section 4 section 4 objectives torque and simple machines · pdf fileaxis of rotation d f the...

Key Termstorque lever arm

Rotational MotionEarlier you studied various examples of uniform circular motion, such as a spinning Ferris wheel or an orbiting satellite. During uniform circular motion, an object moves in a circular path and at constant speed. An object that is in circular motion is accelerating because the direction of the object’s velocity is constantly changing. This centrip-etal acceleration is directed toward the center of the circle. The net force causing the acceleration is a centripetal force, which is also directed toward the center of the circle.

In this section, we will examine a related type of motion: the motion of a rotating rigid object. For example, consider a football that is spinning as it flies through the air. If gravity is the only force acting on the football, the football spins around a point called its center of mass. As the football moves through the air, its center of mass follows a parabolic path. Note that the center of mass is not always at the center of the object.

Rotational and translational motion can be separated.Imagine that you roll a strike while bowling. When the bowling ball strikes the pins, as shown in Figure 4.1, the pins spin in the air as they fly backward. Thus, they have both rotational and linear motion. These types of motion can be analyzed separately. In this section, we will isolate rotational motion. In particular, we will explore how to measure the ability of a force to rotate an object.

Torque and Simple Machines

Objectives

Distinguish between torque and force.

Calculate the magnitude of a torque on an object.

Identify the six types of simple machines.

Calculate the mechanical advantage of a simple machine.

Types of Motion Pins that are spinning and flying through the air exhibit both rotational and translational motion.

FIGURE 4.1

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Differentiated Instruction

Preview VocabularyScientific Meanings The word torque is a technical word with no common usage. It is widely used in physics and mechanics. Torque is used to describe something that generates or tends to generate rotation or torsion.

Misconception Alert!Many students may think that any force acting on an object will produce a torque. You may want to demonstrate that if you push an object at its center of mass without rotating the object, you are applying a force to the object without producing a torque.

FIGURE 4.1 Point out that the pins were initially at rest and were set in motion by the bowling ball.

Ask Is the energy gained by the pins greater than, equal to, or less than the energy lost by the bowling ball?

Answer: The energy gained by the pins is equal to the energy lost by the bowling ball (except for a small loss of energy in the form of sound and a slight temperature increase in the pins).

� Plan and Prepare

� Teach

TEACH FROM VISUALS

PRE-APExpand the definition of rotational motion by including the concept of “circular motion.” A circular motion is a common form of rota-tional motion in which every point on a rigid object moves in a circular path. The axis passing through the center of this circle is called the axis of rotation.

244 Chapter 7

SECTION 4

axis ofrotation

d

F

The Magnitude of a TorqueImagine a cat trying to leave a house by pushing perpendicu-larly on a cat-flap door. Figure 4.2 shows a cat-flap door hinged at the top. In this configuration, the door is free to rotate around a line that passes through the hinge. This is the door’s axis of rotation. When the cat pushes at the bottom edge of the door with a force that is perpendicular to the door, the door opens. The ability of a force to rotate an object around some axis is measured by a quantity called torque.

Torque depends on the force and the lever arm.If a cat pushed on the door with the same force but at a point closer to the hinge, the door would be more difficult to rotate. How easily an object rotates depends not only on how much force is applied but also on where the force is applied. The farther the force is from the axis of rotation, the easier it is to rotate the object and the more torque is produced. The perpendicular distance from the axis of rotation to a line drawn along the direction of the force is called the lever arm.

Figure 4.3 shows a diagram of the force F applied by the pet perpen-dicular to the cat-flap door. If you examine the definition of lever arm, you will see that in this case the lever arm is the distance d shown in the figure, the distance from the pet’s nose to the hinge. That is, d is the perpendicular distance from the axis of rotation to the line along which the applied force acts. If the pet pressed on the door at a higher point, the lever arm would be shorter. As a result, the cat would need to exert a greater force to apply the same torque.

torque a quantity that measures the ability of a force to rotate an object around some axis

lever arm the perpendicular distance from the axis of rotation to a line drawn along the direction of the force

Hinge Rotation The cat-flap door rotates on a hinge, allowing pets to enter and leave a house at will.

FIGURE 4.2

Torque A force applied to an extended object can produce a torque. This torque, in turn, causes the object to rotate.

