Question

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute, rpm) and the cube of its diameter. If a shaft of a certain material 2 inches in diameter can transmit 36 hp at 75 rpm, what diameter must the shaft have in order to transmit 134 hp at 35 rph

Answer 1

the shaft must have 4 inches of diameter

Step-by-step explanation

Step 1

given that The horsepower​ (hp) that a shaft can safely transmit varies jointly with its speed​ (in revolutions per​ minute, rpm) and the cube of its diameter.

`Power\propto diamter^3v`

so

a) If a shaft of a certain material 2 inches in diameter can transmit 36 hp at 75​ rpm

`\begin{gathered} 36\text{ hp=\lparen2 in\rparen}^3*75\text{ rpm*k} \\ 36=600\text{ k} \\ k=(36)/(600)=0.06 \end{gathered}`

b)what diameter must the shaft have in order to transmit 134 hp at 35 rph

`\begin{gathered} Power\propto diamter^3v \\ Power=d^3*v*k \\ replace \\ 134=d^3*35*0.06 \\ 134=d^3*2.1 \\ divide\text{ both sides by 2.1} \\ (134)/(2.1)=(d^3*2.1)/(2.1) \\ 63.80=d^3 \\ cubic\text{ root in both sides} \\ \sqrt[3]{63.80}=\sqrt[3]{d^3} \\ 3.996=d \\ rounded \\ d=4\text{ inches} \end{gathered}`

therefore

the shaft must have 4 inches of diameter

I hope this helps you