Power and Efficiency

In these revision notes for Power and Efficiency, we cover the following key points:

• What is mechanical power and why it is so important in physics?
• What is the difference between output and input power?
• What does wasted energy mean?
• What is efficiency of a machine and how can we calculate it?
• How is mechanical power calculated in terms of force and velocity?

Power and Efficiency Revision Notes

Sometimes it is not enough to do a certain work. The time in which this work is done has also its importance in many cases. Also, it would be good if we are able to do the greatest amount of work possible in a given time.

By definition,

"Power is the amount of work done by a system in the unit of time."

The symbol of power is P. Power is a scalar quantity as both work and time are scalars. The equation of power is

Since work is measured in joules [J] and time in seconds [s], the unit of power is [J/s]. However, the unit of power is often referred as Watt [W]. Therefore,

It is quite impossible that the entire energy produced by a source converts into work done by the system. Therefore, the power we calculate through the above equation represents the output power, or the power that goes for doing work. Output power is often referred as the useful power because only this part is useful for people who manage the system.

On the other hand, if we divide by time the total energy produced by the source, we obtain the total or input power. If we know how to calculate the total energy (or if the value of total energy is given), we calculate the total (input) power using the equation

Theoretically, output and input power can be equal only when the entire energy produced is converted into work. However, this is practically impossible as some of the energy produced turns into other undesirable forms. As a result, we have

and

The energy lost during the process, is often referred as Wasted Energy (E_w). Therefore, it is obvious that in a system,

There is a very simple physical quantity, which allows us to measure numerically the quality of a given machine. It is called "Efficiency" (in short, e) and is calculated by the following formulae:

Or

Efficiency is a dimensionless quantity that is calculated as a percentage.

Another Formula for Calculating the Mechanical Power is

where v⃗ is the velocity of the object in the direction of force F⃗. In this way, we obtain another definition for the mechanical power:

"Power is the scalar product of the force applied by a source and the velocity this object gains due to the action of the force."

Constructors of electrical appliances often write the power of appliance (in watts) on its back or lateral side. This value represents only the output (or useful) power, not the input (or total) one.