Question

A small block of mass 2m initially rests at the bottom of a fixed circular , vertical task ,which has a radius of R. The contact surface between the mass and loop is frictionless. A bullet of mass m strikes the block horizontally with initial speed $v_0$ and remain embedded in the block as the block and the bullet circle the loop. Determine each of the following in terms of m, $v_0$, R and g.
a)The speed of the masses immediately after the impact.
b)The minimum initial speed of the bullet if the block and bullet are to successfully execute a complete ride on the loop.

Answer 1

$\left(a\right)$ Using conservation of momentum:

$m{v}_0=(m+2m)v_f{v}_f=\frac{v_0}{3}$ $=$ speed of the block and the bullet

$\left(b\right)v_t=\sqrt{gr}$ $=$ velocity at top.

$\Rightarrow \frac{1}{2}m{v}_b^2=\frac{1}{2}m{v}_t^2+mg\left(2r\right)$ where $v_b$ is velocity at bottom

$\Rightarrow \frac{1}{2}m{v}_b^2=\frac{1}{2}mgr+2mgr\Rightarrow v_b=\sqrt{5gr}m{v}_{min}=3m\sqrt{5gr}\Rightarrow v_{min}=3\sqrt{5gr}$

$=$ minimum velocity