Work Energy Power

Conservation of Mechanical Energy

If the forces, doing work on a system, are conservative the total mechanical energy of the system is conserved.

This figure shows a ball of mass m being dropped from a height H.

The total mechanical energies E₀, E_h and E_H of the ball at indicated heights are as follows:

`E_H=mg\H`

`E_h=mg\h+1/2mv_h^2`

`E_0=1/2mv_f^2`

As the mechanical energy is conserved, so

`E_H=E_0`

Or, `mg\H=1/2mv_f^2`

Or,`v_f^2=2gH`

Or, `v_f=sqrt(2gH)`

Further,

`E_H=E_h`

So, `v_h^2=2g(H-h)`

The energy is purely potential at height H. At height h it is partially converted to kinetic energy and becomes fully kinetic energy at ground level. This illustrates the conservation of mechanical energy.

Potential Energy of a Spring

This figure shows a block attached to a spring and is resting on a smooth horizontal surface. Other of the spring is attached to a rigid support. Spring is light and can be considered as mass-less. In an ideal spring, the spring force F_s is proportional to x. Where, x is the displacement of block from equilibrium position. The force law for spring is called Hooke’s law and is given by following equation.

`F_s=-kx`

Here, k is constant and is called the spring constant. The unit of spring constant is N m^-1. If k is large then spring is said to be stiff. If k is small then spring is said to be soft.

Let us now pull the block outwards with an extension X_m. In this case, work done by spring force is given by following equation.

Work done by external pulling force F is positive because it overcomes the spring force. Work done in this case is

`W=+(kX_m^2)/2`

Same holds true when spring is compressed with a displacement X_c (<0).

Spring force is position dependent. Work done by spring force depends on initial ad final positions only. So, spring force is a conservative force. When the block and spring system are in equilibrium position, the potential energy of the spring is zero.

The speed and kinetic energy will be maximum at the equilibrium position, X = 0

`1/2mv^2=1/2kx_m^2`

Here, v_m is the maximum speed.

Or, `v_m=sqrt(k/m)x_m`

We have discussed that the total mechanical energy of the system is conserved if forces doing work on it are conservative. If some of the forces are non-conservative, part of the mechanical energy may get transformed into other forms of energy; like heat, light and sound. But the total energy of an isolated system does not change. Energy may be transformed from one form to another but total energy of an isolated system remains constant. Energy can neither be created no destroyed.

Power

The rate of doing work or of transferring energy is called power.

Average Power: The ratio of work W to the total time t is called average power.

`P_av=W/t`

Instantaneous Power: The limiting value of average power as time interval approaches zero is called instantaneous power.

`P=(dW)/(dt)`

The work dW done by a force F for a displacement dr is given as follows:

`dW=F.dr`

So, instantaneous power can also be expressed as follows:

`P=F.(dr)/(dt)=F.v`

Where v is instantaneous velocity when force is F.

Power is a scalar quantity. Its dimensions are [ML²T^-3]. The SI unit of power is Watt (W). 1 W is equal to 1 J s^-1

1 hp = 746 W

1 kWh = 3.6 × 10⁶ J