A small squirrel is put into a circular wheel cage of radius R which has a frictionless central pivot. A horizontal platform of length 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. The maximum speed of squirrel is $\sqrt{nRg}$ . Find n?

Consider squirrel at distance x from midpoint C of platform.

F.B.D. of squirrel :
f = ma  ………………… (1)
N = mg  ………………….(2)

F.B.D. of platform + Wheel cage.
Platform + wheel cage remain at equilibrium only if
${\tau }_{pivot}=0$  which is possible according to following F.B.D. only.
 $\Rightarrow fh=Nx$
  mah = mgx  (from (1) & (2) )
 $\boxed{a=\frac{gx}{h}}$ towards C (restoring acceleration)

  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  $=A\omega =\frac{3R}{5}\sqrt{\frac{g}{h}}$
 $$\begin{gathered} v_{\max }=\frac{3R}{5}\sqrt{\frac{g}{4R/5}} \\ {V}_{\max }=\sqrt{\frac{9Rg}{25}.\frac{5}{4}} \\ {V}_{\max }=\sqrt{\frac{9Rg}{20}} \end{gathered}$$
 Maximum acceleration   $={\omega }^{2}A$
 $a_{\max }=\frac{gA}{h}=\frac{g3R/5}{4R/5}=\frac{3g}{4}$