Question

A horizontal force $F$ is applied at the center of mass of a cylindrical object of mass $m$ and radius $R$, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$. The center of mass of the object has an acceleration $a$. The acceleration due to
gravity is $g$. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

• A. For the same $F$, the value of $a$ does not depend on whether the cylinder is solid or hollow.
• B. For a solid cylinder, the maximum possible value of $a$ is $2\mu g$
• C. The magnitude of the frictional force on the object due to the ground is always $\mu mg$
• D. For a thin-walled hollow cylinder, $a=\frac{F}{2m}$

Answer 1

Answer: (B), (D)

For a thin-walled hollow cylinder, $a=\frac{F}{2m}$

Given situation can be represented as

$F-f=m{a}_C$
and, $fR=I_C\alpha$ where, $R$ is the radius of cylinder.
also, $a_{cm}=R\alpha$
$F=\frac{I_C{a}_C}{R^2}+m{a}_C$
$\Rightarrow a_C=\frac{F}{\frac{I_C}{R^2}+m}$
$fR=\frac{IF}{(\frac{I_C}{R^2}+m)R}$
$f=\frac{I_{C}F}{I_C+m{R}^2}$
depends on $F$
For thin walled cylinder $I_C=m{R}^2,a_C=\frac{F}{2m}$
$f=\frac{I_C{a}_C}{R^2}\le \mu mg$
$a_C\le \frac{\mu mg{R}^2}{I_C}$
For solid cylinder
$I_C=\frac{m{R}^2}{2}$
$a_C\le 2\mu g\Rightarrow (a_C{)}_{max}=2\mu g$