### 4.1 Development of Force Concept

The figure above represents a racetrack with semicircular sections connected by straight sections. Each section has length *d*, and markers along the track are spaced *d*/4 apart. Two people drive cars counterclockwise around the track, as shown. Car *X* goes around the curves at constant speed *v*_{c}, increases speed at constant acceleration for half of each straight section to reach a maximum speed of 2*v*_{c}, then brakes at constant acceleration for the other half of each straight section to return to speed *v*_{c}. Car *Y* also goes around the curves at constant speed *v*_{c}, increases its speed at constant acceleration for one-fourth of each straight section to reach the same maximum speed 2*v*_{c}, stays at that speed for half of each straight section, then brakes at constant acceleration for the remaining fourth of each straight section to return to speed *v*_{c}.

(a) On the figures below, draw an arrow showing the direction of the net force on each of the cars at the positions noted by the dots. If the net force is zero at any position, label the dot with 0.

The position of the six dots on the Car Y track on the right are as follows:

- The first dot on the left center of the track is at the same position as it is on the Car X track.
- The second dot is just slight to the right of the Car X dot (less than a dash) past three perpendicular hash marks moving to the right.
- The third dot is about one and two-thirds perpendicular hash marks to the right of the center top perpendicular has mark.
- The fourth dot is in the same position as the Car X figure (one perpendicular hash mark above the center right perpendicular hash mark).
- The fifth dot is about one and two-third perpendicular hash marks to the right of the center bottom perpendicular hash mark.
- The sixth dot is in the same position as the Car Y dot (one and two third perpendicular hash marks to the left of the center bottom hash mark).

(b)

i. Indicate which car, if either, completes one trip around the track in less time, and justify your answer qualitatively without using equations.

ii. Justify your answer about which car, if either, completes one trip around the track in less time quantitatively with appropriate equations.

Which of the following is an example of a body exerting a force on itself?

- a person standing up from a seated position
- a car accelerating while driving
- both of the above
- none of the above

A hawk accelerates as it glides in the air. Does the force causing the acceleration come from the hawk itself? Explain.

What causes the force that moves a boat forward when someone rows it?

- The force is caused by the rower’s arms.
- The force is caused by an interaction between the oars and gravity.
- The force is caused by an interaction between the oars and the water the boat is traveling in.
- The force is caused by friction.

### 4.4 Newton’s Third Law of Motion: Symmetry in Forces

What object or objects commonly exert forces on the following objects in motion? (a) a soccer ball being kicked, (b) a dolphin jumping, (c) a parachutist drifting to Earth.

A ball with a mass of 0.25 kg hits a gym ceiling with a force of 78.0 N. What is the net force on the ball?

- 2.50 N downward
- 75.5 N downward
- 78.0 N downward
- 80.5 N downward

Which of the following is true?

- Earth exerts a force due to gravity on your body, and your body exerts a smaller force on the Earth, because your mass is smaller than the mass of the Earth.
- The Moon orbits the Earth because the Earth exerts a force on the Moon and the Moon exerts a force equal in magnitude and direction on the Earth.
- A rocket taking off exerts a force on the Earth equal to the force the Earth exerts on the rocket.
- An airplane cruising at a constant speed is not affected by gravity.

Stationary skater A pushes stationary skater B, who then accelerates at 5.0 m/s^{2}. Skater A does not move. Since forces act in action-reaction pairs, explain why Skater A did not move?

The current in a river exerts a force of 9.0 N on a balloon floating in the river. A wind exerts a force of 13.0 N on the balloon in the opposite direction. Draw a free-body diagram to show the forces acting on the balloon. Use your free-body diagram to predict the effect on the balloon.

A force is applied to accelerate an object on a smooth icy surface. When the force stops, which of the following will be true? (Assume zero friction.)

- The object’s acceleration becomes zero.
- The object’s speed becomes zero.
- The object’s acceleration continues to increase at a constant rate.
- The object accelerates, but in the opposite direction.

A parachutist’s fall to Earth is determined by two opposing forces. A gravitational force of 539 N acts on the parachutist. After 2 s, she opens her parachute and experiences an air resistance of 615 N. At what speed is the parachutist falling after 10 s?

A flight attendant pushes a cart down the aisle of a plane in flight. In determining the acceleration of the cart relative to the plane, which factor do you not need to consider?

- The friction of the cart’s wheels.
- The force with which the flight attendant’s feet push on the floor.
- The velocity of the plane.
- The mass of the items in the cart.

A landscaper is easing a wheelbarrow full of soil down a hill. Define the system you would analyze and list all the forces that you would need to include to calculate the acceleration of the wheelbarrow.

Two water-skiers, with masses of 48 kg and 61 kg, are preparing to be towed behind the same boat. When the boat accelerates, the rope the skiers hold onto accelerates with it and exerts a net force of 290 N on the skiers. At what rate will the skiers accelerate?

- 10.8 m/s
^{2} - 2.7 m/s
^{2} - 6.0 m/s
^{2}and 4.8 m/s^{2} - 5.3 m/s
^{2}

A figure skater has a mass of 40 kg and her partner's mass is 50 kg. She pushes against the ice with a force of 120 N, causing her and her partner to move forward. Calculate the pair’s acceleration. Assume that all forces opposing the motion, such as friction and air resistance, total 5.0 N.

