# Power

### What is Power?

You might’ve observed that wrestlers pick up the heavy mass in very little time because they have the power to perform such an activity.

So, what is power?

Power is the rate of doing an activity or work in the minimum possible time. It is the amount of energy transferred or converted per unit time where large power means a large amount of work or energy.

For example, when a powerful car accelerates speedily, it does a large amount of work which means it exhausts large amounts of fuel in a short time.

• What is power?

• Definition of power

• Power formula

• Define average power

• Define electric power

• Power equation

• Work energy and power

### Definition of Power

Example1: Suppose, person A and B are assigned the task of picking up an equal number of boxes to the top floor of the building.

Let’s say there are 10 boxes of 10 kg each to be picked up by both A and B. Each time they walk a distance of 5 m. Since the work done by both is in the form of potential energy mgh is given by,

W =  mgh = 10 x 10 x 5 = 500 J is the work done by you both.

Suppose, A finishes his task in 50 s and B in 100 s.

As you can see the relationship of Work done per unit time is nothing but the Power.

 So Power ∝ Work done

### Definition of Power

The power is the time taken by you to complete any task or activity.

The power doesn’t remain constant, but how?

Let’s consider Example.1,

Suppose the boy A walks at a pace (high power), he slows down (less power), continues with this speed, takes rest in between (P= 0 as W = 0), then walks with the pace.

Here, when he makes variations in the speed, the work done varies too, at an instance time, the power delivered is different at the different instant.

We can conclude that at different instants, the power (Example.1)  doesn’t remain the same.

So the power delivered in a certain period of time is called instantaneous power.

If the  Δt approaches to zero then power will be instantaneous and given by,

 Pav = $\lim_{\Delta t\rightarrow 0}$ $\frac{\Delta W}{\Delta t}$

ΔW is the work done in a short interval of time Δt (instant time).

Power, the least possible time required by a person or an object to do the work.

## Power Formula

 Power = $\frac{\text{Work Done}}{Time}$

### SI Unit of Power

The S.I. unit of power is Watt.

The multiples of power: KW, MW, GW…

Watt: When a body does work of one joule in one second it is called one-watt power.

 1 watt = 1 J/ s

Another unit of power (In British engineering) is Horsepower (hp).

Where 1hp = 746 W

Dimensional Formula of P: [ $M^{1}$$L^{2}$ $T^{3}$]

When a body does work of 550 foot-pounds per second (746 W) is called its one horsepower.

### Average Power

The ratio of the total amount of work done in the total amount of time is called the average power.

There are certain instruments used to compute average power. If we talk about Fibre optic power instruments, they measure the average power of a continuous light beam that is used to test signal power in fiber-optic networks.

 Pav =  $\frac{\Delta W}{\Delta t}$

Note: If the work is done at a uniform rate, then the average and instantaneous power becomes equal, and the common equation comes out to be,

 P =  $\frac{\Delta W}{\Delta t}$

### Electric Power

Electric power is defined as the rate, per unit time at which energy is transformed from the electrical energy of the moving charges to some other form, e.g. heat, mechanical energy, usually created by electric generators.

Electric generators convert mechanical energy obtained from an external source (the power of motion) into electrical energy.

## Electric Power Formula is Stated as,

 P = V I Where P is the power   V is the potential difference in the circuit and I is the electric current.Other formulas of power are:P = $I^{2}R$ = $\frac{V^{2}}{R}$ (This expression is obtained by Ohms’ law V = IR)R = Resistance

P = $I^{2}R$ states that higher the electrical current (I) the higher the heat generated, and so the higher the power/ energy loss since electrical energy is transformed into heat.

### Work, Energy, and Power

Suppose, you want to displace a body (do some work W) to some distance S by applying your energy (Force F).

The mathematical relationship to describe the above scenario is given by,

W = F S …(1)

P = W/ t…..(2)

Using the above two equations,

P = $\frac{\text{F S}}{t}$

We know that V = $\frac{S}{t}$

So

 P = F. V

This is the relation between power and force.

### SAQs

Problem1: Riya has a mass of 60 kg and runs up to 13 m high in 50 seconds. Compute her power. (Take g = 10 $ms^{2}$)

Solution: Given h = 13 m, m = 60 kg and t = 50 s

P = W/t = mgh /t

= 60 x 10 x 13/ 50

On solving we get,

P = 156 W

Problem2: If the current and voltage of an electric circuit are 3 A and 15 V respectively. Calculate the electrical power.

Solution: Given  I = 3 a and V = 15 V

Since P =VI

=  3 x 15

We get, P = 45 W