A Lazlo Holics problem

Relevant wiki: Uniform circular motion - Medium

In the absence of friction, no force acts in the horizontal direction on the cart and ball system. Therefore, their momentum must be conserved along the horizontal direction.

Let the speed of the cart be $v_c$ and the speed of the ball be $v_b$ at the instant the ball reaches the bottommost point of the hemisphere. Thus using the momentum conservation we may write.

$m_{\text{cart}}v_c = m_{\text{ball}}v_b$ $v_b = 4 v_c.$

Again due to no friction on the surface of the hemisphere, we can conserve the mechanical energy, $m_{\text{ball}}g2R = \frac{1}{2}m_{\text{cart}}v_c^2 +\frac{1}{2}m_{\text{ball}}v_b^2$ Substituting the value of $v_b, \, m_{\text{cart}}$ , and $m_{\text{ball}}$ , we get, $v_c =1, v_b = 4 \text{m/s}.$

Now, in the frame of the cart, the path of the ball is circular and its speed equals $v_{bc} = 4+1 =5 \text{m/s}$ .

Applying the equation of centripetal force at the bottommost point of the hemisphere, $N - m_{\text{ball}}g = m_{\text{ball}} \frac{v_{bc}^2}{R}$ Substituting the values, we get, $N = \boxed{30 \text{N}}.$