A plank with a uniform sphere placed on it re | vedclass

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(A) Let $M$ be the mass of the plank and $m$ be the mass of the sphere. Let $a_p$ be the acceleration of the plank and $a_s$ be the acceleration of th

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The acceleration of the centre of the sphere is less than that of the plank.

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(A) Let $M$ be the mass of the plank and $m$ be the mass of the sphere. Let $a_p$ be the acceleration of the plank and $a_s$ be the acceleration of the centre of the sphere.When the plank is pulled to the right with force $F$,friction $f$ acts on the sphere in the forward direction (to the right) to provide the necessary torque for rolling.For the sphere: $f = m a_s$ and $	au = f R = I \alpha = (\frac{2}{5} m R^2) \alpha$. Since there is no slipping,$a_s = R \alpha$,so $f R = \frac{2}{5} m R^2 (\frac{a_s}{R}) = \frac{2}{5} m a_s R$,which implies $f = \frac{2}{5} m a_s$.However,the friction force $f$ acting on the sphere is $m a_s$. This is only possible if the sphere is accelerating. The plank experiences a backward friction force $f$. The equation for the plank is $F - f = M a_p$.Since $f = m a_s$,we have $a_s = \frac{f}{m}$. Since $f = \frac{2}{5} m a_s$ is the condition for pure rolling,the friction force $f$ must be $\frac{2}{7} F \frac{m}{M+m}$ (approx). Comparing $a_s$ and $a_p$,we find $a_s < a_p$ because the sphere is being accelerated only by the friction force,while the plank is being accelerated by the external force $F$ minus the friction force.

$A$ plank with a uniform sphere placed on it rests on a smooth horizontal plane. The plank is pulled to the right by a constant force $F$. If the sphere does not slip over the plank,which of the following is true?

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