Exercises and Problems

Exercises

Section 5.3 Identifying Forces

| A chandelier hangs from a chain in the middle of a dining room. Identify the forces on the chandelier.
| A car is parked on a steep hill. Identify the forces on the car.
| A baseball player is sliding into second base. Identify the forces on the baseball player.
|| An arrow has just been shot from a bow and is now traveling horizontally. Air resistance is not negligible. Identify the forces on the arrow.
|| A jet plane is speeding down the runway during takeoff. Air resistance is not negligible. Identify the forces on the jet.

Section 5.4 What Do Forces Do?

| Two rubber bands pulling on an object cause it to accelerate at $1.2 m / s^{2}$ .
1. What will be the object’s acceleration if it is pulled by four rubber bands?
2. What will be the acceleration of two of these objects glued together if they are pulled by two rubber bands?
| Two rubber bands cause an object to accelerate with acceleration $a$ . How many rubber bands are needed to cause an object with half the mass to accelerate three times as quickly?
|| FIGURE EX5.8 shows an acceleration-versus-force graph for three objects pulled by rubber bands. The mass of object B is 0.20 kg. What are the masses of objects A and C? Explain your reasoning.

FIGURE EX5.8

FIGURE EX5.8 Full Alternative Text
|| FIGURE EX5.9 shows the accelerations when forces $F_{A}$ and $F_{B}$ are applied to a series of different masses constructed by gluing identical blocks together. Suppose $F_{A} = 8 N$ . What is $F_{B}$ ?

FIGURE EX5.9

FIGURE EX5.9 Full Alternative Text
|| For an object starting from rest and accelerating with constant acceleration, distance traveled is proportional to the square of the time. If an object travels 2.0 furlongs in the first $2.0 s,$ how far will it travel in the first $4.0 s ?$
|| You’ll learn in Chapter 13 that the gravitational energy of two masses is inversely proportional to the distance between them. Suppose the gravitational energy of a sun-planet system is $- 4 \times 10^{33} J,$ where the joule (J) is the unit of energy. What would be the gravitational energy if the planet were twice as far from the sun?

Section 5.5 Newton’s Second Law

| FIGURE EX5.12 shows an acceleration-versus-force graph for a 500 g object. What acceleration values go in the blanks on the vertical scale?

FIGURE EX5.12

FIGURE EX5.12 Full Alternative Text
| FIGURE EX5.13 shows an acceleration-versus-force graph for a 200 g object. What force values go in the blanks on the horizontal scale?

FIGURE EX5.13

FIGURE EX5.13 Full Alternative Text
| FIGURE EX5.14 shows an object’s acceleration-versus-force graph. What is the object’s mass?

FIGURE EX5.14

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|| FIGURE EX5.15 shows the acceleration of objects of different mass that experience the same force. What is the magnitude of the force?

FIGURE EX5.15

FIGURE EX5.15 Full Alternative Text
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1. What net force is needed to accelerate a 2500 kg car at one-third of $g$ ?
2. What is the acceleration, as a multiple of $g$ , if this net force is applied to a 110 kg bicyclist? This is the combined mass of the cyclist and the bike.

Section 5.6 Newton’s First Law

Exercises 17, 18, and 19 show two of the three forces acting on an object in equilibrium. Redraw the diagram, showing all three forces. Label the third force $\vec{F}_{3}$ .

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FIGURE EX5.17

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FIGURE EX5.18

FIGURE EX5.18 Full Alternative Text
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FIGURE EX5.19

FIGURE EX5.19 Full Alternative Text

Section 5.7 Free-Body Diagrams

Exercises 20, 21, and 22 show a free-body diagram. For each, write a short description of a real object for which this would be the correct free-body diagram. Use Examples 5.4, 5.5, and 5.6 as examples of what a description should be like.

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FIGURE EX5.20

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FIGURE EX5.21

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FIGURE EX5.22

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Exercises 23, 24, 25, 26, and 27 describe a situation. For each, identify all forces acting on the object and draw a free-body diagram of the object.

| A cat is sitting on a window sill.
| An ice hockey puck glides across frictionless ice.
| Your physics textbook is sliding across the table.
| A steel beam, suspended by a single cable, is being lowered by a crane at a steadily decreasing speed.
| A jet plane is accelerating down the runway during takeoff. Friction is negligible, but air resistance is not.

Problems

| Redraw the two motion diagrams shown in FIGURE P5.28, then draw a vector beside each one to show the direction of the net force acting on the object. Explain your reasoning.

FIGURE P5.28

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| A single force with x-component $F_{x}$ acts on a 2.0 kg object as it moves along the x-axis. The object’s acceleration graph $a_{x}$ versus $t$ ) is shown in FIGURE P5.29. Draw a graph of $F_{x}$ versus $t$ .

FIGURE P5.29

FIGURE P5.29 Full Alternative Text
| A single force with x-component $F_{x}$ acts on a 500 g object as it moves along the x-axis. The object’s acceleration graph $a_{x}$ versus $t$ ) is shown in FIGURE P5.30. Draw a graph of $F_{x}$ versus $t$ .

FIGURE P5.30

FIGURE P5.30 Full Alternative Text
|| A single force with x-component $F_{x}$ acts on a 500 g object as it moves along the x-axis. A graph of $F_{x}$ versus $t$ is shown in FIGURE P5.31. Draw an acceleration graph $a_{x} versus t$ ) for this object.

