Force is the rate of transfer of momentum. The equivalent in the case of angular momentum is called torque (rhymes with “fork”):
Where force tells us how hard we are pushing or pulling on something, torque indicates how hard we are twisting on it. Have you ever had the experience of trying to open a door by pushing on the wrong side, the side near the hinge? It's difficult to do, which apparently indicates that a given amount of force produces less torque when it's applied close to the axis of rotation. To try to pin down this relationship more precisely, let's imagine hitting a tetherball, e.
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
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.
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|
| || || || |
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow><mtext>kg</mtext><mo lspace="0.056em" rspace="0.056em">⋅</mo><mtext>m</mtext><mo lspace="0" rspace="0" stretchy="false">/</mo><mtext>s</mtext></mrow> </math>
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow><mtext>kg</mtext><mo lspace="0.056em" rspace="0.056em">⋅</mo><msup><mtext>m</mtext><mn>2</mn></msup><mo lspace="0" rspace="0" stretchy="false">/</mo><mtext>s</mtext></mrow> </math>
newton-meters <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow><mtext>N</mtext><mo lspace="0.056em" rspace="0.056em">⋅</mo><mtext>m</mtext></mrow> </math>)
| || || || |
Torque vs. Force
Of course a force is necessary in order to create a torque --- you can't twist a screw without pushing on the wrench --- but force and torque are two different things. One distinction between them is direction. We use positive and negative signs to represent forces in the two possible directions along a line. The direction of a torque, however, is clockwise or counterclockwise, not a linear direction.
The other difference between torque and force is a matter of leverage. A given force applied at a door's knob will change the door's angular momentum twice as rapidly as the same force applied halfway between the knob and the hinge. The same amount of force produces different amounts of torque in these two cases.
It is possible to have a zero total torque with a nonzero total force. An airplane with four jet engines, f, would be designed so that their forces are balanced on the left and right. Their forces are all in the same direction, but the clockwise torques of two of the engines are canceled by the counterclockwise torques of the other two, giving zero total torque.
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 g shows an example of a lever within your arm. Different muscles are used to flex and extend the arm, because muscles work only by contraction. The biceps flexes it.
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:
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.
- 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?
- Benjamin Crowell, Conceptual Physics