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

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(D) 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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None of the above.

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(D) 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 sphere. Since the sphere does not slip,the acceleration of the point of contact on the sphere relative to the plank must be zero. The friction $f$ acts on the sphere in the forward direction and on the plank in the backward direction. 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 = \alpha R$,so $f = \frac{2}{5} m a_s$. This implies $a_s = \frac{f}{m}$. The acceleration of the plank is $a_p = \frac{F - f}{M}$. Since $a_s < a_p$ is generally true for such systems,statement $A$ is correct. The work done by friction on the sphere is $W_f = \Delta K_{sphere} = K_{trans} + K_{rot} = \frac{1}{2} m v_s^2 + \frac{1}{2} I \omega^2$. Since $v_s = \omega R$,$W_f = \frac{1}{2} m v_s^2 + \frac{1}{2} (\frac{2}{5} m R^2) (\frac{v_s}{R})^2 = \frac{1}{2} m v_s^2 + \frac{1}{5} m v_s^2 = \frac{7}{10} m v_s^2$,which is the total kinetic energy of the sphere. Thus,statement $B$ is correct. According to the work-energy theorem,the total work done by all forces on the system (plank + sphere) equals the change in total kinetic energy. The only external force doing work is $F$. Friction is an internal force,so its net work on the system is zero. Thus,$W_F = \Delta K_{total}$. Statement $C$ is also correct. Since all statements $A, B,$ and $C$ are correct,the incorrect statement is none of these.

$A$ 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 statements is incorrect?

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