpedo travels at a constant velocity of 20,000 m/s. By that time Alicia has a 30 minute(1800 seconds) head start on the photon torpedo.
You want to visit your friend in Seattle over Winter-quarter break. To save money, you decide totravel there by train. But you are late finishing your physics final, so you are late in arriving at the trainstation. You run as fast as you can, but just as you reach one end of the platform your train departs,30 meters ahead of you down the platform. You can run at a maximum speed of 8 m/s and the train isaccelerating at 1 m/s. You can run along the platform for 50 meters before you reach a barrier. Willyou catch your train? Because of your knowledge of physics, you have been assigned to investigate a train wreck between afast moving passenger train and a slower moving freight train both going in the same direction. Youhave statements from the engineer of each train and the stationmaster as well as some measurementswhich you make. To check the consistency of each person's description of the events leading up to thecollision, you decide to calculate the distance from the station that the collision should have occurred ifeveryone were telling what really happened and compare that with the actual position of the wreckwhich is 0.5 miles from the station. In this calculation you decide that you can ignore all reaction times.
Here is what you know:• The stationmaster claims that she noted that the freight train was behind schedule. As regulationsrequire, she switched on a warning light just as the last car of the freight train passed her.
The freight train engineer says he was going at a constant speed of 10 miles per hour.
The passenger train engineer says she was going at the speed limit of 40 miles per hour when sheapproached the warning light. Just as she reached the warning light she saw it go on andimmediately hit the brakes.
The warning light is located so that a train gets to it 2.0 miles before it gets to the station.
The passenger train slows down at a constant rate of 1.0 mile per hour for each minute as soon asyou hit the brakes.
DO ONLY THE PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE PROBLEM
AND DESCRIBE THE PHYSICS OF THE PROBLEM. DO NOT SOLVE THIS PROBLEM.
Two Dimensional, Constant Acceleration (Projectile Motion)
While on a vacation to Kenya, you visit the port city of Mombassa on the Indian Ocean. On the coastyou find an old Portuguese fort probably built in the 16th century. Large stone walls rise verticallyfrom the shore to protect the fort from cannon fire from pirate ships. Walking around on the ramparts,you find the fort's cannons mounted such that they fire horizontally out of holes near the top of the wallsfacing the ocean. Leaning out of one of these gun holes, you drop a rock which hits the ocean 3.0seconds later. You wonder how close a pirate ship would have to sail to the fort to be in range of thefort's cannon? Of course you realize that the range depends on the velocity that the cannonball leavesthe cannon. That muzzle velocity depends, in turn, on how much gunpowder was loaded into thecannon.
(a) Calculate the muzzle velocity necessary to hit a pirate ship 300 meters from the base of the fort.
(b) To determine how the muzzle velocity must change to hit ships at different positions, make a graph of horizontal distance traveled by the cannonball (range) before it hits the ocean as a function ofmuzzle velocity of the cannonball for this fort.
Because of your knowledge of physics, you have been hired as a consultant for a new James Bondmovie, "Oldfinger". In one scene, Bond jumps horizontally off the top of a cliff to escape a villain. Tomake the stunt more dramatic, the cliff has a horizontal ledge a distance h beneath the top of the cliffwhich extends a distance L from the vertical face of the cliff. The stunt coordinator wants you todetermine the minimum horizontal speed, in terms of L and h, with which Bond must jump so that hemisses the ledge.
You are on the target range preparing to shoot a new rifle when it occurs to you that you would like toknow how fast the bullet leaves the gun (the muzzle velocity). You bring the rifle up to shoulder leveland aim it horizontally at the target center. Carefully you squeeze off the shot at the target which is 300feet away. When you collect the target you find that your bullet hit 9.0 inches below where you aimed.
You have a great summer job working on the special effects team for a Minnesota movie, the sequel toFargo. A body is discovered in a field during the fall hunting season and the sheriff begins herinvestigation. One suspect is a hunter who was seen that morning shooting his rifle horizontally in thesame field. He claims he was shooting at a deer and missed. You are to design the “flashback” scenewhich shows his version of firing the rifle and the bullet kicking up dirt where it hits the ground. Thesheriff later finds a bullet in the ground. She tests the hunter’s rifle and finds the velocity that it shoots abullet (muzzle velocity). In order to satisfy the nitpickers who demand that movies be realistic, thedirector has assigned you to calculate the distance from the hunter that this bullet should hit the groundas a function of the bullet’s muzzle velocity and the rifle’s height above the ground.
The Minneapolis Police Department has hired you as a consultant in a robbery investigation. A thiefallegedly robbed a bank in the IDS Crystal Court. To escape the pursing security guards, the thieftook the express elevator to the roof of the IDS tower. Then, in order to not be caught with theevidence, she allegedly threw the money bag to a waiting accomplice on the roof of Dayton's, which isjust to the west of the IDS tower (they are separated by the Nicollet Mall). The defense attorneycontends that in order to reach the roof of Dayton's, the defendant would have had to throw the moneybag with a minimum horizontal velocity of 10 meters/second. But in a test, she could throw the bagwith a maximum velocity of no more than 5 meters/second. How will you advise the prosecutingattorney? You determine that he IDS tower is 250 meters high, Dayton's is 100 meters high and theMall is 20 meters wide.
You are watching people practicing archery when you wonder how fast an arrow is shot from a bow.
With a flash of insight you remember your physics and see how you can easily determine what youwant to know by a simple measurement. You ask one of the archers to pull back her bow string as faras possible and shoot an arrow horizontally. The arrow strikes the ground at an angle of 86 degreesfrom the vertical at 100 feet from the archer.
You read in the newspaper that rocks from Mars have been found on Earth. Your friend says that therocks were shot off Mars by the large volcanoes there. You are skeptical so you decide to calculatethe magnitude of the velocity that volcanoes eject rocks from the geological evidence. You know thegravitational acceleration of objects falling near the surface of Mars is only 40% that on the Earth. Youassume that you can look up the height of Martian volcanoes and find some evidence of the distancerocks from the volcano hit the ground from pictures of the Martian surface. If you assume the rocksfarthest from a volcano were ejected at an angle of 45 degrees, what is the magnitude of the rock’svelocity as a function of its distance from the volcano and the height of the volcano for the rock furthestfrom the volcano? Watching the world series (only as an example of physics in action), you wonder about the ability ofthe catcher to throw out a base runner trying to steal second. Suppose a catcher is crouched downbehind the plate when he observes the runner breaking for second. After he gets the ball from thepitcher, he throws as hard as necessary to second base without standing up. If the catcher throws theball at an angle of 30 degrees from the horizontal so that it is caught at second base at about the sameheight as that catcher threw it, how much time does it take for the ball to travel the 120 feet from thecatcher to second base? Because of your physics background, you have been hired as a consultant for a new movie aboutGalileo. In one scene, he climbs up to the top of a tower and, in frustration over the people whoridicule his theories, throws a rock at a group of them standing on the ground. The rock leaves hishand at 30° from the horizontal. The script calls for the rock to land 15 m from the base of the towernear a group of his detractors. It is important for the script that the rock take precisely 3.0 seconds tohit the ground so that there is time for a good expressive close-up. The set coordinator is concernedthat the rock will hit the ground with too much speed causing cement chips from the plaza to injure oneof the high priced actors. You are told to calculate that speed.
