Substitute mA Substitute mK

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Define Kariann s distance in terms of the length of the seesaw and Aysha s distance rK 175 m rA When there is no rotation, the sum of the torques is zero FgKrK FgArA FgKrK FgK(175 m Solve for rA FgK(175 m) rA) FgArA FgArA 00 N m 00 N m

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FgK(rA) FgKrA (FgK

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FgK(175 m) (FgK FgA)

Substitute FgK

Math Handbook

Isolating a Variable page 845

(55 102 N)(175 m) (55 102 N 42 102 N)

55 102 N, FgA

42 102 N

099 m

Evaluate the Answer

Are the units correct Distance is measured in meters Do the signs make sense Distances are positive Is the magnitude realistic Aysha is about 1 m from the center, so Kariann is about 075 m away from it Because Kariann s weight is greater than Aysha s weight, the lever arm on Kariann s side should be shorter Aysha is farther from the pivot, as expected

8 Rotational Motion

16 Ashok, whose mass is 43 kg, sits 18 m from the center of a seesaw Steve, whose mass is 52 kg, wants to balance Ashok How far from the center of the seesaw should Steve sit 17 A bicycle-chain wheel has a radius of 770 cm If the chain exerts a 350-N force on the wheel in the clockwise direction, what torque is needed to keep the wheel from turning 18 Two baskets of fruit hang from strings going around pulleys of different diameters, as shown in Figure 8-6 What is the mass of basket A 19 Suppose the radius of the larger pulley in problem 18 was increased to 60 cm What is the mass of basket A now 20 A bicyclist, of mass 650 kg, stands on the pedal of a bicycle The crank, which is 0170 m long, makes a 450 angle with the vertical, as shown in Figure 8-7 The crank is attached to the chain wheel, which has a radius of 970 cm What force must the chain exert to keep the wheel from turning

45 cm 11 cm

Figure 8-6

(Not to scale)

A 023 kg 450

970 cm 0170 m

Figure 8-7

The Moment of Inertia

If you exert a force on a point mass, its acceleration will be inversely proportional to its mass How does an extended object rotate when a torque is exerted on it To observe firsthand, recover the pencil, the coins, and the transparent tape that you used earlier in this chapter First, tape the coins at the ends of the pencil Hold the pencil between your thumb and forefinger, and wiggle it back and forth Take note of the forces that your thumb and forefinger exert These forces create torques that change the angular velocity of the pencil and coins Now move the coins so that they are only 1 or 2 cm apart Wiggle the pencil as before Did the amount of torque and force need to be changed The torque that was required was much less this time Thus, the amount of mass is not the only factor that determines how much torque is needed to change angular velocity; the location of that mass also is relevant The resistance to rotation is called the moment of inertia, which is represented by the symbol I and has units of mass times the square of the distance For a point object located at a distance, r, from the axis of rotation, the moment of inertia is given by the following equation Moment of Inertia of a Point Mass I mr2

The moment of inertia of a point mass is equal to the mass of the object times the square of the object s distance from the axis of rotation

Section 82 Rotational Dynamics

Table 8-2

Moments of Inertia for Various Objects Object Thin hoop of radius r Location of Axis Through central diameter Through center Diagram