When s=0, the spring on the firing mechanism is unstretched....

When $s=0$, the spring on the firing mechanism is unstretched. If the arm is pulled back such that $s=100 \mathrm{mm}$ and released, determine the maximum angle $\theta$ the ball will travel without leaving the circular track. Assume all surfaces of contact to be smooth. Neglect the mass of the spring and the size of the ball.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is associated with an object's position in a gravitational field and changes as the object gains or loses height. In the analysis of the ball's motion on the circular track, variations in gravitational potential energy are taken into account as the ball rises to determine how the energy budget is partitioned between kinetic and potential forms.

Centripetal Force and Critical Motion Conditions

When an object moves in a circular path, a centripetal force is required to maintain its circular trajectory. The critical condition, where the normal force becomes zero, defines the limit at which the ball can continue along the track without losing contact. This concept is essential in determining the maximum angle of travel along the circular path.

Conservation of Energy

This principle states that in the absence of non-conservative forces, the total mechanical energy of a system remains constant. In the context of the problem, the energy stored in the stretched spring converts into kinetic energy and gravitational potential energy as the ball moves along its circular path, enabling the analysis of the ball's motion.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is either compressed or stretched from its natural (unstressed) length, given quantitatively by the formula (1/2) k s². In this problem, the initial energy stored by pulling the mechanism provides the energy input for the motion of the ball.

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