FIGURE 4.3

CHANGING THE LEVER ARM

In this activity, you will explore how the amount of force required to open a door changes when the lever arm changes. Using only perpendicular forces, open a door several times by applying a force at different distances from the hinge. You may have to tape the latch so that the door will open

when you push without turning the knob. Because the angle of the applied force is kept constant, decreasing the distance to the hinge decreases the lever arm. Compare the relative effort required to open the door when pushing near the edge to that required when pushing near

the hinged side of the door. Summarize your findings in terms of torque and the lever arm.

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TEAChER’S NOTESStudents should find that the smallest lever arm (closest to the hinge) requires the greatest force to produce the same torque.

homework Options This QuickLab can easily be performed outside of the physics lab room.

QuickLab


PHYSICS

Spec. Number PH 99 PE C08-001-006-A

Boston Graphics, Inc.

617.523.1333

d

F dF d

F

PHYSICS



617.523.1333

d

F dF d

F

PHYSICS



617.523.1333

d

F dF d

F

F

d

d sin

PH99PE C08-001-007 A7-11

The lever arm depends on the angle.Forces do not have to be perpendicular to an object to cause the object to rotate. Imagine the cat-flap door again. In Figure 4.4(a), the force exerted by the cat is perpendicular to d. When the angle is less than 90°, as in (b) and (c), the door will still rotate, but not as easily. The symbol for torque is the Greek letter tau (τ), and the magnitude of the torque is given by the following equation:

Torqueτ = Fd sin θ

torque = force × lever arm

The SI unit of torque is the N•m. Notice that the inclusion of the factor sin θ in this equation takes into account the changes in torque shown in Figure 4.4.

Figure 4.5 shows a wrench pivoted around a bolt. In this case, the applied force acts at an angle to the wrench. The quantity d is the distance from the axis of rotation to the point where force is applied. The quantity d sin θ, however, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force. Thus, d sin θ is the lever arm. Note that the perpendicular distance between the door hinge and the point of application of force F in Figure 4.4 decreases as the cat goes further through the door.

The Sign of a TorqueTorque, like displacement and force, is a vector quantity. In this textbook, we will assign each torque a positive or negative sign, depending on the direction the force tends to rotate an object. We will use the convention that the sign of the torque resulting from a force is positive if the rotation is counterclockwise and negative if the rotation is clockwise. In calcula-tions, remember to assign positive and negative values to forces and displacements according to the sign convention established in the chapter “Motion in One Dimension.”

Torque and Direction The direction of the lever arm is always perpendicular to the direction of the applied force.

FIGURE 4.5

(a) (b) (c)

Torque and Angles In each example, the cat is pushing on the door at the same distance from the axis. To produce the same torque, the cat must apply greater force for smaller angles.

FIGURE 4.4

Tips and TricksTo determine the sign of a torque, imagine that the force is the only force acting on the object and that the object is free to rotate. Visualize the direction that the object would rotate. If more than one force is acting, treat each force separately. Be careful to associate the correct sign with each torque.

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FIGURE 4.5 Be certain students recog-nize how to identify the force, the distance to the axis, the angle between the force and the axis, and the lever arm.

Ask Mechanics often use “cheater bars” to loosen stubborn bolts. A cheater bar is usually a length of pipe that fits over the handle of a wrench, which makes the handle longer. How do cheater bars help the mechanic?

Answer: Cheater bars increase the lever arm length while keeping the force the same, thus increasing the torque applied to a stubborn bolt.

� Teach continued

TEACH FROM VISUALS

BElOw lEVElHaving some real-life examples can help students understand the concept of torque and how it works. Ask students to find some daily uses of torque and discuss their findings to see whether their examples are appropriate. For example, tell students that rotating the lid of a bottle is an example of applying torque. Moving a seesaw up and down is another example of using torque.

246 Chapter 7

For example, imagine that you are pulling on a wishbone with a perpendicular force F1 and that a friend is pulling in the opposite direction with a force F2. If you pull the wishbone so that it would rotate counterclockwise, then you exert a positive torque of magnitude F1d1. Your friend, on the other hand, exerts a negative torque, –F2 d2. To find the net torque acting on the wishbone, simply add up the individual torques.

τnet = Στ = τ1 + τ2 = F1d1 + (−F2 d2)

When you properly apply the sign convention, the sign of the net torque will tell you which way the object will rotate, if at all.

