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Standard XII

Physics

Physical Quantities

Two rods A an...

Question

Two rods A and B of the same material and the same length have radii $r_{1}$ and $r_{2}$ , respectively. When they are rigidly fixed at one end and twisted by the same couple applied at the other end the ratio of ${\frac{a n g l e o f t w i s t a t t h e e n d A}{a n g l e o f t w i s t a t t h e e n d B}}$ is

\frac{r_{1}^{2}}{r_{2}^{2}}

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\frac{r_{1}^{4}}{r_{2}^{4}}

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\frac{r_{2}^{3}}{r_{1}^{3}}

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\frac{r_{2}^{4}}{r_{1}^{4}}

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Solution

The correct option is D $\frac{r_{2}^{4}}{r_{1}^{4}}$
Tortional couple $G = C θ$ , where $C = \frac{π η r^{4}}{2 l}$ and $θ$ is the angle of twist.
Since, equal couple is applied on both rods A and B, we have
$C_{A} θ_{A} = C_{B} θ_{B}$
$\frac{θ_{A}}{θ_{B}} = \frac{C_{B}}{C_{A}} = \frac{{r_{2}}^{4}}{{r_{1}}^{4}}$

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