E1.5 Conservation of Energy:

Energy is always conserved - the total amount of energy presents stays the same before and after any changes.

In physics, the term conservation refers to something which doesn't change. This means that the variable in an equation which represents a conserved quantity is constant over time. It has the same value both before and after an event.

The principle of conservation of energy states that energy cannot be created or destroyed (i.e. in an isolated system), the total energy before transformation is equal to the total energy transformation. A few examples are as follows:

1) In an electric bulb, electrical energy is converted into light and heat energy.

2) In a hydroelectric plant, water falls from a height on to a turbine causing it to turn. The turbine turns a coil in a magnetic field, thereby generating a electric current. Therefore, potential energy of the water is converted into kinetic energy of the turbine, which is converted into electrical energy.

Source

3) The pendulum shows the principle of conservation of energy in action. The diagram shows a pendulum in three position - two is it's swing position at ends and another as it passes through the middle point. Here, the gravitational potential energy is converted to kinetic energy and back, over and over again, as the pendulum swings.

» Energy Conservation for a Freely Falling Body:

The mechanical energy (total energy) of a freely falling body under the gravity is constant.

At position A, Consider a body of mass $m$, initially at rest ($v_A = u = 0$) at the height $h$ from the ground.

Kinetic Energy of a body =  $\frac{1}{2}\,mv^2 = 0$

Potential Energy of a body = $m\,g\,h$

Total Energy of a body at A = $K.E + P.E$

$ = 0 + m\,g\,h = m\,g\,h $ .......... (i)

At position B, The body falls freely from $A$ to $B$ through a distance $x$ from A. Therefore height of the point $B$ is ($h-x$).

If $v_B$ is the velocity at $B$, then from the kinematics: 

$V{_B}^2 = u^2 + 2\,a\,s = 0 + 2\,g\,x = 2\,g\,x$

Kinetic Energy of a body = $\frac{1}{2}\,mv{_B}^2 = \frac{1}{2}\,m * 2\,g\,x = m\,g\,x$

Potential Energy of a body = $m\,g\,(h-x)$

Total Energy of a body at B = $K.E + P.E$

$ = m\,g\,x\, + m\,g\,(h-x) = m\,g\,h $ .......... (ii)

At position C, If $v_C$ be the velocity of the body at point $C$ (just before striking the ground). Then from the kinematics

$V{_C}^2 = u^2 + 2\,a\,s = 0 + 2\,g\,h = 2\,g\,h$

Kinetic Energy of a body = $\frac{1}{2}\,mv{_C}^2 = \frac{1}{2}\,m * 2\,g\,h = m\,g\,h$

Potential energy of a body = $m\,g\,h = 0$

Total Energy of a body at C = $K.E + P.E$

$ = m\,g\,h\, + 0 = m\,g\,h $ .......... (iii)

From the equation (i), (ii) and (iii), we conclude that:

» The Total Energy (Mechanical Energy) of a body remains the same at all points $A$, $B$ and $C$ i.e. $m\,g\,h$.

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