Torque

Sample Problem E A basketball is being pushed by two players during tip-off. One player exerts an upward force of 15 N at a perpendicular distance of 14 cm from the axis of rotation. The second player applies a downward force of 11 N at a perpendicular distance of 7.0 cm from the axis of rotation. Find the net torque acting on the ball about its center of mass.

ANALYZE Given: F1 = 15 N F2 = 11 N

d1 = 0.14 m d2 = 0.070 m

Unknown: τnet = ?

Diagram: see right

PLAN Choose an equation or situation:Apply the definition of torque to each force, and add up the individual torques.

τ = Fd

τnet = τ1 + τ2 = F1d1 + F2 d2

SOLVE Substitute the values into the equations and solve:First, determine the torque produced by each force. Use the standard convention for signs.

τ1 = F1d1 = (15 N)(−0.14 m) = −2.1 N•m

τ2 = F2 d2 = (−11 N)(0.070 m) = −0.77 N•m

τnet = −2.1 N•m − 0.77 N•m

τnet = −2.9 N•m

CHECK YOUR WORK

The net torque is negative, so the ball rotates in a clockwise direction.

Continued

Interactive DemoHMDScience.com

PREMIUM CONTENT

= 15 N

HRW • Holt PhysicsPH99PE-C08-001-011-A

F 1

= 11 NF 2

= 0.070 md2

= 0.14 md1

Tips and TricksThe factor sin θ is not included because each given distance is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force.


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Problem Solving

Classroom PracticeTorqueA student pushes with a minimum force of 50.0 N on the middle of a door to open it. a. What minimum force must be

applied at the edge of the door in order for the door to open?

b. What minimum force must be applied to the hinged side of the door in order for the door to open?

Answers a. 25.0 N b. The door cannot be opened by a

force at the hinge location. It can be broken but cannot be opened normally.

ProBLeM guide eUse this guide to assign problems.Se = Student Edition TextbookPW = Sample Problem Set I (online)PB = Sample Problem Set II (online)Solving for:

τ Se Sample, 1–2; Ch. Rvw. 37–38, 43 44*, 51a, 54

PW 6, 7aPB 4–6

d PW Sample, 1–2, 3*

PB 7–10

F Se 3; Ch. Rvw. 41–42, 51b, 54

PW 4*, 5*, 7bPB Sample, 1–3

*Challenging ProblemTAke iT FurTherShow students how to solve a torque problem when the force is not applied at a perpendicu-lar angle. For example, the force may be applied at an angle of 60°. In that case, the following formula must be used:τ = Fd sin θ


Types of Simple MachinesWhat do you do when you need to pry a cap off a bottle of soda? You probably use a bottle opener, as shown in Figure 4.6. Similarly, you would probably use scissors to cut paper or a hammer to drive a nail into a board. All of these devices make your task easier. These devices are all examples of machines.

The term machine may bring to mind intricate systems with multicol-ored wires and complex gear-and-pulley systems. Compared with inter-nal-combustion engines or airplanes, simple devices such as hammers, scissors, and bottle openers may not seem like machines, but they are.

A machine is any device that transmits or modifies force, usually by changing the force applied to an object. All machines are combinations or modifications of six fundamental types of machines, called simple machines. These six simple machines are the lever, pulley, inclined plane, wheel and axle, wedge, and screw, as shown in Figure 4.7 on the next page.

Using simple machines.Because the purpose of a simple machine is to change the direction or magnitude of an input force, a useful way of characterizing a simple machine is to compare how large the output force is relative to the input force. This ratio, called the machine’s mechanical advantage, is written as follows:

MA = output force

__ input force

= Fout _ Fin

A Lever Because this bottle opener makes work easier, it is an example of a machine.

FIGURE 4.6

Torque (continued)

1. Find the magnitude of the torque produced by a 3.0 N force applied to a door at a perpendicular distance of 0.25 m from the hinge.

2. A simple pendulum consists of a 3.0 kg point mass hanging at the end of a 2.0 m long light string that is connected to a pivot point.

a. Calculate the magnitude of the torque (due to gravitational force) around this pivot point when the string makes a 5.0° angle with the vertical.

b. Repeat this calculation for an angle of 15.0°.

3. If the torque required to loosen a nut on the wheel of a car has a magnitude of 40.0 N•m, what minimum force must be exerted by a mechanic at the end of a 30.0 cm wrench to loosen the nut?

