* 14-80 . When s=0, the spring on the firing mechanism is...

$* 14-80 . \quad$ 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 speed of the 0.3 -kg ball and the normal reaction of the circular track on the ball when $\theta=60^{\circ} .$ Assume all surfaces of contact to be smooth. Neglect the mass of the spring and the size of the ball.

Key Concepts

Conservation of Energy

This principle states that in an isolated system, the total mechanical energy (sum of potential and kinetic energies) remains constant when no non-conservative forces (like friction) are doing work. It is applied to determine the speed of an object by equating the stored energy (such as elastic potential energy) to the kinetic and other forms of energy as the system evolves.

Spring Potential Energy

This concept refers to the energy stored in a spring when it is deformed from its natural, unstretched (or uncompressed) length. The energy is typically proportional to the square of the displacement from equilibrium, and when the spring is released, this stored energy can be converted to kinetic energy, which is vital in systems involving launching mechanisms or energy transfer.

Circular Motion and Centripetal Force

For an object moving along a circular path, a net inward (centripetal) force is necessary to keep it in circular motion. This force causes the object’s acceleration toward the center of the circle and is determined by the object's mass, speed, and the radius of the path. Understanding centripetal force is essential when analyzing how forces, such as the normal reaction from a track, contribute to the motion of objects on curved paths.

Force Analysis in Curved Motion

This concept involves drawing and analyzing a free-body diagram to identify forces acting on an object moving along a curved path. By resolving forces into their radial and tangential components, one can apply Newton's second law in the radial direction to relate the net force to the required centripetal acceleration, thereby enabling the calculation of unknown forces such as the normal reaction.

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