Torque is a rotating force applied to a lever arm. In the case of a belt and pulley, the force is the tension on the belt and the length of the lever arm is the radius of the pulley. Suppose that a set of scales is attached to the belt, as in Figure 6.9, and assume that the tension on the belt is 100.0 lb.
Because the tension or force in the belt is constant along its length, there is 100.0 lb of force pulling at the edge of both the driver (A) and the driven (B) pulley.
When the driver pulley has a diameter of 10.0 in and the driven pulley has a diameter of 5.0 in, then the torque on the driver pulley in Figure 6.9 is:
To = F x R = 100.0 lb x 5.0 in = 500 lb-ft and the torque on the driven shaft is:
The torque is different on the two pulleys because pulleys behave as class one levers; the radius (one-half the diameter) is the length of the lever arm. A larger diameter pulley will have a longer lever arm. This example also illustrates that because the belt speed (in inches per minute) stays the same; the torque on the driven shaft is inversely proportional to the change in speed between the driver and driven pulleys. When the driven shaft turns at a higher speed, its torque is decreased relative to the driver shaft, and vice versa.
For every speed change, there is a change in the torque. Reviewing the PTO power equation is one way of showing this relationship:
When the horsepower stays the same, assuming that there are no drive train power losses, then as torque increases, the rpm must decrease, and vice versa. This relationship also can be expressed as:
FIGURE 6.9. Speed and torque.
Problem: The driver shaft of a belt power train turns at 300 rpm and applies 20.0 lb-ft of torque, and the driven pulley turns at 50 rpm, how much torque will be developed at the driven pulley?
Solution: Rearranging the torque equation to solve for To2 gives us:
N1 300 rpm
N2 50 rpm
Notice that the speed of the driven shaft is one-sixth that of the driver shaft, but the torque on the driven shaft is six times that on the driver shaft. This relationship suggests the reason for a tractor transmission. The engine turns at high speed with low torque, and the drive axle turns at low speed with high torque. The transmission uses different gear pairs to produce different speed-torque combinations at the drive axle.
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