Tramping through the snow this morning, you were wishing that you were not here taking this test.
Instead, you imagined yourself sitting in the Florida sun watching winter league softball. You have hadbaseball on the brain ever since the Twins actually won the World Series. One of the fielders seemsvery impressive. As you watch, the batter hits a low outside ball when it is barely off the ground. Itlooks like a home run over the left center field wall which is 200 ft from home plate. As soon as theleft fielder sees the ball being hit, she runs to the wall, leaps high, and catches the ball just as it barelyclears the top of 10 ft high wall. You estimate that the ball left the bat at an angle of 30o. How muchtime did the fielder have to react to the hit, run to the fence, and leap up to make the catch ? You are still a member of a citizen's committee investigating safety in the high school sports program.
Now you are interested in knee damage to athletes participating in the long jump (sometimes called thebroad jump). The coach has her best long jumper demonstrate the event for you. He runs down thetrack and, at the take-off point, jumps into the air at an angle of 30 degrees from the horizontal. He comes down in a sand pit at the same level as the track 26 feet away from his take-off point. Withwhat velocity (both magnitude and direction) did he hit the ground? In your new job, you are helping to design stunts for a new movie. In one scene the writers want a carto jump across a chasm between two cliffs. The car is driving along a horizontal road when it goesover one cliff. Across the chasm, which is 1000 feet deep, is another road at a lower height. Theywant to know the minimum value of the speed of the car so that it does not fall into the chasm. Theyhave not yet selected the car so they want an expression for the speed of the car, v, in terms of thecar's mass, m, the width of the chasm, w, and the height of the upper road, h, above the lower road.
The stunt director will plug in the actual numbers after a car is purchased.
Your friend has decided to make some money during the next State Fair by inventing a game of skillthat can be installed in the Midway. In the game as she has developed it so far, the customer shoots arifle at a 5.0 cm diameter target falling straight down. Anyone who hits the target in the center wins astuffed animal. Each shot would cost 50 cents. The rifle would be mounted on a pivot 1.0 meterabove the ground so that it can point in any direction at any angle. When shooting, the customerstands 100 meters from where the target would hit the ground if the bullet misses. At the instant thatthe bullet leaves the rifle (with a muzzle velocity of 1200 ft/sec according to the manual), the target isreleased from its holder 7.0 meters above the ground. Your friend asks you to try out the game whichshe has set up on a farm outside of town. Before you fire the gun you calculate where you should aim.
You have a summer job with an insurance company and have been asked to help with the investigationof a tragic "accident." When you visit the scene, you see a road running straight down a hill which hasa slope of 10 degrees to the horizontal. At the bottom of the hill, the road goes horizontally for a veryshort distance becoming a parking lot overlooking a cliff. The cliff has a vertical drop of 400 feet tothe horizontal ground below where a car is wrecked 30 feet from the base of the cliff. Was it possiblethat the driver fell asleep at the wheel and simply drove over the cliff? After looking pensive, your bosstells you to calculate the speed of the car as it left the top of the cliff. She reminds you to be careful towrite down all of your assumptions so she can evaluate the applicability of the calculation to thissituation. Obviously, she suspects foul play.
You have a summer job with an insurance company and have been asked to help with the investigationof a tragic "accident." When you visit the scene, you see a road running straight down a hill which hasa slope of 10 degrees to the horizontal. At the bottom of the hill, the road goes horizontally for a veryshort distance becoming a parking lot overlooking a cliff. The cliff has a vertical drop of 400 feet tothe horizontal ground below where a car is wrecked 30 feet from the base of the cliff. The onlywitness claims that the car was parked on the hill, he can't exactly remember where, and the car justbegan coasting down the road. He did not hear an engine so he thinks that the driver was drunk andpassed out knocking off his emergency brake. He remembers that the car took about 3 seconds to getdown the hill. Your boss drops a stone from the edge of the cliff and, from the sound of it hitting theground below, determines that it takes 5.0 seconds to fall to the bottom. After looking pensive, shetells you to calculate the car's average acceleration coming down the hill based on the statement of thewitness and the other facts in the case. She reminds you to be careful to write down all of yourassumptions so she can evaluate the applicability of the calculation to this situation. Obviously, shesuspects foul play.
Your group has been selected to serve on a citizen's panel to evaluate a new proposal to search for lifeon Mars. On this unmanned mission, the lander will leave orbit around Mars falling through the atmosphere until it reaches 10,000 meters above the surface of the planet. At that time a parachuteopens and takes the lander down to 500 meters. Because of the possibility of very strong winds nearthe surface, the parachute detaches from the lander at 500 meters and the lander falls freely through thethin Martian atmosphere with a constant acceleration of 0.40g for 1.0 second. Retrorockets then fireto bring the lander to a softly to the surface of Mars. A team of biologists has suggested that Martianlife might be very fragile and decompose quickly in the heat from the lander. They suggest that anysearch for life should begin at least 9 meters from the base of the lander. This biology team hasdesigned a probe which is shot from the lander by a spring mechanism in the lander 2.0 meters abovethe surface of Mars. To return the data, the probe cannot be more than 11 meters from the bottom ofthe lander. Combining the data acquisition requirements with the biological requirements the teamdesigned the probe to enter the surface of Mars 10 meters from the base of the lander. For the probeto function properly it must impact the surface with a velocity of 8.0 m/s at an angle of 30 degrees fromthe vertical. Can this probe work as designed? You have been hired as a technical consultant for a new action movie. The director wants a scene inwhich a car goes up one side of an open drawbridge, leaps over the gap between the two sides of thebridge, and comes down safely on the other side of the bridge. This drawbridge opens in the middleby increasing the angle that each side makes with the horizontal by an equal amount. The directorwants the car to be stopped at the bottom of one side of the bridge and then accelerate up that side inan amount of time which will allow for all the necessary dramatic camera shots. He wants you todetermine the necessary constant acceleration as a function of that time, the gap between the two sidesof the open bridge, the angle that the side of the open bridge makes with the horizontal, and the massof the car.
Two Dimensional, Constant Velocity and Constant Acceleration
The following three problems have a very unfamiliar contexts.
You are sitting in front of your TV waiting for the World Series to begin when your mind wanders.
