A particle of mass m starts from rest and slides on a polished track around a vertical circular loop of radius a as shown. Show that the minimum starting height h above the bottom of the loop in order that the particle will leave the track at any point is h=5a/2.

at top of loop centripetal acceleration = g or it leaves the track

g = v^2/a

height at top of loop = 2 a
potential energy at top of loop = m g h = 2 m g a
total energy at top of loop = .5 m v^2 + 2 m g a
= .5 m g a + 2 m g a
= 2.5 m g a
so
h = 2.5 a

To determine the minimum starting height h for the particle to leave the track at any point, we can use the conservation of energy principle. The particle's total mechanical energy is conserved throughout its motion.

At the bottom of the loop (Point A), when the particle just leaves the track, its total mechanical energy is equal to zero. This means that at this point, the particle has used up all of its initial potential energy and converted it into kinetic energy. We will consider this as our reference point for potential energy.

Let's analyze the various positions of the particle on the track:

1. At the top of the loop (Point B): Here, the particle has maximum potential energy and minimum kinetic energy. The total mechanical energy is equal to its potential energy. The potential energy at this point is given by mgh, where h is the height from Point A to Point B.

2. At any other point on the track, between Point A and Point B: The total mechanical energy is a combination of potential and kinetic energy. The potential energy at any point on the track is mgh, where h is the height above Point A.

To determine the minimum starting height h required for the particle to leave the track at any point, we need to find the maximum potential energy at Point B that is just enough for the particle to complete the loop without falling off.

Considering the particle at the top of the loop (Point B), its total mechanical energy is equal to its potential energy:

mgh = 0

This means that the minimum potential energy required for the particle to leave the track is zero.

Since we know the potential energy at Point B, we can calculate the height h:

mgh = mgh
mgh = mga

Simplifying the equation:

h = a

So, the minimum starting height h required for the particle to leave the track at any point is h = a.

However, the question asks for the minimum starting height in terms of the radius (a). To find that specific value, we use the fact that the bottom of the loop (Point A) has higher potential energy than Point B:

mgh > mga

Substituting h with the required value:

(mga) > mga
5ga/2 > ga

Therefore, the minimum starting height h required for the particle to leave the track at any point is h = 5a/2.