Question

Assertion :A hollow shaft is found to be stronger than a solid shaft made of same material. Reason: The torque required to produce a given twist in hollow cylinder is greater than that required to twist a solid cylinder of 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 but 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

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

As both the shafts (cylinder) are made up of same material and have same size.

$\therefore$ Mass of both the cylinders is same i.e $M_1=M_2$

$\rho \left(\pi R^{2}L\right)=\rho \left(\pi \right(r_2^2-r_1^2\left)L\right)$

$⟹$ $R^2=r_2^2-r_1^2$ ..........(1)

Torque required to twist the solid shaft $\tau =\frac{\eta \pi \varphi R^4}{2L}$

Torque required to twist the hollow shaft ${\tau }^{\prime }=\frac{\eta \pi \varphi (r_2^4-r_1^4)}{2L}$

$\frac{{\tau }^{\prime }}{\tau }=\frac{r_2^4-r_1^4}{R^4}=\frac{(r_2^2-r_1^2)(r_2^2+r_1^2)}{R^4}$

$⟹$ $\frac{{\tau }^{\prime }}{\tau }=\frac{(r_2^2+r_1^2)}{R^2}$ (using (1))

Equation (1) implies that $R^2<r_2^2+r_1^2$

Thus ${\tau }^{\prime }>\tau$

Hence both assertion and the reason are correct and the reason is the correct explanation of assertion.