A small squirrel is put into a circular wheel cage of radius R which has a frictionless central pivot. A horizontal platform of length  $\frac{6R}{5}$ is fixed to the wheel below the pivot as shown. Initially squirrel is at rest at one end of the platform. When the platform is released squirrel starts running but platform and wheel remain stationary. Choose the correct options.

Consider squirrel at distance   from midpoint C of platform.
 
 F.B.D. of squirrel
 $$\begin{gathered} f=ma \mathrm{....}\left(\text{i}\right) \\ N=mg \mathrm{....}\left(\text{ii}\right) \end{gathered}$$
F.B.D. of platform + Wheel cage 
Platform + wheel cage remain at equilibrium only if  ${\tau }_{pivot}=0$, which is possible according to the following F.B.D. only
     [From equation (i) and (ii)]
$\Rightarrow fh=Nx\Rightarrow mah=mgx$

  towards C (restoring acceleration)

$\therefore$   Squirrel performs SHM about mean position C, angular frequency  $\omega =\sqrt{\frac{g}{h}}$ and amplitude  $A=\frac{3R}{5}$ (Half the platform’s length)
In SHM, velocity at extreme should be zero.
Squirrel is at rest at one end of the platform at t = 0.
Maximum velocity  $=A\omega =\frac{3R}{5}\sqrt{\frac{g}{h}}$
 $v_{\max }=\frac{3R}{5}\sqrt{\frac{\frac{g}{4R}}{5}}\Rightarrow v_{\max }=\sqrt{\frac{9Rg}{25}.\frac{5}{4}}\Rightarrow v_{\max }=\sqrt{\frac{9Rg}{20}}$
Maximum acceleration  $={\omega }^{2}A$
$a_{\max }=\frac{gA}{h}=\frac{g\frac{3R}{5}}{\frac{4R}{5}}=\frac{3g}{4}$