LEARNING OBJECTIVE : 3.5.A Power is the rate at which energy changes with respect to time, either by transfer into or out of a system or by conversion from one type to another within a system.
We introduced the meaning and basic definition of power in the last section. In this section we will build on the basic definition and arrive at different equations to express and calculate power.
Average Power
To start with we will first look at Average power. It is simply total work done divided by time it took to do the work. Mathematically,
Average Power = <P> = Total work done / Duration in which this work is done
We also know that work and energy is equivalent, hence we can also write:
Average Power = <P> = Total energy supplied or consumed / Duration in which this energy is supplied or consumed
If a machine supplies energy (generator) it is delivering power if an equipment consumes energy (heater) it is consuming power.
While calculating average power we are not worried about how the power delivery changed every second, we simply take the total work done and divide it by total time. Similar to how we calculate average speed.
For example, if the work done in one second is 100 Joules then the average power delivered is 100 Watts. If the work done in two seconds is 100 joules then the average power delivered is 100/2 = 50 Watts.
Instantaneous Power
In situations where the force changes with time F(t) or location F(x), in such cases work done by force becomes variable, which means it changes every moment. In such cases if we want to define power, we need to define it for a particular instant or instantaneous power.
Let the fore vary with time as F(t)or F(x) then the work done by the force between time t to dt is dW then the instantaneous power is:
P = dW/dt
Or simply rate of doing work.
Relating power with velocity
If a machine exerts a force F and the point of application of this force is displaced during a time t over a distance s in the direction of the force, the work done by the machine is
A = Fs.
The power developed by this machine is P = Fs/ t. Since s / t is the velocity V of motion of the point of application of the force, the power developed by the machine is
P = FV
In other words, if the directions of velocity and the force coincide, the power developed by a machine is equal to the force exerted by this machine multiplied by the velocity of the point of application of the force.
If the velocity is directed against the force, the work done and the power developed are negative: the machine consumes power. For example, if a lift raises a load of mass 400 kg at a constant velocity of 2 m/s, the motor of the lift develops a power P = 4000 N (Force mg) x 2 m/s (Velocity) = 8 kW.
