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प्रश्न
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.
उत्तर
Let M be the mass of the hollow sphere and α be the angular acceleration produced in the sphere by the tangential force F.
Torque due to this force,
\[\tau = F \times R\]
Also, \[\tau = I\alpha\]
\[\text{So, }F \times R = \left( \frac{2}{3} \right)M R^2 \alpha\]
\[ \Rightarrow \alpha = \frac{3F}{2MR}\]
Applying \[\theta = \omega_0 t + \frac{1}{2}\alpha t^2, \] we get
\[2\pi = \frac{1}{2}\alpha t^2 \]
\[ \Rightarrow t^2 = \frac{8\pi MR}{3F}\]
Let d be the distance travelled in this time t.
Acceleration,
\[a = \frac{F}{M}\]
\[\therefore d = \frac{1}{2}a t^2 \]
\[ = \frac{1}{2} \times \frac{F}{M} \times \left( \frac{8\pi MR}{3F} \right)\]
\[ = \frac{4\pi R}{3}\]
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