You know that the image on the screen is created when electrons strike the screen which then gives off
light from that point. In the first TV sets, the electron beam was moved around the screen to make a
picture by passing the electrons between two parallel sheets of metal called electrodes. Before the
electrons entered the gap between the electrodes, which deflect the beam vertically, the electrons had
a velocity of 1.0 x 106 m/s directly toward the center of the gap and toward the center of the screen.
Each electrode was 5.0 cm long (direction the electron was going), 2.0 cm wide and the two were
separated by 0.5 cm. A voltage was applied to the electrodes which caused the electrons passing
between them to have a constant acceleration directly toward one of the electrodes and away from the
other. After the electrons left the gap between the electrodes they were not accelerated and they
continued until they hit the screen. The screen was 15 cm from the end of the electrodes. What
vertical electron acceleration between the electrodes would be necessary to deflect the electron beam
20 cm from the center of the screen?
DO ONLY THE PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE PROBLEM
AND DESCRIBE THE PHYSICS OF THE PROBLEM. DO NOT SOLVE THIS PROBLEM.
You have a summer job in the cancer therapy division of a hospital. This hospital treats cancer byhitting the cancerous region with high energy protons using a machine called a cyclotron. When thebeam of protons leaves the cyclotron it is going at a constant velocity of 0.50 the speed of light. You are in charge of deflecting the beam so it hits the patient. This deflection is accomplished by passing
the proton beam between two parallel, flat, high voltage (HV) electrodes which have a length of 10
feet in the entering beam direction. Initially the beam enters the HV region going parallel to the surface
of the electrodes. Each electrode is 1 foot wide and the two electrodes are separated by 1.5 inches of
very good vacuum. A high voltage is applied to the electrodes so that the protons passing between
have a constant acceleration directly toward one of the electrodes and away from the other electrode.
After the protons leave the HV region between the plates, they are no longer accelerated during the
200 feet to the patient. You need to deflect the incident beam 1.0 degrees in order to hit the patient.
What magnitude of acceleration between the plates is necessary to achieve this deflection angle of 1.0
degree between the incident beam and the beam leaving the HV region? The speed of light is 1.0 foot
per nanosecond
(1 ft /(10-9 sec)).
DO ONLY THE PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE PROBLEM,
DESCRIBE THE PHYSICS OF THE PROBLEM, AND PLAN A SOLUTION. DO NOT
SOLVE THIS PROBLEM.
You have a summer job as an assistant in a University research group that is designing a devise tosample atmospheric pollution. In this devise, it is useful to separate fast moving ions from slow movingones. To do this the ions are brought into the device in a narrow beam so that all of the ions are goingin the same direction. The ion beam then passes between two parallel metal plates. Each plate is 5.0cm long, 4.0 cm wide and the two plates are separated by 3.0 cm. A high voltage is applied to theplates causing the ions passing between them to have a constant acceleration directly toward one of theplates and away from the other plate. Before the ions enter the gap between the plates , they are goingdirectly toward the center of the gap parallel to the surface of the plates. After the ions leave the gapbetween the plates, they are no longer accelerated during the 50 cm journey to the ion detector. Yourboss asks you to calculate the magnitude of acceleration between the plates necessary to separate ionswith a velocity of 100 m/s from those in the beam going 1000 m/s by 2.0 cm? Force Problems
The problems in this section can be solved with the application of Newton's Laws of Motion. The problemsare divided into four groups: (1) linear acceleration, no force components required for solution; (2) linearacceleration, force components required for solution; (3) no acceleration (a = 0), no force componentsrequired for solution; and (4) no acceleration (a = 0), force components required for solution. The specifictypes of forces involved in a problem (e.g., human push or pull, tension, normal, weight, friction, gravitational,electric) are indicated in bold type at the beginning of each problem.
Linear Acceleration, No Force Components
Tension, Weight: PLAN THE SOLUTION FOR THE FOLLOWING PROBLEM. An artist
friend of yours wants your opinion of his idea for a new kinetic sculpture. The basic concept is to
balance a heavy object with two lighter objects using two very light pulleys, which are essentially
frictionless, and lots of string. The sculpture has one pulley hanging from the ceiling by a string attached
to its center. Another string passes over this pulley. One end of this string is attached to a 25 lb object
while the other supports another pulley at its center. This second pulley also has a string passing over it
with one end attached to a 10 lb object and the other to a 15 lb object. Your friend hasn't quite
figured out the rest of the sculpture but wants to know if, ignoring the mass of the pulley and string, the
25 lb object will remain stationary during the time that the 10 and 15 lb objects are accelerating. DO
NOT SOLVE THE PROBLEM.
Weight, Normal: You have always been impressed by the speed of the elevators in the IDS building
in Minneapolis (especially compared to the one in the Physics building). You wonder about the
maximum acceleration for these elevators during normal operation, so you decide to measure it by
using your bathroom scale. While the elevator is at rest on the ground floor, you get in, put down your
scale, and stand on it. The scale reads 130 lbs. You continue standing on the scale when the elevator
goes up, carefully watching the reading. During the trip to the 50th floor, the greatest scale reading
was 180 lbs.
Tension, Weight: You have been hired to design the interior of a special executive express elevator
for a new office building. This elevator has all the latest safety features and will stop with an
acceleration of g/3 in case of any emergency. The management would like a decorative lamp hanging
from the unusually high ceiling of the elevator. You design a lamp which has three sections which hang
one directly below the other. Each section is attached to the previous one by a single thin wire which
also carries the electric current. The lamp is also attached to the ceiling by a single wire. Each section
of the lamp weighs 7.0 N. Because the idea is to make each section appear that it is floating on air
without support, you want to use the thinnest wire possible. Unfortunately the thinner the wire, the
weaker it is. To determine the thinnest wire that can be used for each stage of the lamp, calculate the
force on each wire in case of an emergency stop.
You are investigating an elevator accident which happened in a tall building. An elevator in this buildingis attached to a strong cable which runs over a pulley attached to a steel support in the roof. The otherend of the cable is attached to a block of metal called a counterweight which hangs freely. An electricmotor on the side of the elevator drives the elevator up or down by exerting a force on the side of theelevator shaft. You suspect that when the elevator was fully loaded, there was too large a force on the motor . A fully loaded elevator at maximum capacity weighs 2400 lbs. The counterweight weighs1000 lbs. The elevator always starts from rest at its maximum acceleration of g/4 whether it is goingup or down.
(a) What force does the wall of the elevator shaft exert on the motor if the elevator starts from restand goes up? What force does the wall of the elevator shaft exert on the motor if the elevator starts from restand goes down? Tension, Weight: An artist friend of yours wants your opinion of his idea for a new kinetic sculpture.
