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Work, power, and energy are the fundamentals of classical physics. Power can be defined as the rate of work done by some force. In simple words, power measures the ability to do work by a force. But it is not true that power is only measured in terms of work done and displacement.
In optics, we also measure power but there the idea changes. In general, when we talk about power, we assume that we are tackling something related to mechanics. So, in these notes on the dimensional formula of power, we will limit our discussion to the mechanical aspect of power only.
Power
In physics, power measures the rate at which work is done, or energy is transferred. An excellent example to understand this is to imagine a rock being pushed in a specific direction; the rate at which the rock is being displaced in the direction of the force applied on it would be its power.
The standard unit used to measure power is the watt (W). The formula for power is P= E/t (Where P stands for power, E stands for energy in joule, and T stands for time in seconds). Power is also measured in horsepower (hp).
Based on an interval of time, power is divided into two types:
- The amount of power used at a specific interval of time is known as the average power (Pavg= W/t).
- The amount of time used at a specific moment is known as instantaneous power (Pinst = dW/dt).
Power formula
Power is the rate at which work is done on the object. It is a quantity based on time, which is related to what time a job is done. The formula for calculating power is as follows:
Power = Work/time i.e. P = W/t
Units:
Power’s dimension is the amount of energy divided by the time taken. Moreover, the SI unit of power is watt ‘W’ in the international system, and watt is equal to 1 joule per second. Some other general measures of power are horsepower (hp), where 1 hp = 745 watts approximately, ergs per second ‘erg/s’, foot-pounds/minute, dBm, etc.
Average power:
Just have a look at this simplified example that says; burning 1 kg of coal produces more energy as compared to detonating 1 kg of TNT (Trinitrotoluene). However, the TNT reaction produces energy more rapidly and delivers much more power as compared to coal.
Derivation of the dimensional formula of power:
From the definition, we know that power = work done/time
Dimension of power = dimension of work done / dimension of time
Dimension of force = dimension of mass x dimension of acceleration = [MLT-2]
Dimension of work done = dimension of force x dimension of length
= [MLT-2] X [L] = [ML2T-2]
So , dimension of power = [ML2T-2]/ [T] = [ML2T-3]
Unit: Unit of power is J/sec = Watt in S.I system. In the C.G.S system, it is measured in erg/sec.
Power is measured in terms of value only. The direction doesn’t play a role here. So, it is a scalar quantity.
Numericals on power
- A person of 60 kg climbs up the stairs 20 metres in height in a timespan of 1.5 minutes. Determine the power.
Known:
Mass ‘m’ = 60 kg
Height ‘h’ = 20 metres
‘g’ = 10 m/s2
Time interval ‘t’ = 1.5 minutes = 1.5 X 60 seconds = 90 seconds
Wanted: Power ‘P’
Solution:
Formula of Power:
P = W/t
P = power,
W = work, and t = time
Formula of Work:
W = mgh
W = work,
F = force,
w = weight,
h = height,
d = displacement,
m = mass,
g = 10ms-2
W = mgh = 60 X 10 X 20 = 12000 Joule.
P = W/t = 12000/90 = 133.33 Joule per second
- Calculate the power needed for a person of 60 kg to climb a tree 5 metres in height in 10 seconds.
Known:
Mass ‘m’ = 60 kg
Height ‘h’ = 5 metres
‘g’ = 10 m/s2
Time interval ‘t’ = 10 seconds
Wanted: Power
Solution:
Work:
W = mgh = 60 X 10 X 5 = 3000 Joule
Power:
P = W/t = 3000/10 = 300 Joule per second.
Conclusion
From the above discussion, it is clear how to define power and deduce its dimensional formula. This dimensional formula is very helpful while calculating the dimensions of other quantities related to power (like energy). The concept of power varies according to its context and the available information. Work is accomplished at a rate represented by power (P). In other words, power is the rate at which work is done. Therefore, calculating its power by dividing the work done by the time taken can be used.
