Final answer:
In this case, to transmit hp at rpm, the diameter of the shaft should be approximately 4.84 inches.
Step-by-step explanation:
The problem states that the horsepower (hp) that a shaft can transmit varies jointly with its speed in revolutions per minute (rpm) and the cube of the diameter. This means that the horsepower is directly proportional to the product of the speed and the cube of the diameter.
Let's say the initial shaft has a diameter of 4 inches and can transmit 128 hp at 120 rpm. We can set up a proportion to find the diameter needed to transmit hp at rpm.
First, we can write the proportion as:
(128 hp) / (120 rpm) = (hp) / (rpm)
To find the diameter, we need to consider the cube of the diameter. Let's call the diameter we want to find as D.
So, the proportion can be written as:
(128 hp) / (120 rpm) = (hp) / (rpm) * (D³) / (4³)
Simplifying the equation:
(128 hp) / (120 rpm) = (hp) / (rpm) * (D³) / (64)
Now, we can solve for D:
(D³) = (128 hp * 64) / (120 rpm)
D³ = 68.2667 hp/rpm
D = (68.2667 hp/rpm)⁽¹/³⁾
D ≈ 4.84 inches
Therefore, to transmit hp at rpm, the diameter of the shaft should be approximately 4.84 inches when rounded to 2 decimal places.