The basic concept is to balance a heavy object with two lighter objects using two very light pulleys,
which are essentially frictionless, and lots of string. The sculpture has one pulley hanging from the
ceiling by a string attached to its center. Another string passes over this pulley. One end of this string
is attached to a 25-lb object while the other supports another pulley at its center. This second pulley
also has a string passing over it with one end attached to a 10-lb object and the other to a 15-lb
object. Your friend hasn't quite figured out the rest of the sculpture but wants to know if, ignoring the
mass of the pulley and string, the 25-lb object will remain stationary during the time that the 10-lb and
15-lb objects are accelerating.
DO ONLY THE PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE PROBLEM,
DESCRIBE THE PHYSICS OF THE PROBLEM, AND PLAN A SOLUTION. DO NOT
SOLVE THIS PROBLEM.
Weight, Normal, Friction: Because of your physics background, you have been able to get a job
with a company devising stunts for an upcoming adventure movie being shot in Minnesota. In the
script, the hero has been fighting the villain on the top of the locomotive of a train going down a straight
horizontal track at 20 mph. He has just snuck on the train as it passed over a lake so he is wearing his
rubber wet suit. During the fight, the hero slips and hangs by his fingers on the top edge of the front of
the locomotive. The locomotive has a smooth steel vertical front face. Now the villain stomps on the
hero's fingers so he will be forced to let go and slip down the front of the locomotive and be crushed
under its wheels. Meanwhile, the hero's partner is at the controls of the locomotive trying to stop the
train. To add to the suspense, the brakes have been locked by the villain. It will take her 10 seconds
to open the lock. To her horror, she sees the hero's fingers give way before she can get the lock off.
Since she is the brains of the outfit, she immediately opens the throttle causing the train to accelerate
forward. This causes the hero to stay on the front face of the locomotive without slipping down giving
her time to save the hero's life. The movie company wants to know what minimum acceleration is
necessary to perform this stunt. The hero weighs 180 lbs. in his wet suit. The locomotive weighs 100
tons. You look in a book giving the properties of materials and find that the coefficient of kinetic
friction for rubber on steel is 0.50 and its coefficient of static friction is 0.60.
Weight, Normal, Friction: While working in a mechanical structures laboratory, your boss assigns
you to test the strength of ropes under different conditions. Your test set-up consists of two ropes
attached to a 30 kg block which slides on a 5.0 m long horizontal table top. Two low friction, light
weight pulleys are mounted at opposite ends of the table. One rope is attached to each end of the 30
kg block. Each of these ropes runs horizontally over a different pulley. The other end of one of the
ropes is attached to a 12 kg block which hangs straight down. The other end of the second rope is
attached to a 20 kg block also hanging straight down. The coefficient of kinetic friction between the
block on the table and the table's surface is 0.08. The 30 kg block is initially held in place by a
mechanism that is released when the test begins so, that the block is accelerating during the test.
During this test, what is the force exerted on the rope supporting the 12 kg block?
Linear Acceleration, Force Components
Human, Weight, Normal: You are taking care of two small children, Sarah and Rachel, who are
twins. On a nice cold, clear day you decide to take them ice skating on Lake of the Isles. To travel
across the frozen lake you have Sarah hold your hand and Rachel's hand. The three of you form a
straight line as you skate, and the two children just glide. Sarah must reach up at an angle of 60
degrees to grasp your hand, but she grabs Rachel's hand horizontally. Since the children are twins,
they are the same height and the same weight, 50 lbs. To get started you accelerate at 2.0 m/s2. You
are concerned about the force on the children's arms which might cause shoulder damage. So you
calculate the force Sarah exerts on Rachel's arm, and the force you exert on Sarah's other arm. You
assume that the frictional forces of the ice surface on the skates are negligible.
Tension, Weight, Normal, and Friction: You are planning to build a log cabin in northern
Minnesota. You will pull the logs up a long, smooth hill to the building site by means of a rope
attached to a winch. You need to buy a rope for this purpose, so you need to know how strong the
rope must be. Stronger ropes cost more. You know that the logs weigh a maximum of 200 kg. You
measure that the hill is at an angle of 30o with respect to the horizontal, and the coefficient of kinetic
friction between a log and the hill is 0.90. When pulling a log up the hill, you will make sure that the
rope stays parallel to the surface of the hill and the acceleration of the log is never more than 0.80 m/s2.
How strong a rope should you buy?
10. Tension, Weight, Normal, Friction: You have taken a summer job at a warehouse and have
designed a method to help get heavy packages up a 15° ramp. In your system a package is attachedto a rope which runs parallel to the ramp and over a pulley at the top of the ramp. After passing overthe pulley the other end of the rope is attached to a counterweight which hangs straight down. In yourdesign the mass of the counterweight is always adjusted to be twice the mass of the package. Yourboss is worried about this pulley system. In particular, she is concerned that the package will be toodifficult to handle at the top of the ramp and tells you to calculate its acceleration. To determine theinfluence of friction between the ramp and the package you run some tests. You find that you can pusha 50 kg package with a horizontal force of 250 Newtons at a constant speed along a level floor madeof the same material as the ramp.
Tension, Weight, Normal, Friction: After graduating you get a job in Northern California. To
move there, you rent a truck for all of your possessions. You also decide to take your car with you by
towing it behind the truck. The instructions you get with the truck tells you that the maximum truck
weight when fully loaded is 20,000 lbs. and that the towing hitch that you rented has a maximum
strength of 1000 lbs. Just before you leave, you weigh the fully loaded truck and find it to be 15,000
lbs. At the same time you weigh your car and find it to weigh 3000 lbs. You begin to worry if the
hitch is strong enough. Then you remember that you can push your car and can easily keep it moving
at a constant velocity. You know that air resistance will increase as the car goes faster but from your
experience you estimate that the sum of the forces due to air resistance and friction on the car is not
more than 300 lbs. If the largest hill you have to go up is sloped at 10o from the horizontal, what is the
maximum acceleration you can safely have on that hill?
DO ONLY THE PROBLEM SOLVING STEPS NECESSARY TO FOCUS THE PROBLEM,
DESCRIBE THE PHYSICS OF THE PROBLEM, AND PLAN A SOLUTION. DO NOT
SOLVE THIS PROBLEM.
Weight, Normal, Friction: Because of your physics background, you have been able to get a job
with a company devising stunts for an upcoming adventure movie being shot in Minnesota. In the
script, the hero has been fighting the villain on the top of the locomotive of a train going down a straight
horizontal track at 20 mph. He has just snuck on the train as it passed over a lake so he is wearing his
rubber wet suit. During the fight, the hero slips and hangs by his fingers on the top edge of the front of
the locomotive. The locomotive has a smooth steel front face sloped at 20o from the vertical so that
the bottom of the front is more forward that the top. Now the villain stomps on the hero's fingers so he
will be forced to let go and slip down the front of the locomotive and be crushed under its wheels.
