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Systems of Particles and Rotational Motion — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsSystems of Particles and Rotational Motion3 Marks
Question
Equal torqueses are applied on a cylinder and a sphere. Both have same mass and radius. The cylinder rotates about its axis and the sphere rotates about one of its diameters. Which will acquire greater speed? Explain why.
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Answer

We know, $\tau=\text{l}\alpha\text{ or }\alpha=\frac{\tau}{\text{l}}$ 
$\therefore$ Angular acceleration produced in the rylinder is $\alpha_\text{c}=\frac{\tau}{\text{I}_\text{c}}$
Similarly, acceleration produced in the sphere is
$\alpha_\text{s}=\frac{\tau}{\text{I}_\text{s}}$
$\therefore\frac{\alpha_\text{c}}{\alpha_\text{s}}=\frac{\text{I}_\text{s}}{\text{I}_\text{c}}$
Now $\text{I}_\text{s}=\frac{2}{3}\text{MR}^2$ and $\text{I}_\text{}C=\frac{1}{2}\text{MR}^2$
$\therefore\frac{\alpha_\text{c}}{\alpha_\text{s}}=\frac{4}{3}$
or $\alpha_\text{c}=\frac{4}{3}\alpha_\text{s}\text{ and }\alpha_\text{c}>\alpha_\text{s}.$
Thus cylinder will acquire greater speed than that of the sphere.

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