Question

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm)) and the cube of the diameter. If the shaft of a certain material 4 inches in diameter can transmit 128 hp at 120 rpm, what must the diameter be in order to transmit 30 hp at 100 rpm? Round to 2 decimal places.

Answer 1

To transmit 30 hp at 100 rpm, the diameter of the shaft must be approximately 2.31 inches.

Step-by-step explanation:

The problem states that the horsepower (hp) a shaft can transmit varies jointly with its speed (rpm) and the cube of the diameter. This can be expressed as an equation: hp = k * speed * diameter^3, where k is a constant.

Given that the 4-inch diameter shaft transmits 128 hp at 120 rpm, we can use these values to find the constant k. Substituting these values into the equation, we get 128 = k * 120 * 4³. Solving for k, we find k = 128 / (120 * 4³).

Now, with the new requirement of 30 hp at 100 rpm, we can use the constant k to find the corresponding diameter. The equation becomes 30 = (128 / (120 * diameter^3)) * 100. Solving for the diameter, we find it to be approximately 2.31 inches when rounded to two decimal places.

In summary, the diameter of the shaft needed to transmit 30 hp at 100 rpm is approximately 2.31 inches.