Meanwhile, the hero's partner is at the controls of the locomotive trying to stop the train. To add to
the suspense, the brakes have been locked by the villain. It will take her 10 seconds to open the lock.
To her horror, she sees the hero's fingers give way before she can get the lock off. Since she is the
brains of the outfit, she immediately opens the throttle causing the train to accelerate forward. This
causes the hero to stay on the front face of the locomotive without slipping down giving her time to
save the hero's life. The movie company wants to know what minimum acceleration is necessary to
perform this stunt. The hero weighs 180 lbs. in his wet suit. The locomotive weighs 100 tons. You
look in a book giving the properties of materials and find that the coefficient of kinetic friction for
rubber on steel is 0.50 and its coefficient of static friction is 0.60.
Gravitational: You have been hired as a consultant for the new Star Trek TV series to make sure
that any science on the show is correct. In this episode, the crew of the Enterprise discovers an
abandoned space station in deep space far from any stars. This station is obviously the work of an
advanced race and consists of four identical 3 x 1020 kg asteroids configured so that each is at the
corner of a square with 200 km sides. According to the tricorder, the station has been abandoned for
at least two centuries. You know that such a configuration is unstable and worry whether there would
be observable motion of the asteroids after two hundred years so you calculate the acceleration of one
of the asteroids in the proposed configuration. Make sure you give both the magnitude and the
direction of the acceleration.
Gravitational: Because the movie industry is trying to make the technical details of movies as correct
as possible, you have been made a member of a panel reviewing the details of a new science fiction
script. Although neither astronomy nor navigation is your field, you are disturbed by one scene in which
a space ship which is low on fuel is attempting to land on the Earth. As the ship approaches, it is
heading straight for the center of the Earth. The commander cuts off the ship's engines so that it will be
pulled in by the Earth's gravitational force. As the commander looks in the viewer, she sees the Earth
straight ahead and the Moon off to the left at an angle of 30o. The line between the centers of theMoon and Earth is at right angles to the initial path of the space ship. Under these conditions you don'tthink the ship will continue heading toward the Earth, so you calculate the component of its accelerationwhich is perpendicular to the initial path of the ship. First you look up the distance between the Earthand the Moon (3.8 x 105 km), the mass of the Earth (6.0 x 1024 kg), the mass of the Moon (7.3 x 1022 kg), the radius of the Earth (6.4 x 103 km), the radius of the Moon (1.7 x 103 km), and the universalgravitational constant (6.7 x 10-11 N m2/kg2). As a first approximation, you decide to neglect the effectof the Sun and the other planets in the solar system. You guess that a space ship such as described inthe script might have a mass of about 100,000 kg.
No Acceleration (a = 0), No Force Components
Weight - Buoyancy, Normal, Friction, Electric: The quarter is almost over so you decide to have
a party. To add atmosphere to your otherwise drab apartment, you decide to decorate with balloons.
You buy about fifty and blow them up so that they are all sitting on your carpet. After putting most of
them up, you decide to play with the few balloons left on the floor. You rub one on your sweater and
find that it will "stick" to a wall. Ah ha, you know immediately that you are observing the electric force
in action. Since it will be some time before you guests arrive and you have already made the onion dip,
you decide to calculate the minimum electric force of the wall on the balloon. You know that the air
exerts a net upward force (the "buoyant" force) on the balloon which makes it almost float. You
measure that the weight of the balloon minus the buoyant force of the air on the balloon is 0.05 lb. By
reading your physics book, you estimate that the coefficient of static friction between the wall and the
balloon (rubber and concrete) is 0.80.
Tension, Weight, Electric: While working in a University research laboratory you are given the job
of testing a new device for precisely measuring the weight of small objects. The device consists of two
very light strings attached at one end to a support. An object is attached to the other end of each
string. The strings are far enough apart so that objects hanging on them don’t touch. One of the
objects has a very accurately known weight while the other object is the unknown. A power supply is
slowly turned on to give each object an electric charge which causes the objects to slowly move away
from each other (repel) because of the electric force. When the power supply is kept at its operating
value, the objects come to rest at the same horizontal level. At that point, each of the strings supporting
them makes a different angle with the vertical and that angle is measured. To test the device, you want
to calculate the weight of an unknown sphere from the measured angles and the weight of a known
sphere. You use a standard sphere with a known weight of 2.000 N supported by a string which
makes an angle of 10.0° with the vertical. The unknown sphere's string makes an angle of 20.0° with
the vertical.
Gravitational: You are writing a short science fiction story for your English class. You get your idea
from the fact that when people cross the Earth's equator for the first time, they are awarded a
certificate to commemorate the experience. In your story it is the 21st Century and you are the tour
director for a trip to the moon. Transplanetary Tours promises tour participants a certificate to
commemorate their passage from the stronger influence of the Earth's gravitational pull to the stronger
gravitational pull of the moon. To finish the story, you need to figure out where on the trip you should
award the certificate. In your physics book you look up the distance between the Earth and the Moon
(3.8 x 105 km), the mass of the Earth (6.0 x 1024 kg), the mass of the Moon (7.3 x 1022 kg), the
radius of the Earth
(6.4 x 103 km), the radius of the Moon (1.7 x 103 km), and the universal gravitational constant (6.7 x10-11 N m2/kg2).
Gravitational: You have been hired as a consultant for the new Star Trek TV series to make sure
that the science in the show is correct. In this episode, the crew of the Enterprise goes into standard
orbit around a newly discovered planet. The plot requires that the planet is hollow and contains the
underground cities of a lost civilization. From orbit the science officer determines that the radius of the
planer is 1/4 (one-fourth) that of Earth. The first officer beams down to the surface of the planet and
measures that his weight is only 1/2 (one-half) of his weight on Earth. How does the mass of this
planet compare with the mass of the Earth? If it were hollow, its density would be less than Earth.
Are the measurements consistent with a hollow planet? Gravitational, Electric: You and a friend are reading a newspaper article about nuclear fusion
energy generation in stars. The article describes the helium nucleus, made up of two protons and two
neutrons, as very stable so it doesn't decay. You immediately realize that you don't understand why
the helium nucleus is stable. You know that the proton has the same charge as the electron except that
the proton charge is positive. Neutrons you know are neutral. Why, you ask your friend, don't the
protons simply repel each other causing the helium nucleus to fly apart? Your friend says she knows
why the helium nucleus does not just fly apart. The gravitational force keeps it together, she says. Her
model is that the two neutrons sit in the center of the nucleus and gravitationally attract the two protons.
Since the protons have the same charge, they are always as far apart as possible on opposite sides of
the neutrons. What mass would the neutron have if this model of the helium nucleus works? Is that a
reasonable mass? Looking in your physics book, you find that the mass of a neutron is about the same
as the mass of a proton and that the diameter of a helium nucleus is 3.0 x 10-13 cm.