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Problem Solving

AnswersPractice E 1. 0.75 N·m

2. a. 5.1 N·m

b. 15 N·m

3. 133 N

FIGURE 4.6 Point out that the reason the bottle opener is so useful is that the length of the handle is longer than the distance between the rim, which pries the bottle cap up, and the cap itself. The amount of force needed to produce a large torque on the cap is small because the lever arm is long.

Ask Would the bottle opener still provide a mechanical advantage if the “prying arm” were equal in length to the handle?

Answer: No, if both the “prying arm” and the handle were the same length, the force input would equal the force output, and the mechanical advantage would equal 1.

� Teach continued

TEACH FROM VISUALS

AlTERNATIVE APPROAChESAnother way to calculate torque is to use the full distance from the pivot point to the applied force and to use only the perpendicu-lar component of the applied force. The magnitude of this “effective force” is equal to F sin θ. The resulting torque will be equal to the result that would be obtained using the applied force and lever arm.

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Lever

Fulcrum

Inclined planeWheel

Axle

Screw

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Lever

Fulcrum

Inclined planeWheel

Axle

Screw

PulleysWedge

Spec# PH99PE C08-004-001-A

Lever

Fulcrum

Inclined planeWheel

Axle

Screw

PulleysWedge

Spec# PH99PE C08-004-001-A

Lever

Fulcrum

Inclined planeWheel

Axle

Screw

PulleysWedge

Spec# PH99PE C08-004-001-A

Lever

Fulcrum

Inclined planeWheel

Axle

Screw

PulleysWedge

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Lever

Fulcrum

Inclined planeWheel

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Screw

PulleysWedge

One example of mechanical advantage is the use of the back of a hammer to pry a nail from a board. In this example, the hammer is a type of lever. A person applies an input force to one end of the handle. The handle, in turn, exerts an output force on the head of a nail stuck in a board. If friction is disregarded, the input torque will equal the output torque. This relation can be written as follows:

τin = τout

Fin din = Fout dout

Substituting this expression into the definition of mechanical advantage gives the following result:

MA = Fout

_ Fin

= din

_ dout

The longer the input lever arm as compared with the output lever arm, the greater the mechanical advantage is. This in turn indicates the factor by which the input force is amplified. If the force of the board on the nail is 99 N and if the mechanical advantage is 10, then an input force of 10 N is enough to pull out the nail. Without a machine, the nail could not be removed unless the input force was greater than 99 N.

Tips and TricksThis equation can be used to predict

the output force for a given input

force if there is no friction. The

equation is not valid if friction is

taken into account. With friction,

the output force will be less than

expected, and thus d in

___ d out

will not

equal F out ___ F in

.

FIGURE 4.7

SIX SIMPLE MACHINES


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The language of PhysicsThe ideal mechanical advantage (IMA)

refers to the mechanical advantage that

would exist if there were no friction.

Thus, IMA = d

in

___ d

out

. The actual mechanical

advantage (AMA) takes friction into

account. Because there is always some

friction, the ideal mechanical advantage is

always greater than the actual mechanical

advantage. If Fout

is the meas ured output

force, then the ratio F

out

___ Fin

will give the

actual mechanical advantage, and F

out

___ Fin

will

not equal d

in

___ d

out

. On the other hand, if the

equality on this page is used to calculate

Fout

, then Fout

is the predicted output

force in an ideal situation with no

friction.

ENGlISh lEARNERSBuilding acrostics may help students remember lists, such as a list of the six simple machines. Students should be encouraged to create their own acrostics, because they will be more likely to recall words of their own choosing. You may want to give them an example like the follow-ing: “Let People Wander When the Party Starts,” for Lever, inclined Plane, Wheel and axle, Wedge, Pulley, and Screw.


Small distance—Large force

Large distance—Small force

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Machines can alter the force and the distance moved.You have learned that mechanical energy is conserved in the absence of friction. This law holds for machines as well. A machine can increase (or decrease) the force acting on an object at the expense (or gain) of the distance moved, but the product of the two—the work done on the object—is constant.

For example, Figure 4.8 shows two examples of a trunk being loaded onto a truck. Figure 4.9 illustrates both examples schematically. In one example, the trunk is lifted directly onto the truck. In the other example, the trunk is pushed up an incline into the truck.

