Amarble of mass m initially at rest at height h moves along a frictionless rail that forms a hoop of radius r. at the end of the rail the marble hits a spring of elastic constant k. calculate:

a. the minimum value of h for which the marble hits the spring.
b. the maximum compression of the spring for that value of h.

a. The minimum value for h is 2R.

b. The maximum compression of the spring is .

Explanation:

a. If the marble is initially at rest, then its total mechanical energy is equal to its potential energy at that point, mgh. The rail forms a hoop of radius R. In order for the marble to hit the spring, it must reach the top of the hoop, otherwise it falls down.

The height of top of the hoop is 2R. So, in order for the marble to reach the top of the hoop, the height h must be greater than or equal to height of the hoop, 2R.

In this case, the minimum value for h is 2R.

b. Since the system is frictionless, the compression of the spring depends on the initial potential energy.

the order for strong to weak forces such as

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force

the fundamental forces are those forces that act between to objects or element to make interactions. there are four types of fundamental forces such as strong and weak nuclear forces (due to protons and electrons), electromagnetic force (due to the motion of electrons) and gravitational force (due to orbital movement of the planet). if these forces are arranged on the basis of their maximum strength then the order is

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force