Key Termsinertianet forceequilibrium

InertiaA hovercraft, such as the one in Figure 2.1, glides along the surface of the water on a cushion of air. A common misconception is that an object on which no force is acting will always be at rest. This situation is not always the case. If the hovercraft shown in Figure 2.1 is moving at a constant velocity, then there is no net force acting on it. To see why this is the case, consider how a block will slide on different surfaces.

First, imagine a block on a deep, thick carpet. If you apply a force by pushing the block, the block will begin sliding, but soon after you remove the force, the block will come to rest. Next, imagine pushing the same block across a smooth, waxed floor. When you push with the same force, the block will slide much farther before coming to rest. In fact, a block sliding on a perfectly smooth surface would slide forever in the absence of an applied force.

In the 1630s, Galileo concluded correctly that it is an object’s nature to maintain its state of motion or rest. Note that an object on which no force is acting is not necessarily at rest; the object could also be moving with a constant velocity. This concept was further developed by Newton in 1687 and has come to be known as Newton’s first law of motion.

Newton’s First Law

An object at rest remains at rest, and an object in motion continues in motion with constant velocity (that is, constant speed in a

straight line) unless the object experiences a net external force.

Inertia is the tendency of an object not to accelerate. Newton’s first law is often referred to as the law of inertia because it states that in the absence of a net force, a body will preserve its state of motion. In other words, Newton’s first law says that when the net external force on an object is zero, the object’s acceleration (or the change in the object’s velocity) is zero.

Newton’s First Law Main Ideas

Explain the relationship between the motion of an object and the net external force acting on the object.

Determine the net external force on an object.

Calculate the force required to bring an object into equilibrium.

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Hovercraft on Air A hovercraft floats on a cushion of air above the water. Air provides less resistance to motion than water does.

FIGURE 2.1

inertia the tendency of an object to resist being moved or, if the object is moving, to resist a change in speed or direction

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Differentiated Instruction

Preview VocabularyLatin Word Origins The root of inertia comes from Latin. The root word can mean “idleness,” “unskilled,” or “inactive.” Inertia has the same root word as the term inert, a word used in chemistry to describe something that is chemically inactive. The Group 18 elements, such as neon and argon, are sometimes called the inert gases.

InertIaPurpose Help students develop a kinesthetic sense of inertia.

Materials physics book, calculator

Procedure Tell students that they will be able to feel the effects of inertia. First, tell them to hold the physics book upright between their hands, palms facing inward. Have them move the book from side to side (oscillating a distance of 30 cm) at regular time intervals. Tell the students to note the effort involved in changing the motion of the book. Repeat the demonstration with the calculator and have students note the much smaller effort required.

� Plan and Prepare

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Demonstration

IncLusIOnKinesthetic learners may benefit from a simple activity which demonstrates inertia. Take students outside or into the gymnasium. Mark off a 25-meter race course. Set cones at 23 and 25 meters. Tell students that the goal is for them to run as fast as possible and then to come to a complete stop between the cones. Allow students to warm up and then start from one end.

After students have completed their run, ask them to describe what they experienced. They should have noticed that while it was easy to stop their feet, they may have felt as if their upper body was still moving forward. Explain to them that they were experiencing inertia. Use this experience as a starting point for a classroom discussion about Newton’s first law.

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PHYSICSSpec. Number PH 99 PE C04-002-002-ABoston Graphics, Inc.617.523.1333

resistanceF

gravityF

forwardF

ground-on-carF

The sum of forces acting on an object is the net force.Consider a car traveling at a constant velocity. Newton’s first law tells us that the net external force on the car must be equal to zero. However, Figure 2.2 shows that many forces act on a car in motion. The vector Fforwardrepresents the forward force of the road on the tires. The vector Fresistance, which acts in the opposite direction, is due partly to friction between the road surface and tires and is due partly to air resistance. The vector Fgravity represents the downward gravitational force on the car, and the vector Fground-on-car represents the upward force that the road exerts on the car.