No Acceleration (a = 0), Force Components
Tension, Weight, Friction: You are taking advantage of an early snow to go sledding. After a long
afternoon of going up and down hills with your sled, you decide it is time to go home. You are thankful
that you can pull your sled without climbing any more hills. As you are walking home, dragging the
sled behind you by a rope fastened to the front of the sled, you wonder what the coefficient of friction
of the snow on the sled is. You estimate that you are pulling on the rope with a 2 pound force, that the
sled weighs 10 pounds, and that the rope makes an angle of 25 degrees to the level ground.
Human, Weight, Normal, Friction: You are helping a friend move into a new apartment. A box
weighing 150 lbs. needs to be moved to make room for a couch. You are taller than the box, so you
reach down to push it at an angle of 50 degrees from the horizontal. The coefficient of static friction
between the box and the floor is 0.50 and the coefficient of kinetic friction between the box and the
floor is 0.30.
(a) If you want to exert the minimum force necessary, how hard would you push to keep the box
(b) Suppose you bent your knees so that your push were horizontal. How hard would you push to Human, Weight, Normal, Friction: You are helping an investigation of back injuries in the
construction industry. Your assignment is to determine why there is a correlation of the height of the
worker to the likelihood of back injury. You suspect that some back injuries are related to the way
people push heavy objects in order to move them. When people push an object, such as a box,
across the floor they tend to lean down and push at an angle to the horizontal. Taller people push at a
larger angle with respect to the horizontal than shorter people. To present your ideas to the rest of the
research team, you decide to calculate the force a 200-lb box exerts on a 150-lb person when they
push it across a typical floor at a constant velocity of 7.0 ft/s as a function of the angle with respect to
the horizontal at which the person pushes the box. Once you have your function, you will use angles of
0o, 10o, 20o, 30o, and 40o to make a graph of the result for the presentation. One of your coworkers
tells you that a typical coefficient of static friction between a box and a floor of 0.60 and while a typical
coefficient of kinetic friction between a box and a floor is 0.50. (Don't forget to make the graph).
Tension, Weight: Your are part of a team to help design the atrium of a new building. Your boss,
the manager of the project, wants to suspend a 20-lb sculpture high over the room by hanging it from
the ceiling using thin, clear fishing line (string) so that it will be difficult to see how the sculpture is held
up. The only place to fasten the fishing line is to a wooden beam which runs around the edge of the
room at the ceiling. The fishing line that she wants to use will hold 20 lbs. (20-lb test) so she suggests
attaching two lines to the sculpture to be safe. Each line would come from the opposite side of the
ceiling to attach to the hanging sculpture. Her initial design has one line making an angle of 20o with the
ceiling and the other line making an angle of 40o with the ceiling. She knows you took physics, so she
asks you if her design can work.
Electric, Weight, Tension: While working in a University research laboratory you are given the job
of testing a new device, called an electrostatic scale, for precisely measuring the weight of small
objects. The device is quite simple. It consists of two very light but strong strings attached to a
support so that they hang straight down. An object is attached to the other end of each string. One of
the objects has a very accurately known weight while the other object is the unknown. A power
supply is slowly turned on to give each object an electric charge which causes the objects to slowly
move away from each other (repel) because of the electric force. When the power supply is kept at its
operating value, the objects come to rest at the same horizontal level. At that point, each of the strings
supporting them makes a different angle with the vertical and that angle is measured. To test the
device, you want to calculate the weight of an unknown sphere from the measured angles and the
weight of a known sphere. You use a standard sphere with a known weight of 2.00000 N supported
by a string which makes an angle of 10.00o with the vertical. The unknown sphere's string makes an Force and Linear Kinematics
The following problems require both Newton's Laws of Motion and one or more kinematics relationship fora solution. The specific types of forces involved in a problem (e.g., human push or pull, tension, normal,weight, friction, gravitational, electric) are indicated in bold type at the beginning of each problem.
Weight, Normal: While driving in the mountains, you notice that when the freeway goes steeply
down hill, there are emergency exits every few miles. These emergency exits are straight dirt ramps
which leave the freeway and are sloped uphill. They are designed to stop trucks and cars that lose
their breaks on the downhill stretches of the freeway even if the road is covered in ice. You are
curious, so you stop at the next emergency road. You estimate that the road rises at an angle of 10o
from the horizontal and is about 100 yards (300 ft) long. What is the maximum speed of a truck that
you are sure will be stopped by this road, even if the frictional force of the road surface is negligible?
Weight, Normal: While driving in the mountains, you notice that when the freeway goes steeply
down hill, there are emergency exits every few miles. These emergency exits are straight dirt ramps
which leave the freeway and are sloped uphill. They are designed to stop trucks and cars that lose
their breaks on the downhill stretches of the freeway even if the road is covered in ice. You wonder at
what angle from the horizontal an emergency exit should rise to stop a 50 ton truck going 70 mph up a
ramp 100 yards (300 ft) long, even if the frictional force of the road surface is negligible.
Weight, Normal: You and a few friends have decided to open a small business called Wee Deliver.
The business will guarantee to deliver any box between 5 lbs. and 500 lbs. to any location in the Twin
City area by the next day. At your distribution center, boxes slide down a ramp between the delivery
area and the sorting area. In designing the distribution center, you must determine the angle this ramp
should have with the horizontal so that a 500-lb box takes 5.0 seconds to slide down the ramp starting
from rest at the top. When the box arrives at the bottom of the ramp, its speed should not be too large
or the contents of the box might be damaged. You decide that this speed should be 10 ft/s. Using the
latest technology, your ramp will have a very slippery surface so you make the approximation that the
frictional force between the ramp and the box can be neglected.
Weight, Normal: You are watching a ski jump contest on television when you wonder how high the
skier is when she leaves the starting gate. In the ski jump, the skier glides down a long ramp. At the
end of the ramp, the skier glides along a short horizontal section which ends abruptly so that the skier
goes into the air. You measured that the skier was in the air for 2.3 seconds and landed 87 meters, in
the horizontal direction, from the point she went into the air. Make the best estimate of the height of
the starting gate at the top of the ramp from the horizontal section from which the skier takes off into
the air. Make clear on what assumptions your answer depends (this is why it is an estimate).
Weight, Normal, Friction: You are passing a construction site on the way to physics class, and stop
to watch for awhile. The construction workers appear to be going on coffee break, and have left a
large concrete block resting at the top of a wooden ramp. As soon as their backs are turned, the
block begins to slide down the ramp. You quickly clock the time for the block to reach the bottom of
the ramp at 10 seconds. You wonder how long the ramp is. You estimate that the ramp is at an angle
of about 20o to the horizontal. In your physics book you find that the coefficient of kinetic friction
between concrete and wood is 0.35.
