A body is allowed to slide down a frictionless track freely under gravity.The track ends in a semicircular shaped part of diameter d.What should be the height(minimum) from which the body must fall so that it completes the circle?

Assuming the "semicircular shape" is like the loop of a roller-coaster, and we want the body to complete the circle.

At the top of the loop, of diameter d (radius d/2), the velocity must be such that the centripetal force exerted by the rails ≥0, or
mv²/(d/2)-mg ≥0

At this position, the kinetic energy is (1/2)mv^2=mgh, where h is the height of release of the body above the top of the loop.
Substitute in above, we find:
2mgh/(d/2)-mg ≥0
or simply
h > d/4.
So the body must be released at a point h> d/4+d, above the lowest part of the loop, or
h > 5d/4

To determine the minimum height from which the body must fall so that it completes the circle, we can consider the conservation of energy.

The initial potential energy of the body at the starting point is given by mgh, where m is the mass of the body, g is the acceleration due to gravity, and h is the height from which the body falls.

When the body reaches the highest point of the circular path, it has zero kinetic energy and maximum potential energy. The potential energy at this point is given by mgh_max.

The conservation of energy allows us to equate the initial potential energy to the maximum potential energy:
mgh = mgh_max

Since the kinetic energy is zero at the highest point, the total mechanical energy (sum of potential and kinetic energy) is equal to the maximum potential energy:
mgh = mgh_max + 0
mgh = mgh_max

From this equation, it is clear that the initial height h must be equal to the maximum height, h_max, from which the body reaches the top of the circular path.

In the circular path, the body experiences a centripetal force that provides the necessary centripetal acceleration for the circular motion. This centripetal force is the gravitational force, given by mg.

At the highest point of the circular path, the force experienced by the body is only the centripetal force, since there is no normal force or gravitational force acting in the vertical direction. Therefore, we have:
mg = mv^2/r

Where v is the velocity of the body at the top of the circular path, and r is the radius of the circular path, which is equal to half the diameter (r = d/2).

From this equation, we can solve for v:
v^2 = rg

To complete the circle, the body must have a minimum velocity at the top of the circular path such that it does not lose contact with the track. This minimum velocity is zero, since the body needs to stop momentarily at the top and change direction.

Substituting v = 0 into the equation above, we get:
(0)^2 = rg
0 = rg

Therefore, the minimum height h from which the body must fall so that it completes the circle is zero. This means that even if the body starts from rest at the bottom of the track, it will still be able to complete the circular path without losing contact with the track.

To determine the minimum height from which the body must fall in order to complete the circle, we need to consider the forces acting on the body at different points along the track.

When the body is at the highest point of the track (the starting point), it has gravitational potential energy. As it slides down the track, this potential energy is converted into kinetic energy.

At the bottom of the track (the starting point of the semicircular part), the body has maximum kinetic energy and zero potential energy. At the topmost point of the semicircle, the body will have maximum potential energy and zero kinetic energy.

Since the track is frictionless, the total mechanical energy (potential energy + kinetic energy) of the body remains constant throughout the motion. Therefore, we can equate the initial potential energy to the final potential energy at the topmost point of the semicircle.

The initial potential energy is given by mgh, where m is the mass of the body, g is the acceleration due to gravity, and h is the height from which the body falls.

The final potential energy at the topmost point of the semicircle is given by 2mgr, where r is the radius of the semicircle (which is half the diameter d). The factor of 2 accounts for the full circle, as the body must complete one full revolution.

Setting the initial potential energy equal to the final potential energy:

mgh = 2mgr

Simplifying the equation:

h = 2r

Therefore, the minimum height from which the body must fall to complete the circle is twice the radius of the semicircular track.

Anything falling from a height of h will rise to a height of h so h = d Period