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Conservation of Energy

Theory

Principle of Conservation of Energy

According to the principle of conservation of energy, energy can neither be created nor can it be destroyed. It only changes from one form to another.

  • The sum of all forms of energy in the universe remains constant.
  • When there is a transformation of energy from one form to another, the total energy always remains the same, i.e. it remains conserved.
  • If there is only an interchange between potential energy and kinetic energy, the total mechanical energy (the sum of kinetic energy KK and potential energy UU) remains constant, i.e. K+UK + U = constant when there are no frictional forces.

Mechanical energy

Gravitational potential energy is the energy of position; kinetic energy is the energy of motion. Their sum is the mechanical energy of the body.

U=mghU = mgh
K=12mv2K = \tfrac{1}{2} m v^2
E=K+UE = K + U
K+U=ConstantK + U = \text{Constant}

Let a body of mass mm fall freely under gravity from a height hh above the ground (position A). As it falls, its potential energy changes into kinetic energy, and at every point the sum of the two remains unchanged.

Position A — at height h above the ground

Initial velocity of the body is zero, since the body is at rest at A.

u=0u = 0
K=0K = 0
U=mghU = mgh
K+U=0+mgh=mgh…(i)K + U = 0 + mgh = mgh \qquad \dots (i)

Position B — after falling a distance x

Let v1v_1 be the velocity at B. Here u=0u = 0, S=xS = x, a=ga = g.

v1 2=u2+2aS=0+2gx=2gxv_1^{\,2} = u^2 + 2aS = 0 + 2gx = 2gx
K=12mv1 2=12m(2gx)=mgxK = \tfrac{1}{2} m v_1^{\,2} = \tfrac{1}{2} m (2gx) = mgx

The height of the body above the ground at B is h−xh - x.

U=mg(h−x)U = mg(h-x)
K+U=mgx+mg(h−x)=mgh…(ii)K + U = mgx + mg(h-x) = mgh \qquad \dots (ii)

Position C — on the ground

Let vv be the velocity on reaching the ground, with u=0u = 0, S=hS = h, a=ga = g.

v2=u2+2aS=2ghv^2 = u^2 + 2aS = 2gh
K=12mv2=12m(2gh)=mghK = \tfrac{1}{2} m v^2 = \tfrac{1}{2} m (2gh) = mgh
U=0(h=0 at the ground)U = 0 \quad (h = 0 \text{ at the ground})
K+U=mgh+0=mgh…(iii)K + U = mgh + 0 = mgh \qquad \dots (iii)

From equations (i), (ii) and (iii), the total mechanical energy remains constant at each point of motion and equals the initial potential energy at height hh. As the body falls its potential energy decreases and its kinetic energy increases; just as it strikes the ground the whole of the potential energy has changed into kinetic energy.