Power

Many times, when the work is being done, we are not only interested in the amount of work done but also in the time in which it is being done. In such cases, we use the term power. In physics, it is defined as the rate at which energy is converted or transferred over time. It's important to note that power is a scalar quantity, meaning it only has magnitude and no direction.

1.0What is Power?

Power, as a physical quantity, indeed measures the rate of energy transfer. It is defined as the rate at which work is done with respect to time. In other words, power represents the amount of energy consumed or transferred per unit of time. 

Mathematically, it is expressed as the ratio of work (W) done to the time taken (t) to do that work.

Therefore, it is the rate of energy transfer or work done with respect to time, and the symbol P denotes it.

2.0Average Power

If a force performs a certain amount of work within a given time interval, we can compute the average power generated by the force. It is a scalar quantity.

$P_{avg}=\frac{W}{\Delta t}or{P}_{avg}=\frac{\Delta E}{\Delta t}$

3.0Instantaneous power

Instantaneous power refers to the power consumed or generated at a specific moment in time. It is given by the scalar or dot product of force and velocity.

$P=\frac{dW}{dt}$

or

$P=\vec{F}.\vec{v}$

Instantaneous power = slope of work-time curve = tanθ

When the rate of work done by a body is uniform or constant, the average power and the instantaneous power are inherently equal. This holds true because the average power over any interval of time is calculated by dividing the total work done during that interval by the duration of the interval. Since the rate of work is constant, this average power value remains constant throughout the interval. Consequently, at any moment within that interval, the instantaneous power – which represents the power at a specific instant in time – is also equal to the constant average power.

So, we can state that when the rate of work done by a body is uniform or constant, the average power and the instantaneous power are indeed equal throughout that period.

4.0Unit of Power

The SI unit of power is the joule per second, also known as the watt, named after James Watt.

Another unit of power is Horsepower.

1 Horsepower = 746 watts

(for motors and engines, power is usually measured in Horsepower)

5.0Power Formula

For average power,

$P_{avg}=\frac{W}{\Delta t}$

$P_{avg}=\frac{\Delta E}{\Delta t}$

For instantaneous power,

$P=\frac{dW}{dt}$

or

$P=\vec{F}.\vec{v}$

6.0Power Efficiency

Machines are designed to convert energy into useful work; however, because of frictional effects and other dissipative forces, work performed by the machine is always less than the energy supplied to the machine. Thus, we define the efficiency of a machine, which denotes how effective a machine is in converting energy into useful work. 

The efficiency of a machine is given by

$\eta =\frac{workdone}{eneryinput}$

7.0Important Points on Power

Certainly, here are the important points about power summarized based on the concepts of work and energy:

(a)  Rate of Energy Usage: Power represents the rate at which energy is being utilized or transferred. This means that it quantifies how quickly energy is being consumed or transformed into other forms.

(b)  Determinant of Work Rate: Power determines how rapidly work can be accomplished. Higher power implies that work can be done more quickly, while lower power indicates a slower rate of work completion.

(c) Measurement of Power: Power can vary over time, so there are two important ways to measure it:

• Instantaneous Power: This refers to the power at any given moment in time. It captures the power output at a specific instant, which may fluctuate over time depending on the task or process.
• Average Power: The average power is calculated by dividing the total energy utilized or work done by the total time taken. It provides an overall measure of power over a certain duration, taking into account variations in instantaneous power.

These points encapsulate the essential aspects of power, its significance in work and energy contexts, and the methods used to measure it, both instantaneously and over an extended period.

8.0Solved Examples

Example:

A girl lifts a box of 5 kg up to a height of 20 m for 10 seconds. Calculate the power delivered to the box.

Solution:

Work done by the girl, W = F d = 50 N × 20 m = 1000 J

Power delivered, P = W ⁄ t = 1000 / (10s) = 100 J/s

Hence, the power delivered to the box is 100 J/s.

Example:

A motor is used to power a lift that raises a load of bricks weighing 800N to a height of 20m in 40s. What is the minimum power motor needed?

Solution:

Assuming that the bricks are lifted without acceleration, the upward force is equal to the force of gravity 800N. The speed of the bricks is = $\frac{20m}{40s}$ = 0.5 m/s.

Now, power, P = Fv = (800N)(0.5 m/s) = 400 W

If there are no energy losses, e.g. to frictional forces, the motor must have a power output of 400W.

Example:

A pump can take out water at the rate of 7200 kg/hr from a 100 m deep well. Calculate the power of the pump, assuming that its efficiency is 50%. (g = 10 m/s²)

Solution:

Output power = $\frac{\mathrm{mgh}}{\mathrm{t}}=\frac{7200\times 10\times 100}{3600}=2000\mathrm{W}Efficiency\eta =\frac{\text{ output power }}{\text{ input power }}$ 

Input power = $\frac{\text{ output power }}{\eta }=\frac{2000\times 100}{50}=4\mathrm{kW}$