Weight, Normal, Friction: You have a summer job at a company that specializes in the design of
sports facilities. The company has been given the contract to design a new hockey rink to try to keep
the North Stars in town. The rink floor is very flat and horizontal and covered with a thick coat of ice.
Your task is to determine the refrigeration requirements which gives best temperature for the ice. You
have a table which gives the coefficient of static and kinetic friction between ice and the standard NHL
hockey puck as a function of ice temperature. You have been told that the hockey game will be more
exciting if passes are swift and sure. Experts say that the passing game is best if, after it goes 5.0 m, a
puck has a speed which is 90% of the speed with which it left the hockey stick. A puck typically has a
speed of 20 km/hr when it leaves the hockey stick for a pass.
Weight, Normal, Friction: You and some friends visit the Minnesota State Fair and decide to play a
game on the Midway. To play the game you must slide a metal hockey-type puck up a wooden ramp
so that it drops through a hole at the top of the ramp. Your prize, if you win, is a large, pink, and
rather gaudy, stuffed poodle. You realize the secret to winning is giving the puck just enough velocity
at the bottom of the ramp to make it to the hole. You estimate the distance from the bottom of the
ramp to the hole at about 10 feet, and the ramp appears to be inclined with an angle of 10o from the
horizontal. You just got out of physics class and recall the coefficient of static friction between steel
and wood is 0.1 and the coefficient of kinetic friction between steel and wood is 0.08. The mass of
the puck is about 2.5 lbs. You decide to impress your friends by sliding the puck at the precise speed
on the first try so as to land it in the hole. You slide the puck at 8.0 ft/sec. Do you win the stuffed
poodle?
Weight, Normal, Tension, Friction: Finally you are leaving Minneapolis to get a few days of Spring
break, but your car breaks down in the middle of nowhere. A tow truck weighing 4000 lbs. comes
along and agrees to tow your car, which weighs 2000 lbs., to the nearest town. The driver of the truck
attaches his cable to your car at an angle of 20o to the horizontal. He tells you that his cable has a
strength of 500 lbs. He plans to take 10 seconds to tow your car at a constant acceleration from rest
in a straight line along the flat road until he reaches the maximum speed limit of 45 miles/hour. Can the
driver carry out his plan? You assume that rolling friction behaves like kinetic friction, and the
coefficient of rolling friction between your tires and the road is 0.10.
Weight, Normal, Friction: While visiting a friend in San Francisco you decide to drive around the
city. You turn a corner and are driving up a steep hill. Suddenly, a small boy runs out on the street
chasing a ball. You slam on the brakes and skid to a stop leaving a 50 foot long skid mark on the
street. The boy calmly walks away but a policeman watching from the sidewalk walks over and gives
you a ticket for speeding. You are still shaking from the experience when he points out that the speed
limit on this street is 25 mph. After you recover your wits, you examine the situation more closely.
You determine that the street makes an angle of 20o with the horizontal and that the coefficient of static
friction between your tires and the street is 0.80. You also find that the coefficient of kinetic friction
between your tires and the street is 0.60. Your car's information book tells you that the mass of your
car is 1570 kg. You weigh 130 lbs. Witnesses say that the boy had a weight of about 60 lbs. and
took 3.0 seconds to cross the 15 foot wide street. Will you fight the ticket in court?
Weight, Lift, Thrust, Drag: One morning while waiting for class to begin, you are reading a
newspaper article about airplane safety. This article emphasizes the role of metal fatigue in recent
accidents. Metal fatigue results from the flexing of airframe parts in response to the forces on the plane
especially during take off and landings. As an example, the reporter uses a plane with a take off weight
of 200,000 lbs. and take off speed of 200 mph which climbs at an angle of 30o with a constant acceleration to reach its cruising altitude of 30,000 feet with a speed of 500 mph. The three jetengines provide a forward thrust of 240,000 lbs. by pushing air backwards. The article then goes onto explain that a plane can fly because the air exerts an upward force on the wings perpendicular totheir surface called "lift." You know that air resistance is also a very important force on a plane and isin the direction opposite to the velocity of the plane. The article tells you this force is called the "drag."Although the reporter writes that some metal fatigue is primarily caused by the lift and some by thedrag, she never tells you their size for her example plane. Luckily the article contains enoughinformation to calculate them, so you do.
Force and Circular Motion at a Constant Speed
The problems in this section require the application of Newton's Laws of Motion as well as the relationshipsbetween speed, frequency, and radial acceleration for circular motion at a constant speed. The problems aredivided into two groups: (1) No radial force components required for solution; and (2) Radial forcecomponents required for solution. The specific types of forces involved in a problem (e.g., tension, normal,weight, friction, gravitational, electric) are indicated in bold type at the beginning of each problem.
No Radial Force Components
Weight, Normal: Just before finals you decide to visit an amusement park set up in the Metrodome.
Since it is a weekend, you invite your favorite niece along. She loves to ride on a Ferris wheel, and
there is one at the amusement park. The Ferris wheel has seats on the rim of a circle with a radius of
25 m. The Ferris wheel rotates at a constant speed and makes one complete revolution every 20
seconds. While you wait, your niece who has a mass of 42 kg, rides the Ferris wheel. To kill time you
decide to calculate the total force (both magnitude and direction) on her when she is one quarter
revolution past the highest point. Because the Ferris wheel can be run at different speeds, you also
decide to make a graph which gives the magnitude of the force on her at that point as a function of the
period of the Ferris wheel.
Weight, Normal: While relaxing from studying physics, you watch some TV. While flipping through
channels you see a circus show in which a woman drives a motorcycle around the inside of a vertical
ring. You determine that she goes around at a constant speed and that it takes her 4.0 seconds to get
around when she is going her slowest. If she is going at the minimum speed for this stunt to work, the
motorcycle is just barely touching the ring when she is upside down at the top. At that point she is in
free fall so her acceleration is just g. She just makes it around without falling off the ring but what if she
made a mistake and her motorcycle fell off at the top? How high up is she?
Weight, Normal, Friction: The producer of the last film you worked on was so impressed with the
way you handled a helicopter scene that she hired you again as technical advisor for a new "James
Bond" film. The scene calls for 007 to chase a villain onto a merry-go-round. An accomplice starts
the merry-go-round rotating in an effort to toss 007 (played in this new version by Billy Crystal) off
into an adjacent pool filled with hungry sharks. You must determine a safe rate of rotation such that
the stunt man (you didn't think Billy would do his own stunts did you?) will not fly off the merry-go-
round and into the shark-infested pool. (Actually they are mechanical sharks, but the audience doesn't
know that.) You measure the diameter of the merry-go-round as 50 meters. You determine that the
coefficient of static friction between 007's shoes and the merry-go-round surface is 0.7 and the
coefficient of kinetic friction is 0.5.
