Power (3.3.3)
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Power is defined as the rate of doing work. In other words, power is the amount of work done per unit of time:
Where:
- is power in watts ,
- is work in joules , and
- is time in seconds .
Another common unit of power is the joules per second
Differentiating between energy, work, and power can be challenging. The table below compares the three quantities and provides an example of each.
Question walkthrough
Calculating a Lift's Work and Power
Uses a lift's constant driving force and floor height to find distance travelled, then applies W = Fd and P = W/t to calculate work done and power output.
Power is the rate of doing work :
When a constant force moves an object at velocity in the same direction as the force, this simplifies to:
Where:
- is the force in newtons , and
- is velocity in metres per second .
This is one of the most widely used mechanical equations for engines, vehicles, and motors.
You should be able to derive from first principles. Firstly, the expression for power and work done , respectively are:
Where:
- is the force,
- is the displacement,
- is time, and
- is the angle between the force and displacement vectors.
Replacing the expression of the work done in :
For a constant velocity :
Hence, the power can be expressed as:
When the displacement and force vectors are pointing in the same direction, the power equation reduces into:
Question walkthrough
Finding Engine Power from Resistive Forces
Applies Newton's first law at constant velocity to equate driving force with resistive force, then uses P = Fv to calculate a car engine's power output in watts and kilowatts.
Efficiency is a term used to describe how well energy is converted in a mechanical system:
- A system with high efficiency converts most of the input energy into useful output energy.
- A system with low efficiency converts most of the input energy into wasted output energy.
Efficiency can be expressed as a decimal between 0 and 1 or as a percentage between 0% and 100%. A system with an efficiency of zero wastes all the input energy, and an efficiency of one (100%) is considered ideal. It converts all the input energy into useful energy. In reality, a system can never be 100% efficient.
Mathematically, the efficiency of a mechanical system is defined as the ratio of the useful output energy to the input energy:
To get the efficiency in percentage form, we apply the ratio below:
It is important to note that effiency can be anything between zero and one or 0% and 100%.
The efficiency can also be described in terms of power. Efficiency can also be defined as the ratio of the output power to the input power. In decimal form, the efficiency is:
And in percentage form, the efficiency is:
Question walkthrough
Calculating Elevator Motor Power and Efficiency
Calculates an elevator motor's output and input energy, its input and output power, and verifies the result matches the stated efficiency.