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(A) For translational motion of the center of mass $(C.M.)$:$F + f = ma$  (where $f$ is the friction force)For rotational motion about the $C.M.$:$R(F

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thin pipe

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(A) For translational motion of the center of mass $(C.M.)$:$F + f = ma$  (where $f$ is the friction force)For rotational motion about the $C.M.$:$R(F - f) = I\alpha$For pure rolling,the condition is $a = R\alpha$,which implies $\alpha = a/R$.Substituting $\alpha$ in the rotational equation:$R(F - f) = I(a/R) \implies F - f = \frac{I}{R^2} a$From the translational equation,$a = \frac{F + f}{m}$. Substituting this:$F - f = \frac{I}{R^2} \left( \frac{F + f}{m} \right)$$mR^2(F - f) = I(F + f)$$F(mR^2 - I) = f(mR^2 + I)$$f = F \frac{mR^2 - I}{mR^2 + I}$For pure rolling to occur on a smooth surface without needing external friction,we require $f = 0$.This implies $mR^2 - I = 0$,or $I = mR^2$.Among the given options,a thin pipe (or ring) has a moment of inertia $I = mR^2$ about its central axis.

$A$ body kept on a smooth horizontal surface is pulled by a constant horizontal force applied at the top point of the body. If the body rolls purely on the surface,its shape can be:

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