Weight, Normal, Friction: A new package moving system in the new, improved post office consists
of a large circular disc (i.e. a turntable) which rotates once every 3.0 seconds at a constant speed in
the horizontal plane. Packages are put on the outer edge of the turntable on one side of the room and
taken off on the opposite side. The coefficient of static friction between the disc surface and a
package is 0.80 while the coefficient of kinetic friction is 0.60. If this system is to work, what is the
maximum possible radius of the turntable?
Force and Circular Motion at a Constant Speed Weight, Normal, Friction: You are driving with a friend who is sitting to your right on the passenger
side of the front seat. You would like to be closer to your friend and decide to use your knowledge of
physics to achieve your romantic goal. So you'll make a sharp turn. Which direction should you turn
so as to make your friend slide closer to you? If the coefficient of static friction between your friend
and the seat of the car is 0.40, and you drive at a constant speed of 18 m/s, what is the maximum
radius you could make your turn and still have your friend slide your way?
Weight, Normal, Friction: On a trip through Florida, you find yourself driving in your 3000-lb car
along a flat level road at 50 mph. The road makes a turn which you take without changing your speed.
The curve is approximately an arc of a circle with a radius of 0.05 miles. You notice that the curve is
flat and level with no sign of banking. There are no warning signs but you wonder if it would be safe to
try to go 50 mph around the curve in the rain when the wet surface has a lower coefficient of friction.
What is the minimum coefficient of static friction between the road and your car's tires which will allow
your car to make the turn?
Weight, Tension: After watching the movie "Crocodile Dundee," you and some friends decide to
make a communications device invented by the Australian Aborigines. It consists of a noise-maker
swung in a vertical circle on the end of a string. Your design calls for a 400 gram noise-maker on a 60
cm string. You are worried about whether the string you have will be strong enough, so you decide to
calculate the tension in the string when the device is swung with an acceleration which has a constant
magnitude of 20 m/s2 . You and your friends can't agree whether the maximum tension will occur
when the noise maker is at the highest point in the circle, at the lowest point in the circle, or is always
the same. To settle the argument you decide to calculate the tension at the highest point and at the
lowest point and compare them.
You are watching a TV news program when they switch to some scenes taken aboard the spaceshuttle which circles 500 miles above the Earth once every 95 minutes. To allow the audience toappreciate the distances involved, the announcer tells you that the radius of the Earth is about 4000miles and the distance from the Earth to the Moon is about 250,000 miles. When an astronaut dropsher pen it floats in front of her face. You immediately wonder how the acceleration of the dropped pencompares to the acceleration of a pen that you might drop here on the surface of the Earth.
Gravitational: You are still a consultant for the new Star Trek TV series. You were hired to make
sure that any science on the show is correct. In this episode, the crew of the Enterprise discovers an
abandoned space station in deep space far from any stars. This station, which was built by Earth in the
21st century, is a large wheel-like structure where people live and work in the rim. In order to create
"artificial gravity," the space station rotates on its axis. The special effects department wants to know
at what rate a space station 200 meters in diameter would have to rotate to create "gravity" equal to
0.7 that of Earth.
Gravitational: You did so well in your physics course that you decided to try to get a summer job
working in a physics laboratory at the University. You got the job as a student lab assistant in a
research group investigating the ozone depletion at the Earth's poles. This group is planning to put an
atmospheric measuring device in a satellite which will pass over both poles. To collect samples of the
upper atmosphere, the satellite will be in a circular orbit 200 miles above the surface of the Earth. To
adjust the instruments for the proper data taking rate, you need to calculate how many times per day
the device will sample the atmosphere over the South pole. Using the inside cover of your trusty
Force and Circular Motion at a Constant Speed Physics text you find that the radius of the Earth is 6.38 x 103 km, the mass of the Earth is 5.98 x 1024kg, and the universal gravitational constant is 6.7 x 10-11 N m2/kg2.
Gravitational: You did so well in your physics course that you decided to try to get a summer job
working in a physics laboratory at the University. You got the job as a student lab assistant in a
research group investigating the ozone depletion at the Earth's poles. This group is planning to put an
atmospheric measuring device in a satellite which will pass over both poles. To collect samples of the
upper atmosphere, the satellite will be in a circular orbit 200 miles above the surface of the Earth
where g is 95% of its value on the Earth's surface. To adjust the instruments for the proper data taking
rate, you need to calculate how many times per day the device will sample the atmosphere over the
South pole. Using the inside cover of your trusty Physics text you find that the radius of the Earth is
6.38 x 103 km and the mass of the Earth is 5.98 x 1024 kg.
Gravitational: You are reading a magazine article about pulsars. A few years ago, a satellite in orbit
around the Earth detected X-rays coming from sources in outer space. The X-rays detected from one
source, called Cygnus X-3, had an intensity which changed with a period of 4.8 hours. This type of
astronomical object emitting periodic signals is called a pulsar. One popular theory holds that the
pulsar is a normal star (similar to our Sun) which is in orbit around a much more massive neutron star.
The period of the X-ray signal is then the period of the orbit. In this theory, the distance between the
normal star and the neutron star is approximately the same as the distance between the Earth and our
Sun. You realize that if this theory is correct, you can determine how much more massive the neutron
star is than our Sun. All you need to do is first find the mass of the neutron star in terms of two
unknowns, the universal gravitational constant G and the radius of the Earth's orbit. Then find the mass
of our Sun in terms of the same two unknowns, G and the radius of the Earth's orbit. (The period of
the Earth's orbit is 1 year). Then you can calculate how many times more massive the neutron star is
than our Sun.
Radial Force Components
Weight, Lift: You are reading an article about the aesthetics of airplane design. One example in the
article is a beautiful new design for commercial airliners. You are worried that this light wing structure
might not be strong enough to be safe. The article explains that an airplane can fly because the air
exerts a force, called "lift," on the wings such that the lift is always perpendicular to the wing surface.
For level flying, the wings are horizontal. To turn , the pilot "banks" the plane so that the wings are
oriented at an angle to the horizontal. This causes the plane to have a trajectory which is a horizontal
circle. The specifications of the 100 x 103 lb plane require that it be able to turn with a radius of 2.0
miles at a constant speed of 500 miles/hr. The article states that tests show that the new wing structure
will support a force 4 times the lift necessary for level flight. Is the wing structure sufficiently strong for
the plane to make this turn?

Source: http://fisica.fcyt.umss.edu.bo/tl_files/Fisica/Biblioteca%20Fisica/FISICA%20BASICA%20I/cooperative%20group%20problem%20solving%20in%20physics/chapter4.pdf