Force can be interpreted as the rate of transfer of momentum. The equivalent in the case of angular momentum is called torque. Where force tells us how hard we are pushing or pulling on something, torque indicates how hard we are twisting on it. Torque is represented by the Greek letter tau, ττ, and the rate of change of an object’s angular momentum equals the total torque acting on it:
(If the angular momentum does not change at a constant rate, the total torque equals the slope of the tangent line on a graph of LL versus tt.)
As with force and momentum, it often happens that angular momentum recedes into the background and we focus our interest on the torques. The torque-focused point of view is exemplified by the fact that many scientifically-untrained but mechanically-apt people know all about torque, but none of them have heard of angular momentum. Car enthusiasts eagerly compare engines’ torques, and there is a tool called a torque wrench that allows one to apply a desired amount of torque to a screw and avoid overtightening it.
Torque distinguished from force
Of course a force is necessary in order to create a torque—you cannot 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, o, 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!
Relationship between force and torque
How do we calculate the amount of torque produced by a given force? Since it depends on leverage, we should expect it to depend on the distance between the axis and the point of application of the force. We will derive an equation relating torque to force for a particular, simple situation, and state without proof that the equation applies to all situations.
To try to pin down this relationship more precisely, let us imagine hitting a tetherball. The boy applies a force FF to the ball for a short time ΔtΔt, accelerating the ball from rest to a velocity vv. Since force is the rate of transfer of momentum, we have:
Since the initial velocity is zero, Δv is the same as the final velocity v. Multiplying both sides by r givesFr=mΔvΔt.=mvrΔt.
Since the initial velocity is zero, Δv is the same as the final velocity v. Multiplying both sides by r givesFr=mvrΔt.
But mvrmvr is simply the amount of angular momentum he has given the ball, so mvr/Δtmvr/Δt also equals the amount of torque he applied. The result of this example is:
If the boy had applied a force parallel to the radius line, either directly inward or outward, then the ball would not have picked up any clockwise or counterclockwise angular momentum.
If a force acts at an angle other than 0 or 90° with respect to the line joining the object and the axis, it would be only the component of the force perpendicular to the line that would produce a torque:
Although this result was proved under a simplified set of circumstances, it is more generally valid.
The rate at which a force transfers angular momentum to an object, i.e., the torque produced by the force, is given by:
where rr is the distance from the axis to the point of application of the force, and F⊥F⊥ is the component of the force that is perpendicular to the line joining the axis to the point of application.
The equation is stated with absolute value signs because the positive and negative signs of force and torque indicate different things, so there is no useful relationship between them. The sign of the torque must be found by physical inspection of the case at hand.
From the equation, we see that the units of torque can be written as newtons multiplied by meters. Metric torque wrenches are calibrated in N⋅mN⋅m, but American ones use foot-pounds, which is also a unit of distance multiplied by a unit of force. Newtons multiplied by meters equal joules, but torque is a completely different quantity from work, and nobody writes torques with units of joules, even though it would be technically correct.
Sometimes torque can be more neatly visualized in terms of the quantity r⊥r⊥, which gives us a third way of expressing the relationship between torque and force:
Of course, you would not want to go and memorize all three equations for torque. Starting from any one of them, you could easily derive the other two using trigonometry. Familiarizing yourself with them can however clue you in to easier avenues of attack on certain problems.
The torque due to gravity
Up until now, we have been thinking in terms of a force that acts at a single point on an object, such as the force of your hand on the wrench. This is of course an approximation, and for an extremely realistic calculation of your hand’s torque on the wrench, you might need to add up the torques exerted by each square millimeter where your skin touches the wrench. This is seldom necessary. But in the case of a gravitational force, there is never any single point at which the force is applied. Our planet is exerting a separate tug on every brick in the Leaning Tower of Pisa, and the total gravitational torque on the tower is the sum of the torques contributed by all the little forces. Luckily there is a trick that allows us to avoid such a massive calculation. It turns out that for purposes of computing the total gravitational torque on an object, you can get the right answer by pretending that the whole gravitational force acts at the object’s center of mass.
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