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# Torque Figure $$\PageIndex{1}$$: Figure e: The boy makes a torque on the tetherball.

The boy applies a force F to the ball for a short time t, accelerating the ball to a velocity v. Since force is the rate of transfer of momentum, we have

$F=mvt$

and multiplying both sides by r gives

$Fr =mvrt$

But ± mvr is simply the amount of angular momentum he's given the ball, so ± mvr/t also equals the amount of torque he applied. The result of this example is

torque = ± Fr ,

where the plus or minus sign indicates whether torque would tend to create clockwise or counterclockwise motion. This equation applies more generally, with the caveat that F should only include the part of the force perpendicular to the radius line.

self-check:

There are four equations on this page. Which ones are important, and likely to be useful later?

To summarize, we've learned three conserved quantity, each of which has a rate of transfer:

 conserved quantity rate of transfer name units name units energy joules (J) power watts (W) momentum kgm/s [/itex] force newtons (N) angular momentum kgm2/s [/itex] torque newton-meters Nm [/itex])

## Torque vs. Force Figure $$\PageIndex{1}$$: Figure f: The plane's four engines produce zero total torque but not zero total force.

Conversely, we can have zero total force and nonzero total torque. A merry-go-round's engine needs to supply a nonzero torque on it to bring it up to speed, but there is zero total force on it. If there was not zero total force on it, its center of mass would accelerate!

### Example 5: A lever Figure $$\PageIndex{1}$$: Figure g: the biceps muscle flexes the arm.

There are three forces acting on the forearm: the force from the biceps, the force at the elbow joint, and the force from the load being lifted. Because the elbow joint is motionless, it is natural to define our torques using the joint as the axis. The situation now becomes quite simple, because the upper arm bone's force exerted at the elbow has r=0, and therefore creates no torque. We can ignore it completely. In general, we would call this the fulcrum of the lever.

If we restrict ourselves to the case in which the forearm rotates with constant angular momentum, then we know that the total torque on the forearm is zero, so the torques from the muscle and the load must be opposite in sign and equal in absolute value:

$r_{muscle} F_{muscle} = r_{load} F_{load}$

where rmuscle, the distance from the elbow joint to the biceps' point of insertion on the forearm, is only a few cm, while rload might be 30 cm or so. The force exerted by the muscle must therefore be about ten times the force exerted by the load. We thus see that this lever is a force reducer. In general, a lever may be used either to increase or to reduce a force.

Why did our arms evolve so as to reduce force? In general, your body is built for compactness and maximum speed of motion rather than maximum force. This is the main anatomical difference between us and the Neanderthals (their brains covered the same range of sizes as those of modern humans), and it seems to have worked for us.

As with all machines, the lever is incapable of changing the amount of mechanical work we can do. A lever that increases force will always reduce motion, and vice versa, leaving the amount of work unchanged.

## Discussion Questions

• You whirl a rock over your head on the end of a string, and gradually pull in the string, eventually cutting the radius in half. What happens to the rock's angular momentum? What changes occur in its speed, the time required for one revolution, and its acceleration? Why might the string break?
• A helicopter has, in addition to the huge fan blades on top, a smaller propeller mounted on the tail that rotates in a vertical plane. Why?
• The photo shows an amusement park ride whose two cars rotate in opposite directions. Why is this a good design? Discussion question C.