Question

A mass m rotates in a vertical circle of radius R and has a circular speed $v_c$ at the top. If the radius of the circle is decreased by a factor of 4, derive the difference of velocities at the top of circle in terms of R

Answer 1

The radius of orbit decreases by $R^{\prime }=R/4$

velocity in new orbit is $t$

Apply concentration of energy:

$MgR+\frac{1}{2}M{V}_C^2=Mg\frac{R}{4}+\frac{1}{2}M{V}_C^2$

$\Rightarrow \frac{1}{2}(V_c^2-{V^{\prime }}_c^2)=\frac{3gR}{4}\Rightarrow V_c^2-{V^{\prime }}_c^2=\frac{3gR}{2}$