13. A solid shaft with a diameter of  is designed to tranismit a torque. The design team considers to replace the solid shaft with a hollow shaft having an outside diameter of  and an inside diameter of . If the two shafts have the same length and density, determine the mean and standard deviation of the percentage reduction in the shaft weight using FOSM. Note that ,  and are independent.

Solution

The weight of the solid shaft is

where  is the length of the shaft, and  is the density.

The weight of the hollow shaft is

So the percentage reduction in shaft weight is

Let

where .

Using FOSM, we have

Ans.