Question

Assertion:
A hollow shaft is found to be stronger than a solid shaft made of the same size and material.
Reason:
The torque required to produce a given twist in a hollow cylinder is greater than that required to twist a solid cylinder of the same size and material.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B. Both Assertion and Reason are correct and Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect.
D. Both Assertion and Reason are incorrect.

Answer 1

Hint: The torque is the force required to rotate an object around an axis, then clearly torque depends on the perpendicular distance from the point and the force exerted, which in turn depends on the mass of the object.
Formula used:
$\Delta L=\dfrac{\rho g{{L}^{2}}}{2Y}$

Complete step by step answer:
Let us consider the shaft as a cylinder. Then we have a solid cylinder $A$ with radius $D$ and a hollow cylinder $B$ with inner radius $d$ and outer radius $D$ as shown in the figure.

If both the cylinders are made of the same material, clearly the mass of the $A$ is greater than the mass of $B$ as cylinder $B$ is hollow.
Now if we assume that both the cylinders are made of the same mass, then the cylinder $B$ is stronger than the cylinder $A$ as the mass in the centre of cylinder $B$ is spread at a smaller area.
Thus we can say that a hollow shaft is found to be stronger than a solid shaft made of the same size and material or that the assertion is true.
If the force is applied on the external diameter of both the cylinders, then the distance where the force is applied will remain constant. But we saw that the cylinder $B$ is stronger than the cylinder $A$ then we can say that cylinder $B$ needs more force and hence more torque as compared to the cylinder $A$.
Then, the reason is also correct.
Hence, the answer is A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

Note:
Since both the cylinders are made of the same mass it may seem that we need to apply the same force and hence the same torque on them. But the concentration of the mass is varied in two cases as discussed above and hence there is a difference in the torque required to rotate the shafts.