Question

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

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

Answer 1

The correct option is D $2\mu g$


Sphere is rotating clockwise, therefore friction will act rightward on the sphere.
$f_k=\mu N=\mu mg...\left(1\right)$
Assume $a_1$ and $a_2$ are the accelerations of the sphere and plank respectively w.r.t ground as shown.

For sphere: $f_k=m{a}_1$
Using Eq. (1) $\mu mg=m{a}_1\Rightarrow a_1=\mu g$

For plank: $-f_k=m{a}_2$
From Eq. (1) $-\mu mg=m{a}_2$
$a_2=-\mu g$
[-ve sign indicates direction is opposite to that assumed]
Acceleration of sphere w.r.t plank $\left(a_{sp}\right)$
$a_{sp}=a_{sg}-a_{pg}$
$=a_1-a_2$
$=\left(\mu g\right)-(-\mu g)$
$=2\mu g$