To understand how a car under the influence of so many forces can maintain a constant velocity, you must understand the distinction between external force and net external force. An external force is a single force that acts on an object as a result of the interaction between the object and its environment. All four forces in Figure 2.2 are external forces acting on the car. The net force is the vector sum of all forces acting on an object.

When many forces act on an object, it may move in a particular direction with a particular velocity and acceleration. The net force is the force, which when acting alone, produces exactly the same change in motion. When all external forces acting on an object are known, the net force can be found by using the methods for finding resultant vectors. Although four forces are acting on the car in Figure 2.2, the car will main-tain a constant velocity if the vector sum of these forces is equal to zero.

Mass is a measure of inertia.Imagine a basketball and a bowling ball at rest side by side on the ground. Newton’s first law states that both balls remain at rest as long as no net external force acts on them. Now, imagine supplying a net force by pushing each ball. If the two are pushed with equal force, the basketball will accelerate more than the bowling ball. The bowling ball experiences a smaller acceleration because it has more inertia than the basketball.

As the example of the bowling ball and the basketball shows, the inertia of an object is proportional to the object’s mass. The greater the mass of a body, the less the body accelerates under an applied force. Similarly, a light object undergoes a larger acceleration than does a heavy object under the same force. Therefore, mass, which is a measure of the amount of matter in an object, is also a measure of the inertia of an object.

net force a single force whose external effects on a rigid body are the same as the effects of several actual forces acting on the body

Net Force Although several forces are acting on this car, the vector sum of the forces is zero, so the car moves at a constant velocity.

FIGURE 2.2

INERTIA

Place a small ball on the rear end of a skateboard or cart. Push the skateboard across the floor and into a wall. You may need to either hold the ball in place while push-ing the skateboard up to speed or accelerate the skateboard slowly so that friction holds the ball

in place. Observe what happens to the ball when the skateboard hits the wall. Can you explain your observation in terms of inertia? Repeat the procedure using balls with different masses, and compare the results.

MATERIALSskateboard or cart•

toy balls with various masses•

SAFETY Perform this experiment away from walls and furniture that can be damaged.

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Problem solving

teacher’s nOtesIf students have trouble keeping the ball in place while accelerating the skate-board, they can tape a wooden block onto the skateboard to keep the ball from rolling off the back. Students should recognize that when the skate-board hits the wall, the ball continues moving forward due to its inertia.

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QuickLab

take It FurtherDuring liftoff, astronauts on a space shuttle experience tremendous forces. Challenge students to research the forces that act on astronauts as they lift off the launch pad and leave Earth’s atmosphere. Ask students to choose one segment of the astronauts’ path and create a drawing illustrating the net forces working at that point in time. Suggest that students research education websites hosted by NASA in order to get started.

124 Chapter 4

Determining Net Force

Sample Problem B Derek leaves his physics book on top of a drafting table that is inclined at a 35° angle. The free-body diagram at right shows the forces acting on the book. Find the net force acting on the book.

ANALYZE Define the problem, and identify the variables.Given: Fgravity-on-book = Fg = 22 N

Ffriction = Ff = 11 N Ftable-on-book = Ft = 18 N

Unknown: Fnet = ?

Select a coordinate system, and apply it to the free-body diagram.

Choose the x-axis parallel to and the y-axis perpendicular to the incline of the table, as shown in (a). This coordinate system is the most convenient because only one force

needs to be resolved into x and y components.

PLAN Find the x and y components of all vectors.Draw a sketch, as shown in (b), to help find the components of the vector Fg. The angle θ is equal to 180°- 90° - 35° = 55°.

cos θ = Fg, x

_ Fg

sin θ = Fg, y

_ Fg

Fg,x = Fg cos θ Fg,y = Fg sin θ

Fg,x = (22 N)(cos 55°) = 13 N Fg,y = (22 N)(sin 55°) = 18 N

Add both components to the free-body diagram, as shown in (c).

SOLVE Find the net force in both the x and y directions.Diagram (d) shows another free-body diagram of the book, now with forces acting only along the x- and y-axes.

