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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