Question

A sphere of mass m is given some angular velocity about a horizontal axis through its center and gently placed on the plank of mass m. The coefficient of friction between the two is $\mu$. The plank rests on a smooth horizontal surface. The initial acceleration of centre of mass of the sphere relative to the plank will be:

• A. zero
• B. $\mu g$
• C. $\frac{7}{5}\mu g$
• D. $2\mu g$

Answer 1

The correct option is D $2\mu g$

$\tau =I\alpha \to$ on the sphere against a friction which is backlab

Assume coeff. of fric = M

$F_{fric}R=I\alpha$

$\Rightarrow \alpha =\frac{MmgR}{\frac{2}{5}m{R}^2}=\frac{5}{2}\frac{Mg}{R}$

Now the initial acc. of COM of of sphere is,

a sphere = $\alpha R=\frac{5}{2}Mg$

$aplant=\frac{Mmg}{m+m}=\frac{Mg}{2}$

$\therefore$ in ace. of COM of sphere rel. to plant is,

$a_{rel}=a_{sphere}-a_{plant}=\frac{5}{2}Mg-\frac{Mg}{2}=2Mg$

Since we know,

Coefficient of friction = u

$\therefore a_{rel}=2ug$