Answer to Question #274495 in Physics for Kim

Question

A block of mass 500 g presses a spring with distance X, as shown in figure. The spring is negligible mass and has spring constant 4500 mN/m. When it is released, the block travels along a frictionless horizontal surface to point A, the bottom of frictionless circular track of radius R = 2 m and continue to move up the track. The block’s speed at point A is 12 m/s. What is X? and What is minimum X to complete the circular motion?

Answer 1

If the block’s speed at point A is 12 m/s, the compression is

$$X=v\sqrt{m/k}=4\text{ m}.$$

The minimum X to complete the circular motion can be found from the minimum value of velocity to complete that motion:

$$mg=m\frac {v^2}R,\\\space\\ v=\sqrt{gR}.$$

Substitute this into the equation for X:

$$X=v\sqrt{m/k}=\sqrt{gmR/k}=1.5\text{ m}.$$