### 4.5 Normal, Tension, and Other Examples of Forces

An archer shoots an arrow straight up with a force of 24.5 N. The arrow has a mass of 0.4 kg. What is the force of gravity on the arrow?

- 9.8 m/s
^{2} - 9.8 N
- 61.25 N
- 3.9 N

A cable raises a mass of 120.0 kg with an acceleration of 1.3 m/s2. What force of tension is in the cable?

A child pulls a wagon along a grassy field. Define the system, the pairs of forces at work, and the results.

Two teams are engaging in a tug–of-war. The rope suddenly snaps. Which statement is true about the forces involved?

- The forces exerted by the two teams are no longer equal; the teams will accelerate in opposite directions as a result.
- The forces exerted by the players are no longer balanced by the force of tension in the rope; the teams will accelerate in opposite directions as a result.
- The force of gravity balances the forces exerted by the players; the teams will fall as a result
- The force of tension in the rope is transferred to the players; the teams will accelerate in opposite directions as a result.

The following free-body diagram represents a toboggan on a hill. What acceleration would you expect, and why?

- Acceleration down the hill; the force due to being pushed, together with the downhill component of gravity, overcomes the opposing force of friction.
- Acceleration down the hill; friction is less than the opposing component of force due to gravity.
- No movement; friction is greater than the force due to being pushed.
- No movement; friction is greater than the sum of the downhill forces.

Draw a free-body diagram to represent the forces acting on a kite on a string that is floating stationary in the air. Label the forces in your diagram.

A car is sliding down a hill with a slope of 20°. The mass of the car is 965 kg. When a cable is used to pull the car up the slope, a force of 4215 N is applied. What is the car’s acceleration, ignoring friction?

### 4.6 Problem-Solving Strategies

A toboggan with two riders has a total mass of 85.0 kg. A third person is pushing with a force of 42.5 N in the direction of motion on a toboggan moving on a sloped surface at the top of a hill that has a downward angle of 15°. The force of friction on the toboggan is 31.0 N. Which statement describes an accurate free-body diagram to represent the situation?

- An arrow of magnitude 10.5 N points down the slope of the hill.
- An arrow of magnitude 833 N points straight down.
- An arrow of magnitude 833 N points perpendicular to the slope of the hill.
- An arrow of magnitude 73.5 N points down the slope of the hill.

A mass of 2.0 kg is suspended from the ceiling of an elevator by a rope. What is the tension in the rope when the elevator (i) accelerates upward at 1.5 m/s^{2}? (ii) accelerates downward at 1.5 m/s^{2}?

- (i) 22.6 N; (ii) 16.6 N
- Because the mass is hanging from the elevator itself, the tension in the rope will not change in either case.
- (i) 22.6 N; (ii) 19.6 N
- (i) 16.6 N; (ii) 19.6 N

Which statement is true about drawing free-body diagrams?

- Drawing a free-body diagram should be the last step in solving a problem about forces.
- Drawing a free-body diagram helps you compare forces quantitatively.
- The forces in a free-body diagram should always balance.
- Drawing a free-body diagram can help you determine the net force.

### 4.7 Further Applications of Newton’s Laws of Motion

A basketball player jumps as he shoots the ball. Describe the forces that are acting on the ball and on the basketball player. What are the results?

Two people push on a boulder to try to move it. The mass of the boulder is 825 kg. One person pushes north with a force of 64 N. The other pushes west with a force of 38 N. Predict the magnitude of the acceleration of the boulder. Assume that friction is negligible.

The figure shows the forces exerted on a block that is sliding on a horizontal surface: the gravitational force of 40 N, the 40 N normal force exerted by the surface, and a frictional force exerted to the left. The coefficient of friction between the block and the surface is 0.20. The acceleration of the block is most nearly

- 1.0 m/s
^{2}to the right - 1.0 m/s
^{2}to the left - 2.0 m/s
^{2}to the right - 2.0 m/s
^{2}to the left

### 4.8 Extended Topic: The Four Basic Forces—An Introduction

Which phenomenon correctly describes the direction and magnitude of normal forces?

- electromagnetic attraction
- electromagnetic repulsion
- gravitational attraction
- gravitational repulsion

Explain which of the four fundamental forces is responsible for a ball bouncing off the ground after it hits, and why this force has this effect.

Which of the basic forces best explains tension in a rope being pulled between two people? Is the acting force causing attraction or repulsion in this instance?

- gravity; attraction
- electromagnetic; attraction
- weak and strong nuclear; attraction
- weak and strong nuclear; repulsion

Explain how interatomic electric forces produce the normal force, and why it has the direction it does.

The gravitational force is the weakest of the four basic forces. In which case can the electromagnetic, strong, and weak forces be ignored because the gravitational force is so strongly dominant?

- a person jumping on a trampoline
- a rocket blasting off from Earth
- a log rolling down a hill
- all of the above

Describe a situation in which gravitational force is the dominant force. Why can the other three basic forces be ignored in the situation you described?