Question
A hollow sphere of radius R lies on a smooth horizontal surface. It is pulled by a horizontal force acting tangentially from the highest point. Find the distance travelled by the sphere during the time it makes one full rotation.
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Answer
Taking moment about the centre of hollow sphere we will get
$\text{F}\times\text{R}=\frac{2}{3}\text{MR}^2\alpha$
$\Rightarrow\alpha=\frac{3\text{F}}{2\text{MR}}$
Again, $2\pi=\Big(\frac{1}{2}\Big)\alpha\text{t}^2$ $\Big(\text{From}\ \theta=\omega_0\text{t}+\Big(\frac{1}{2}\Big)\alpha\text{t}^2\Big)$
$\Rightarrow\text{t}^2=\frac{8\pi\text{MR}}{\text{3F}}$
$\Rightarrow\text{a}_{\text{c}}=\frac{\text{F}}{\text{m}}$
$\Rightarrow\text{X}=\Big(\frac{1}{2}\Big)\text{a}_{\text{c}}\text{t}^2=\Big(\frac{1}{2}\Big)=\frac{4\pi\text{R}}{3}$
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