FIGURE P5.31

FIGURE P5.31 Full Alternative Text
| A single force with x-component $F_{x}$ acts on a 2.0 kg object as it moves along the x-axis. A graph of $F_{x}$ versus $t$ is shown in FIGURE P5.32. Draw an acceleration graph $a_{x} versus t$ ) for this object.

FIGURE P5.32

FIGURE P5.32 Full Alternative Text
| A constant force is applied to an object, causing the object to accelerate at $8.0 m / s^{2}$ . What will the acceleration be if
1. The force is doubled?
2. The object’s mass is doubled?
3. The force and the object’s mass are both doubled?
4. The force is doubled and the object’s mass is halved?
| A constant force is applied to an object, causing the object to accelerate at $10 m / s^{2}$ . What will the acceleration be if
1. The force is halved?
2. The object’s mass is halved?
3. The force and the object’s mass are both halved?
4. The force is halved and the object’s mass is doubled?

Problems 35, 36, 37, 38, 39, and 40 show a free-body diagram. For each:

Identify the direction of the acceleration vector $\vec{a}$ and show it as a vector next to your diagram. Or, if appropriate, write $\vec{a} = \vec{0}$ .
If possible, identify the direction of the velocity vector $\vec{v}$ and show it as a labeled vector.
Write a short description of a real object for which this is the correct free-body diagram. Use Examples 5.4, 5.5, and 5.6 as models of what a description should be like.

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FIGURE P5.35

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FIGURE P5.36

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FIGURE P5.37

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FIGURE P5.38

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FIGURE P5.39

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FIGURE P5.40

FIGURE P5.40 Full Alternative Text
|| In lab, you propel a cart with four known forces while using an ultrasonic motion detector to measure the cart’s acceleration. Your data are as follows:

Force (N) Acceleration $(m / s^{2})$

0.25 0.5

0.50 0.8

0.75 1.3

1.00 1.8
1. How should you graph these data so as to determine the mass of the cart from the slope of the line? That is, what values should you graph on the horizontal axis and what on the vertical axis?
2. Is there another data point that would be reasonable to add, even though you made no measurements? If so, what is it?
3. What is your best determination of the cart’s mass?

Force (N)	Acceleration $(m / s^{2})$
0.25	0.5
0.50	0.8
0.75	1.3
1.00	1.8

Problems 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, and 52 describe a situation. For each, draw a motion diagram, a force-identification diagram, and a free-body diagram.

| A Styrofoam ball has just been shot straight up. Air resistance is not negligible.
| An elevator, suspended by a single cable, has just left the tenth floor and is speeding up as it descends toward the ground floor.
|| A rocket is being launched straight up. Air resistance is not negligible.
| A lawnmower is pushed at a constant speed.
| You’ve just kicked a rock on the sidewalk and it is now sliding along the concrete.
|| You’ve slammed on the brakes and your car is skidding to a stop while going down a $20 °$ hill.

Video Tutor Solution: Problem 5.47
| You’ve jumped down from a platform. Your feet are touching the ground and your knees are flexing as you stop.
|| You are bungee jumping from a high bridge. You’re slowing down as the bungee cord approaches its maximum stretch.
|| A spring-loaded gun shoots a plastic ball. The trigger has just been pulled and the ball is starting to move down the barrel. The barrel is horizontal.
|| Your friend went for a loop-the-loop ride at the amusement park. Her car is upside down at the top of the loop.
|| A model rocket is fired straight down from the top of a tower.
|| BIO The leaf hopper, champion jumper of the insect world, can jump straight up at $4 m / s^{2}$ . The jump itself lasts a mere 1 ms before the insect is clear of the ground.
1. Draw a free-body diagram of this mighty leaper while the jump is taking place.
2. While the jump is taking place, is the force of the ground on the leaf hopper greater than, less than, or equal to the force of gravity on the leaf hopper? Explain.
|| A heavy box is in the back of a truck. The truck is accelerating to the right. Draw a motion diagram, a force-identification diagram, and a free-body diagram for the box.
|| A bag of groceries is on the seat of your car as you stop for a stop light. The bag does not slide. Draw a motion diagram, a force-identification diagram, and a free-body diagram for the bag.
|| A rubber ball bounces. We’d like to understand how the ball bounces.
1. A rubber ball has been dropped and is bouncing off the floor. Draw a motion diagram of the ball during the brief time interval that it is in contact with the floor. Show 4 or 5 frames as the ball compresses, then another 4 or 5 frames as it expands. What is the direction of $\vec{a}$ during each of these parts of the motion?
2. Draw a picture of the ball in contact with the floor and identify all forces acting on the ball.
3. Draw a free-body diagram of the ball during its contact with the ground. Is there a net force acting on the ball? If so, in which direction?
4. Write a paragraph in which you describe what you learned from parts a to c and in which you answer the question: How does a ball bounce?
|| If a car stops suddenly, you feel “thrown forward.” We’d like to understand what happens to the passengers as a car stops. Imagine yourself sitting on a very slippery bench inside a car. This bench has no friction, no seat back, and there’s nothing for you to hold onto.
1. Draw a picture and identify all of the forces acting on you as the car travels at a perfectly steady speed on level ground.
2. Draw your free-body diagram. Is there a net force on you? If so, in which direction?
3. Repeat parts a and b with the car slowing down.
4. Describe what happens to you as the car slows down.
5. Use Newton’s laws to explain why you seem to be “thrown forward” as the car stops. Is there really a force pushing you forward?
6. Suppose now that the bench is not slippery. As the car slows down, you stay on the bench and don’t slide off. What force is responsible for your deceleration? In which direction does this force point? Include a free-body diagram as part of your answer.