Question

The hollow shaft is __________ than a solid of the same mass, material and length.
A) Less stiff
B) More stiff
C) Equally stiff
D) None of these

Answer 1

Hint:The stiffness of the material can be calculated using the torsion couple of the shaft. Greater the value of the torsion couple of the object, more will be the stiffness. The torsion torque is dependent on the internal torque and the radius of the shaft.

Step by step solution:

According to the question it is given that a solid has the same mass, same material and the same length as the hollow shaft.

Let the mass of both shafts be $m$, the Length of the shaft $l$ and the radius of the shaft is $R$.

Write the formula for the torsion couple of the shaft as shown below.
$ \Rightarrow \tau = \dfrac{{TR}}{J}$

Where, $\tau $ is the torsion torque of the shaft, $T$ is internal torque and $R$ is the radius of the sphere and $J$ is known as the constant of torsion torque.

Write the formula for the value of the constant of torsion torque as shown below.

$ \Rightarrow J = \dfrac{{\pi {d^4}}}{{32}}$
Where, $d$ is the diameter of the shaft.

So, the expression for the torsion torque becomes:
$ \Rightarrow \tau = \dfrac{{32TR}}{{\pi {d^4}}}$

For Hollow cylinder,
$
   \Rightarrow {\tau _1} = \dfrac{{TR}}{{\pi \dfrac{{d_1^4 - d_2^4}}{{32}}}} \\
   \Rightarrow {\tau _1} = \dfrac{{32TR}}{{\pi \left( {d_1^4 - d_2^4} \right)}} \\
 $
Where, ${d_1}\;{\text{and}}\;{d_2}$ are the outer and the inner diameter of the shaft.

For solid cylinder:

$
   \Rightarrow {\tau _2} = \dfrac{{TR}}{{\pi \dfrac{{{d^4}}}{{32}}}} \\
   \Rightarrow {\tau _2} = \dfrac{{32TR}}{{\pi {d^4}}} \\
 $

As, The value$d_1^4 - d_2^4 < {d^4}$
So, the torsion couple for the hollow shaft is more.

So, the stiffness of the hollow shaft is also more.

Thus, a hollow shaft is more stiff than a solid of the same mass, material and length.

So, option B is correct.

Note:Assume that ${d_1}$ and ${d_2}$ are the outer and inner diameter of the shaft. The thickness of the hollow shaft is given as \[{d_1} - {d_2}\]. This thickness of the shaft is always less than the diameter of the solid shaft. This is observed only for the shaft of the same dimensions.