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Newton’s laws of motion describe how objects move when forces are applied to them. The first law states that an object’s velocity won’t change unless you apply a force to it. One of the most famous and important equations in physics is Newton’s second law: Force equals mass times acceleration. The third law says that whenever you exert a force on an object, that object will exert an equal force on you.

The Problems You’ll Work On

In this chapter you’ll apply Newton’s laws to the following types of problems:

✓ Using Newton’s second law to relate force and acceleration

✓ Drawing free-body diagrams

✓ Determining equal and opposite forces with Newton’s third law

✓ Redirecting forces with pulleys

What to Watch Out For

You’ll be forced to try the problem again unless you keep the following in mind:

✓ Drawing a free-body diagram to make sure you include all the forces

✓ Determining the components of the forces on your free-body diagrams with the correct signs

✓ Remembering that the equal and opposite forces in Newton’s third law always act on different objects

✓ Recalling that the magnitude of the tension of a massless rope is the same all along the rope

Resisting Motion with Newton’s First Law

106–107

106. What are the SI units of mass?

107. What property of an object does mass measure?

Forcing a Massive Object to Accelerate

108–109

108. What is the acceleration of a 0.25-kilogram particle subject to a single force of 10 newtons eastward?

109. A 300-gram block slides across a ceramic floor at a speed of 13.5 meters per second. If no forces act on the block along the axis of its motion, what is the block’s speed 1 second later? Round your answer to the nearest tenth of a meter per second.

Drawing Free-Body Diagrams

110–113 Use the following force diagram of a mass hanging from a pulley by a massless rope to answer Questions 110–113. Letters on the diagram signify vectors.

Illustration by Thomson Digital

110. Which vector represents the gravitational force Earth exerts on the mass?

111. Which vector(s) represent the force of tension?

112. Which two vectors always have equal magnitudes?

113. In terms of B, C, D, and E – the magnitudes of their respective vectors – what inequality must be true if the mass accelerates downward?

Use the following force diagram, showing a box of mass m sitting on a tabletop, to answer Questions 114–116. Letters on the diagram signify vectors.

Illustration by Thomson Digital

114. Which vector represents the normal force exerted on the box?

115. Assuming the table completely supports the box’s weight, what equality must be true in terms of A, B, C, and D – the magnitudes of the vectors displayed?

116. A cat sees the box on the table and hops up to start pushing it to the right. After it starts moving, the cat exerts a force K on the box that keeps it moving at a constant velocity. If K is parallel to the table’s surface, write an expression for the magnitude of K in terms of A, B, C, and D – the magnitudes of the four vectors shown.

Adding Forces Together to Obtain the Net Force

117–119

117. One force pulls a brick with a force of 12 newtons due west, while another force pulls the brick with a force of 8 newtons due east. What is the magnitude of the net force exerted on the brick?

118. What is the net force in the east-west direction on a crate being pushed with 58 newtons of force 12 degrees north of east by one worker and with 30 newtons of force 64 degrees south of east by a second worker? Round your answer to the nearest integer.

119. Two forces act on a cardboard box: a 340-newton force directed 25 degrees south of east and a 300-newton force directed 85 degrees north of west. What net force does the cardboard box experience? Round your answer to the nearest integers, in units of newtons and degrees.

Moving a Distance with the Net Force

120–122

120. A force of 50 newtons provides a constant acceleration to a 25-kilogram crate originally at rest. How many meters does the crate move in 3 seconds? Round your answer to the nearest tenth.

121. Given a box of mass M initially at rest, write an expression for the displacement the box experiences when a force F constantly accelerates it to a speed of v meters per second. Your answer should contain only variables given here, but it does not have to use all of them.

122. Starting from rest, a 5,850-kilogram sports car goes from 0 to 200 kilometers per hour. It proceeds at that same speed for 10 seconds. The driver then slams on the brakes, producing a constant acceleration until the car is once again stationary. If the engine and brakes provided forces of 40,600 newtons and 31,800 newtons, respectively, what is the sports car’s displacement during the trip? Round your answer to the nearest tenth of a kilometer.

Finding the Needed Force to Speed Up

123–125

123. Johnny guns the engine on a 180-kilogram vehicle currently rolling backward at 10.8 kilometers per hour. If the engine provides 450 newtons of force (10 percent of which is lost to frictional forces), what is the vehicle’s velocity 4 seconds later? Round your answer to the nearest meter per second.

124. A 7,200-kilogram car is accelerated from rest at a constant rate by an engine producing 96 kilonewtons of force for 7.5 seconds. How fast is the car traveling – in kilometers per hour – after that time? Round your answer to two significant digits.

125. An 800-kilogram elevator cab is attached to a cable rising high into a skyscraper. The sides of the cab contain brake pads that provide the force for slowing the cab down when it approaches its destination. Taking the elevator down from the tenth floor to the fifth floor, Floyd obtains a maximum vertical velocity of 5 meters per second downward before the cab starts braking halfway between the sixth and seventh floors. Assume that the tension in the cable when the brakes are off is twice the tension in the cable when the brakes are on. If each floor is 20 meters apart, how many newtons of force do the elevator’s brakes exert? Round your answer to three significant digits.

Pairing Up Equal and Opposite Forces

126–128

126. Joe pushes against a brick wall with a force of 12 newtons. If Joe’s mass is 120 kilograms and the brick wall’s mass is 12,000 kilograms, with how much force does the brick wall push against Joe? Give your answer in newtons using two significant digits.

127. An angry, 120-kilogram astronaut punches a 3,040-kilogram space shuttle with 45 newtons of force. What is the magnitude of the astronaut’s acceleration – in meters per second squared – as a result of his punch? Round your answer to two significant digits.

128. Four boxes of varying masses are dragged along a frictionless surface by a force of 85 newtons, as shown here:

Illustration by Thomson Digital

How many more times powerful is the contact force between the 2- and 4-kilogram boxes than the contact force between the 5- and 1-kilogram boxes? Round your answer to the nearest integer.

Overcoming Friction by Pulling Hard

129–131

129. Two cartons of milk are dragged across a table. They are identical except for their masses: One has a mass of 1.2 kilograms and the other has a mass of 0.6 kilograms. If the magnitude of the force of friction acting on the more massive carton is denoted z, what is the magnitude of the force of friction on the other carton, in terms of z?