For the x direction: For the y direction:

ΣFx = Fg,x - Ff ΣFy = Ft - Fg,y

ΣFx = 13 N - 11 N = 2 N ΣFy = 18 N - 18 N = 0 N

Find the net force.Add the net forces in the x and y directions together as vectors to find the total net force. In this case, Fnet = 2 N in the +x direction, as shown in (e). Thus, the book accelerates down the incline.

CHECK YOUR WORK

The box should accelerate down the incline, so the answer is reasonable.

Continued

35°

11 N

18 N

18 N

13 N

22 N

11 N13 N

18 N

18 N

= 2 NFnet

= 18 Ntable-on-book F

= 22 Ngravity-on-book F

= 11 NfrictionF

11 N 18 N

22 N

TSI GraphicsHRW • Holt Physics

PH99PE-C04-002-007-A

Tips and TricksTo simplify the problem, always choose the coordinate system in which as many forces as possible lie on the x- and y-axes.

(a)

(b)

(c)

(d)

(e)

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Classroom PracticeDetermining net ForCeAn agriculture student is designing a support to keep a tree upright. Two wires have been attached to the tree and placed at right angles to each other. One wire exerts a force of 30.0 N on the tree; the other wire exerts a 40.0 N force. Determine where to place a third wire and how much force it should exert so that the net force acting on the tree is equal to zero.Answer: 50.0 N at 143° from the 40.0 N force and at 127° from the 30.0 N force

A flying, stationary kite is acted on by a force of 9.8 N downward. The wind exerts a force of 45 N at an angle of 50.0° above the horizontal. Find the force that the string exerts on the kite.Answer: 38 N, 40° below the horizontal

AlternAtive APProAChesFor free-body diagrams, it sometimes helps to try a different arrangement of the vectors. Show students that, by arranging the force vectors on the coordinate system in such a way that more vectors lie on one of the axes, there will be fewer vectors to resolve into compo-nents. As a result, calculating the solution will take fewer steps.

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Determining Net Force (continued)

1. A man is pulling on his dog with a force of 70.0 N directed at an angle of +30.0° to the horizontal. Find the x and y components of this force.

2. A gust of wind blows an apple from a tree. As the apple falls, the gravitational force on the apple is 2.25 N downward, and the force of the wind on the apple is 1.05 N to the right. Find the magnitude and direction of the net force on the apple.

3. The wind exerts a force of 452 N north on a sailboat, while the water exerts a force of 325 N west on the sailboat. Find the magnitude and direction of the net force on the sailboat.

Astronaut Workouts

G ravity helps to keep bones strong. Loss of bone density is a serious outcome of time spent in space. Astronauts routinely exercise on treadmills

to counteract the effects of microgravity on their skeletal systems. But is it possible to increase the value of their workouts by increasing their mass? And does it matter if they run or walk?

A team of scientists recruited runners to help find out. The runners used treadmills that measured the net force on their legs, or ground reaction force, while they ran and walked. The runners’ inertia was changed by adding masses to a weighted vest. A spring system supported them as they exercised. Although the spring system did not simulate weightless conditions, it kept their weight the same even as their inertia was changed by the added mass. This mimicked the situation in Earth orbit, where a change in mass does not result in a change in weight.

The scientists were surprised to discover that ground reaction force did not increase with mass while the subjects were running. Ground reaction force did increase with mass while the subjects were walking. But overall, ground reaction force for running was still greater. So astronauts still need to run, not walk—and they can’t shorten their workouts by carrying more mass.

Tips and TricksIf there is a net force in both the x and y directions, use vector addition to find the total net force.

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Problem Solving

TAKE IT FURTHERAfter solving Practice B(2), ask students to identify the x and y components of the net force. The x component of the net force is +1.05 N and the y component is –2.25 N. Such extension of the problem can be considered as the evaluation method.

