Energy in Circular Motion and Loops

When an object travels along a vertical circular path, energy conservation relates speeds at different heights without tracking forces continuously. Since mechanical energy is conserved when only gravity and the normal force act, we write:

This equation allows us to solve for the speed at any point if we know the speed at another point.

For a vertical loop of radius $R$, choosing the bottom as reference height ($h = 0$), the top of the loop is at height $h = 2R$. Energy conservation then relates the speed at the bottom $v_b$ to the speed at the top $v_t$:

Finding the Minimum Speed at the Top

From our study of vertical circular motion, we know that at the top of the loop, the minimum speed occurs when the normal force becomes zero and gravity alone provides the centripetal force. At this critical condition:

Solving for the minimum speed at the top: