State the law of conservation of mechanical energy. Show that the total mechanical energy of a freely falling body is conserved.
In a conservative force system, the total mechanical energy i.e. the sum of kinetic and potential energy is always constant – this is known as the law of conservation of mechanical energy.
The above law is valid for a freely falling body under gravity.
Let a body of mass m is raised to a vertical height h above the ground and it is kept rest at A. At point A, the energy of the body is entirely potential and it is EA = mgh.
Now, when the body falls vertically downloads, its velocity goes on increasing and when the body is at point B (when it falls a vertical height x from A) let its velocity become VB. From the relation v2 = u2 + 2gh
we get, vB2 = 02 + 2gx ∴ vB2 = 2gx
At point B, the energy of the body is partly potential and partly kinetic.
∴ Total energy at B is EB = mg(h-x) + 1\over 2mvB2
= mg(h-x) + 1\over 2m2gx
= mgh – mgx + mgx
Again, when the body reaches point C (just before hitting the ground) it possesses kinetic energy only and if the velocity of the body at that point C be then v then vc2 = 2gh
∴ Total Energy at C, Ec = 1\over 2 mv2
= 1\over 2m2gh
∴ EA = EB = EC = mgh
i.e. total mechanical energy at all points during the free fall of a body under gravity remains constant. Hence the principle of conservation of energy is valid for a freely falling body.