In the first example, a force (F1) of 360 N is required to lift the trunk, which moves through a distance (d1) of 1.0 m. This requires 360 N•m of work (360 N × 1 m). In the second example, a lesser force (F2) of only 120 N would be needed (ignoring friction), but the trunk must be pushed a greater distance (d2) of 3.0 m. This also requires 360 N•m of work (120 N × 3 m). As a result, the two methods require the same amount of energy.

Efficiency is a measure of how well a machine works.The simple machines we have considered so far are ideal, frictionless machines. Real machines, however, are not frictionless. They dissipate energy. When the parts of a machine move and contact other objects, some of the input energy is dissipated as sound or heat. The efficiency of a machine is the ratio of useful work output to work input. It is defined by the following equation:

eff = Wout _ Win

If a machine is frictionless, then mechanical energy is conserved. This means that the work done on the machine (input work) is equal to the work done by the machine (output work) because work is a measure of energy transfer. Thus, the mechanical efficiency of an ideal machine is 1, or 100 percent. This is the best efficiency a machine can have. Because all real machines have at least a little friction, the efficiency of real machines is always less than 1.

An Inclined Plane Lifting this trunk directly up requires more force than pushing it up the ramp, but the same amount of work is done in both cases.

FIGURE 4.8

Changing Force or Distance Simple machines can alter both the force needed to perform a task and the distance through which the force acts.

FIGURE 4.9

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Misconception Alert!Reinforce the idea that machines do not create something from nothing. If friction is disregarded, machines use the same amount of energy to achieve the goal. Use numerical examples to illustrate that the work done on the objects is the same. Show the trade-off between force and distance.

Teaching TipIf you have discussed the difference between ideal and actual mechanical advantage with students, point out that efficiency can also be calculated with the ratio AMA

____ IMA because this ratio is

equivalent to the ratio W

in ____ W

out

.

� Teach continued

BElOw lEVElHave students write a brief description of the differences between force and torque. Ask them to share their descriptions with the class. Encourage other students to ask questions.

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O 23°

25.0 N

10.0 N

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Reviewing Main Ideas

1. Determine whether each of the following situations involves linear motion, rotational motion, or a combination of the two.a. a baseball dropped from the roof of a houseb. a baseball rolling toward third basec. a pinwheel in the windd. a door swinging open

2. What quantity describes the ability of a force to rotate an object? How does it differ from a force? On what quantities does it depend?

3. How would the force needed to open a door change if you put the handle in the middle of the door?

4. What are three ways that a cat pushing on a cat-flap door can change the amount of torque applied to the door?

5. The efficiency of a squeaky pulley system is 73 percent. The pulleys are used to raise a mass to a certain height. What force is exerted on the machine if a rope is pulled 18.0 m in order to raise a 58 kg mass a height of 3.0 m?

6. A person lifts a 950 N box by pushing it up an incline. If the person exerts a force of 350 N along the incline, what is the mechanical advantage of the incline?

7. You are attempting to move a large rock by using a long lever. Will the work you do on the lever be greater than, the same as, or less than the work done by the lever on the rock? Explain.

Interpreting Graphics

8. Calculate the torque for each force acting on the bar in Figure 4.10. Assume the axis is perpendicular to the page and passes through point O. In what direction will the object rotate?

9. Figure 4.11 shows an example of a Rube Goldberg machine. Identify two types of simple machines that are included in this compound machine.

Critical Thinking

10. A bicycle can be described as a combination of simple machines. Identify two types of simple machines that are used to propel a typical bicycle.

FIGURE 4.10

FIGURE 4.11


SECTION 4 FORMATIVE ASSESSMENT

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Answers to Section Assessment

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.

� Assess and Reteach

1. a. linear motion

b. linear and rotational motion

c. rotational motion

d. rotational motion

2. torque; It is measured in units of N·m, not N. It is a measure of the ability of a force to accelerate an object around an axis; It depends on the force and the lever arm.

3. Twice as much force would be needed to open the door.

4. by changing the point at which the force is applied, the angle at which the force is applied, or the magnitude of the applied force

5. 130 N

6. 2.7

7. Ideally, the amounts of work will be the same, but because no machine is perfectly efficient, the work you do will be greater.

8. τ30

= 0 N·m, τ25

= +43 N·m, τ10

= −16 N·m; The bar will rotate counterclockwise.

9. Possible answers include lever, wedge, and pulley.

10. the wheels (wheel and axle) and the gear system (pulley)


section 4 section 4 objectives torque and simple machines · pdf fileaxis of rotation d f the...

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