Solution

Suppose that the speed of the body at its lowest point is $v_2$ and the speed of the support moving in the opposite direction is $v_1$. Then, by conservation of momentum $m_1v_1=m_2v_2$, because both objects are initially at rest. From conservation of energy, the gravitational potential energy of the body transforms into kinetic energy of both objects, so

$$\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2=m_2gR.$$

Solving the two equations from conservation laws, we have

$$v_1=\frac{m_2}{m_1}\cdot\sqrt{\frac{2m_1gR}{m_1+m_2}},\qquad v_2=\sqrt{\frac{2m_1gR}{m_1+m_2}}.$$

The body moves along a circular path relative to the support, and its relative speed at its lowest point is

$$v=v_1+v_2=\sqrt{\frac{2(m_1+m_2)gR}{m_1}}.$$

At this point, the centripetal force is $N-m_2g$, where $N$ is the normal force on the body from the support (which has the same magnitude as the force that the body exerts on the support). Consequently,

$$N=m_2g+\frac{m_2v^2}{R}=\frac{m_2(3m_1+2m_2)g}{m_1}.$$

Note that while the support is a non-inertial frame of reference, the pseudo-force in this frame is horizontal and does not affect $N$.