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Torque is a measure of how effectively a force causes rotation The magnitude of torque is the product of the force and the lever arm Because force is measured in newtons, and distance is measured in meters, torque is measured in newton-meters (N m) Torque is represented by the Greek letter tau, The equation for torque is shown below Torque
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Torque is equal to the force times the lever arm
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Lever Arm A bolt on a car engine needs to be tightened with a torque of 35 N m You use a 25-cm-long wrench and pull on the end of the wrench at an angle of 600 from the perpendicular How long is the lever arm, and how much force do you have to exert
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Analyze and Sketch the Problem
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Sketch the situation Find the lever arm by extending the force vector backwards until a line that is perpendicular to it intersects the axis of rotation Known:
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r 025 m 600 35 N m
600
Unknown:
L F
600
25 cm
Solve for the Unknown
Solve for the length of the lever arm L r sin (025 m)(sin 600 ) Substitute r 022 m Solve for the force Fr sin F
r sin 35 N m (025 m)(sin 600 )
Substitute 35 N m, r 025 m, 600 025 m, 600
Math Handbook
Trigonometric Ratios page 855
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16 102 N
Evaluate the Answer
Are the units correct Force is measured in newtons Does the sign make sense Only the magnitude of the force needed to rotate the wrench clockwise is calculated
8 Rotational Motion
11 Consider the wrench in Example Problem 1 What force is needed if it is applied to the wrench at a point perpendicular to the wrench 12 If a torque of 550 N m is required and the largest force that can be exerted by you is 135 N, what is the length of the lever arm that must be used 13 You have a 0234-m-long wrench A job requires a torque of 324 N m, and you can exert a force of 232 N What is the smallest angle, with respect to the vertical, at which the force can be exerted 14 You stand on the pedal of a bicycle If you have a mass of 65 kg, the pedal makes an angle of 35 above the horizontal, and the pedal is 18 cm from the center of the chain ring, how much torque would you exert 15 If the pedal in problem 14 is horizontal, how much torque would you exert How much torque would you exert when the pedal is vertical
Finding Net Torque
Try the following experiment Get two pencils, some coins, and some transparent tape Tape two identical coins to the ends of the pencil and balance it on the second pencil, as shown in Figure 8-5 Each coin exerts a torque that is equal to its weight, Fg, times the distance, r, from the balance point to the center of the coin, as follows: Fgr But the torques are equal and opposite in direction Thus, the net torque is zero:
or Fg1 r1 Fg2 r2 0
How can you make the pencil rotate You could add a second coin on top of one of the two coins, thereby making the two forces different You also could slide the balance point toward one end or the other of the pencil, thereby making the two distances different
Figure 8-5 The torque exerted by the first coin, Fg1r1, is equal and opposite in direction to the torque exerted by the second coin, Fg2r2, when the pencil is balanced
Section 82 Rotational Dynamics
Balancing Torques Kariann (56 kg) and Aysha (43 kg) want to balance on a 175-m-long seesaw Where should they place the pivot point
Analyze and Sketch the Problem
Sketch the situation Draw and label the vectors Known:
mK 56 kg mA 43 kg rK rA 175 m
rK rA
Unknown:
rK rA
Solve for the Unknown
Find the two forces Kariann: FgK mKg (56 kg)(980 m/s2) 55 102 N Aysha: FgA mAg (43 kg)(980 m/s2) 42 102 N
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