The 05lb ball is shot from the spring device shown Determine

The 0.5-lb ball is shot from the spring device shown Determine the smallest stiffness k which is required to shoot the ball a maximum distance of 30 in. up the smooth plane after the spring is pushed back 3 in. and the ball it released from rest. The four cords C and plate P keep the spring compressed 2 in. when no load is on the plate.

Solution

The daturn is set at the lowest point

= 30/12 * sin 30

= 1.25 ft

GPE = 0.5 * 1.25 = 0.625 ft.lb

The initial elastic potential energy = 1/2 * k * [2 + 3 / 12]² = 0.0868 k

The final elastic potential energy = 1/2 * k * [2 / 12]² = 0.0139 k

From conservation of energy

0 + 0.0868 k = 0 + 0.625 + 0.0139 k

0.0729 k = 0.625

k = 8.57 lb/ft