0.33 Phy1330: angular momentum -- torque, work and energy  (Page 3/4)

Torque

We know that torque is equal to

T = r * Fp

where

• T represents torque
• r represents the distance from the center of the wheel to the point where the perpendicular force is applied
• Fp represents the perpendicular force

Arc length

We also know that the arc length is given by

s = r * A

where

• s represents the arc length
• r represents the distance from the center of the wheel to the point where the perpendicular force is applied as before
• A represents the angle of displacement measured in radians

Through substitution

W = Fp * s, or

W = (T/r)*r*A, or

The work done by a constant torque is given by the equation shown in Figure 2.

Power

As in the translational case, power is a measure of the work done per unit of time. If we divide both sides of the above equation by time, we get

(W/t) = T*(A/t)

where

• W/t = work per second or power
• A is the angular displacement in radians
• t is time in seconds
• A/t is the displacement in radians per second, which we recognize as angular velocity

Thus, the power generated or consumed by applying a constant torque is given by the equation shown in Figure 3 .

Variable torque

A torque doesn't have to be constant to do work. In fact, the torque generated by the user with the starter rope on the power mower discussed in theprevious section probably isn't constant.

However, if the torque is not constant, you cannot use the equations developed in the previous section to compute the work done by the torque.

Maybe you can use calculus

If the torque as a function of time can be described by a function that you can integrate using integral calculus, you can use calculus to compute the workdone by the torque. However, in the real word, this is probably rarely the case.

Maybe you can use a computer

If you are in the business of computing work done by a variable torque, the most likely case is that you will have equipment that allows you to sample thetorque and displacement values at uniform intervals of time and to save the values of the samples for digital processing. Then you can use any one ofseveral digital methods to approximately integrate the product of the torque function and the displacement function.

Example scenario

I once visited a factory where mirrors were made. At one of the stations on the manufacturing line, a person used a large horizontal grinding wheel to grinda bevel on the edge of the mirror.

Assume that the grinding wheel is a uniform disk with: