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A solid sphere can rotate about its diameter. When a torque is applied to this sphere about this diameter, it starts rotation with angular acceleration of $8 rad / s^{2}$ . What will be the angular acceleration when radius of the sphere is doubled and torque applied is also doubled ?

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$2 rad / s^{2}$

$0.5 rad / s^{2}$

$4 rad / s^{2}$

$0.25 rad / s^{2}$

answer is B.

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Detailed Solution

$τ = Iα = (\frac{2}{5} {mr}^{2}) α = \frac{2}{5} . (\frac{4}{3} . {πr}^{3} ρ) r^{2} . α$

$\Rightarrow α = (\frac{15}{8 π ρ}) . \frac{τ}{r^{5}}$

$∴ \frac{α_{2}}{α_{1}} = (\frac{τ_{2}}{τ_{1}}) {(\frac{r_{1}}{r_{2}})}^{5} = (2) {(\frac{1}{2})}^{5} = \frac{1}{16}$

$\Rightarrow α_{2} = 8 x \frac{1}{16} rad / s^{2} = \frac{1}{2} rad / s^{2}$

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