AnswersPractice B 1. Fx = 60.6 N; Fy = 35.0 N 2. 2.48 N at 25.0° counterclockwise

from straight down 3. 557 N at 35.7° west of north

PROBLEM GUIDE BUse this guide to assign problems.SE = Student Edition TextbookPW = Sample Problem Set I (online)PB = Sample Problem Set II (online)Solving for:

Fx , FySE Sample, 1;

Ch. Rvw. 11–12PW 3, 4*, 5*

PB 7–10

FnetSE Sample, 2–3;

Ch. Rvw. 10, 22a*

PW Sample, 1–2PB 1–6

*Challenging Problem

Why It MattersScientists have determined that the body adjusts its stride to changes in inertia differently than it adjusts to changes in weight. Point out that this experiment suggests that changing inertia (mass) does not increase the net force on runners in microgravity. Ask students to think about resistance training and list ways that astronauts could replace the resistance of gravity while in space. Then explain that astronauts use bungee cords to replace the resistance of gravity during exercise. However, bungees replace only about 60% of the astronauts’ weight on Earth.

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Reviewing Main Ideas

1. If a car is traveling westward with a constant velocity of 20 m/s, what is the net force acting on the car?

2. If a car is accelerating downhill under a net force of 3674 N, what addi-tional force would cause the car to have a constant velocity?

3. The sensor in the torso of a crash-test dummy records the magnitude and direction of the net force acting on the dummy. If the dummy is thrown forward with a force of 130.0 N while simultaneously being hit from the side with a force of 4500.0 N, what force will the sensor report?

4. What force will the seat belt have to exert on the dummy in item 3 to hold the dummy in the seat?

Critical Thinking

5. Can an object be in equilibrium if only one force acts on the object?

EquilibriumObjects that are either at rest or moving with constant velocity are said to be in equilibrium. Newton’s first law describes objects in equilibrium, whether they are at rest or moving with a constant velocity. Newton’s first law states one condition that must be true for equilibrium: the net force acting on a body in equilibrium must be equal to zero.

The net force on the fishing bob in Figure 2.3(a) is equal to zero because the bob is at rest. Imagine that a fish bites the bait, as shown in Figure 2.3(b). Because a net force is acting on the line, the bob accelerates toward the hooked fish.

Now, consider a different scenario. Suppose that at the instant the fish begins pulling on the line, the person reacts by applying a force to the bob that is equal and opposite to the force exerted by the fish. In this case, the net force on the bob remains zero, as shown in Figure 2.3(c), and the bob remains at rest. In this example, the bob is at rest while in equilib-rium, but an object can also be in equilibrium while moving at a constant velocity.

An object is in equilibrium when the vector sum of the forces acting on the object is equal to zero. To determine whether a body is in equilibrium, find the net force, as shown in Sample Problem B. If the net force is zero, the body is in equilibrium. If there is a net force, a second force equal and opposite to this net force will put the body in equilibrium.

equilibrium the state in which the net force on an object is zero

Forces on a Fishing Line (a) The bob on this fishing line is at rest. (b) When the bob is acted on by a net force, it accelerates. (c) If an equal and opposite force is applied, the net force remains zero.

FIGURE 2.3

(a)

(b) (c)

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SECTION 2 FORMATIVE ASSESSMENT

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answers to section assessment

FIgure 2.3 Point out that in order for the bob to be in equilibrium, all the forces must cancel. You may want to diagram this situation on the board and include the force of the water on the bob (buoyant force).

ask Other than the forces applied by the person and the fish, do any other forces act on the bob?

answer: yes, the upward (buoyant) force of the water on the bob and the downward gravitational force

assess Use the Formative Assessment on this page to evaluate student mastery of the section.

reteach For students who need additional instruction, download the Section Study Guide.

response to Intervention To reassess students’ mastery, use the Section Quiz, available to print or to take directly online at hMDscience.com.

TEACH FROM VISUALS

� Assess and Reteach

1. zero

2. -3674 N

3. 4502 N at 1.655° forward of the side

4. the same magnitude as the net force in item 3 but in the opposite direction

5. No, either no force or two or more forces are required for equilibrium.

Forces and the Laws of Motion 127