Power is defined as the rate at which energy is converted or transferred over time. In the International System of Units (SI), the unit is the watt (W), where 1 watt equals 1 joule per second. Power is a scalar quantity, which means it has a size but no direction. It is usually written as P.
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The image shows a man lifting a weight, demonstrating muscular power, the physical power generated by muscles to perform work efficiently.
\boxed {P = \frac{W}{t}} where,
- P = Power gain or loss.
- W = Work done,
- t =Time for which the work is done.
Hence, the above relation of Power is called the work-time equation.
Work done (W) by an object can be defined as the product of force and the displacement of the object, then the power formula in terms of force is given by,
Since,
W = F × s
Therefore,
where,
F is the force required,
s is the displacement of the object,
It is the time taken.
Other Power Formulas
The other formulas for calculating the power are discussed below:
Since the velocity of the object,
v =\frac{s}{t}
Then,
P = F × v
where,
F is the Force required,
v is the velocity of the object.
Hence, the above relation is known as the Force-Velocity Equation.
Units of Power
The SI unit of Power is the watt (W).
- A Watt is defined as, when a body does work of one joule in one second it is called One-Watt Power.
1 Watt (W) = 1 Joule (J) / 1 Second (s)
Other Units :
- Horsepower (hp), commonly used in engines: 1 hp = 746 W
- Dimensional Formula of Power is [ML2T−3].
Horsepower
Interestingly, despite its name, horsepower doesn't directly relate to horses in modern usage. It is a unit used to measure the power output of machines or engines, such as those in cars or bikes. It is denoted by 'hp'. It is mathematically equal to,
1 hp = 746 watts
Horsepower is commonly used in the automobile industry to describe engine performance.
Measuring Power
Power is calculated using the formula:
P = \frac{\Delta E}{\Delta t} where,
- P is the power,
- ΔE is the change in energy (in joules),
- Δt is the time taken (in seconds).
Since energy is measured in joules and time in seconds, power is measured in:
1Watt=1 Joule/second
Average Power
Average Power is defined as the total work done divided by the total time taken. It represents the overall rate at which work is performed over a period of time. Mathematically, it is expressed as:
\text{Average Power} = \frac{\text{Total Work Done}}{\text{Total Time Taken}} or
P_{\text{av}} = \frac{\Delta W}{\Delta T} where,
Pav is the Average Power,
ΔW is the Total Work Done, and
ΔT is the total amount of time taken.
In the case when the rate of work done by the body is uniform or constant, then the average and instantaneous power become equal.
Mechanical Power
In mechanical systems, power is the rate at which work is done. It is generally calculated as the product of force and velocity for linear motion, or the product of torque and angular velocity for rotational motion. Hence, the mechanical power is given by,
Mathematically,
Linear motion:
P=F⋅v
where F is the force and v is the velocity.Rotational motion:
P=τ⋅ω
where τ torque and ω is the angular velocity.
Electrical Power
Electric power is the rate at which electrical energy is converted into another form of energy (such as heat, light, or mechanical energy) per unit time.
Mathematically, electric power is defined as the product of voltage and the current flowing, given as:
P = V × I
According to Ohm's Law. V = I × R, therefore,
P = I2 × R
or
P = V2 / R
where,
P is the electric power,
I is the current flowing,
R is the resistance and
V is the voltage.
Read More, Electric Energy and Power
Calculating Power and Energy Consumption
Energy consumption is calculated using the relationship between power and time. It is found by multiplying the amount of power used by the duration for which it is consumed:
Energy = Power × Time
This gives the total energy consumed over a specific period.
Hence, the energy consumption formula or the power consumption formula can be stated as:
E = \frac {P × t}{1000} where,
- E is the Energy consumed or power consumed
- P is the power and
- t is the time over which the power or energy was consumed.
Energy consumed or power consumed is generally measured in Joules or kilowatt-hours (kWh)
Solved Problems
Question 1: A boy pushes a box of 20 kg up to a distance of 5 m for 10 seconds. Calculate the power delivered to the box.
Solution: Given,
Mass of the box, m = 20 kg
Displacement covered, d = 5 m
Time of displacement, t = 10 s
Weight of the box, F = mg = 20 ×10 N = 200 N
Work done by the boy, W = F d = 200 N × 5 m = 1000 J
Power delivered, P = W ⁄ t = 1000 / (10) = 100 J/s
Hence, power delivered to the box is 100 W
Question 2: A pump is required to lift 500 kg of water per minute from an 8 m deep well and eject it with a speed of 25 m/s. Calculate the power of the pump.
Solution: Given,
Mass of the water, m = 500 kg
Height covered, h = 8 m
Eject velocity of water, v = 25 m/s
Delivery time, t = 1 min = 60 s
Total energy is converted into work,W = E = m.g.h+(1/2) m v2
= (500×10×8)+(500×25×25)/2= (40000+156250) J
=196250 J
Power delivered,
P = W / t= 196250 / 60
= 3271 W
Hence, the power delivered by the pump is 3271 W.
Question 3: An elevator is designed to lift a load of 500 kg through 5 floors of a building, averaging 3 m per floor in 5 seconds. Calculate the power of the elevator.
Solution: Given:
Mass of the load, m = 500 kg
Total height covered, h = 5 × 3 m =15 m
Time taken, t = 5 s
Power delivered by elevator,P = W ⁄ t = m.g.h ⁄ t
= (500 × 10 × 15) / 5 W
= 15000 W
= 1.5×104 W
Hence, the power of the elevator is 1.5×104 W.
Question 4: A force of 5 N is required to move a body on a frictionless floor with a constant velocity of 5 m/s. Find the power generated by the force.
Solution: Given:
Velocity of body, v = 5 m/s
Force required to maintain the velocity, F = 5 N
Power generated,
P = 5 × 5 W
= 25 W
Hence, the power generated by the force is 25 W.
Unsolved Problem
Question 1: A man lifts a 50 kg crate vertically upward through a height of 4 m in 8 seconds. Calculate the power delivered by the man.
Question 2: A motor raises a 200 kg load to a height of 10 m in 20 seconds. Find the power developed by the motor.
Question 3: A car engine applies a force of 400 N to move a car at a constant speed of 20 m/s. Determine the power produced by the engine.
Question 4: A water pump lifts 1000 kg of water from a 12 m deep tank in 1 minute. Calculate the power of the pump.
Question 5: A cyclist exerts a force of 150 N to move a bicycle 30 m in 10 seconds. Find the power output of the cyclist.