# Physics Tutorial: Power and Efficiency

In this Physics tutorial, you will learn:

• 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?

## Introduction

Suppose you are the owner of a company that imports goods using trade ships and you are looking to hire a stevedore. Two candidates have applied for this job. The first candidate manages to discharge the goods from the ship in 8 hours while the other candidate completes the same job in 6 hours. Which candidate you are most likely to hire? Why?

Now, think about having a limited time to discharge as much goods as possible from the ship. Which from the abovementioned stevedores is more suitable for this task? Why?

## The Meaning of Power

As seen in the above example, 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. Every company boss prefers workers that can do a certain work in the least time possible or if the time is equal for all, a boss prefers to hire workers who are able to do the greatest amount of work possible in a given time. Such outstanding workers have a common ability compared to other workers - we say they are more powerful.

From the above reasoning, we can deduce another fundamental concept in physics: the Mechanical Power, or simply, Power. 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

### Equation 1

P = W/t

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,

[1 W] = [1 J/s]

### Example 1

An object moves horizontally by 12 m when applying a 25 N horizontal force on it as shown in the figure.

The entire process takes 15 s to complete. What is the power delivered to the object by the force source?

### Solution 1

We have the following clues in this exercise:

Δx = 12 m
F = 25 N
t = 15 s
P = ?

First, we must calculate the work done on the object. Thus,

W = F × ∆x
= 25 N × 12 m
= 300 J

Now, let's calculate the power delivered by the source. We have

P = W/t
= 300 J/15 s
=20 W

## Output and Input Power

It is quite impossible that the entire energy produced by a source converts into work done by the system. For example, if we consider the example of the previous paragraph, it is clear that the energy delivered by the source is not 300 J but much more as some part of it, goes for other purposes, such as to overcome friction etc.

Therefore, the power we calculate through the equation (1) 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

### Equation 2

Pinput = Etotal/t

It is obvious that the input power has always a greater value that the output power (at most they are equal) because the total energy produced by the source is always greater than the part of it that converts into work.

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 (look at the Physics tutorial "Elastic Potential Energy. Combination of Springs"). As a result, we have

Poutput < Pinput

and

W < Etotal

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

Etotal = W + Ew

## The Meaning of Efficiency

In general, work is done by machines, which take their energy from a power source. Even the human body can be though as a machine which takes its energy from food.

The quality of machines depend on how complicated they are or how much they meet our demands. These are general features, which are difficult to measure. However, 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 formula:

### Equation 4

e = Poutput/Pinput × 100%

Efficiency is a dimensionless quantity that is calculated as a percentage of the output power to the total (or input) one.

We have

Poutput = W/t ⟹ W = Poutput × t

and

Pinput = Etotal/t ⟹ Etotal = Pinput × t

Therefore, when multiplying both numerator and denominator in the equation (4) by time t, we obtain

e = Poutput/Pinput × 100%
= Poutput × t/Pinput × t × 100%

In this way, we obtain another equation for the efficiency of a machine:

### Equation 5

e = W/Etotal × 100%

### Example 2

A crane can lift a 200 kg load at 12 m above the ground in 6 s. During this process, the crane draws 30 000 J of electrical energy from the power source. Calculate the efficiency of this crane. For convenience, take g = 10 N/kg.

### Solution 2

As stated in the Physics tutorial "Gravitational Potential Energy. The Law of Mechanical Energy Conservation", the work done on an object by a source of energy when raising it from the ground level to a certain height h, contributes in the increase of its gravitational potential energy from GPE1 = 0 to GPE2 = m × g × h. Therefore, we can write

Wsource = ∆GPEobject
= GPE2 - GPE1
= m × g × h - 0
= m × g × h
= 200 kg × 10 N/kg × 12 m
= 24 000 J

Now, there are two ways for calculating the efficiency of the crane.

Method I - Using the equation (4). For this, we have to calculate the output and input power by divining both work and total energy by time. Thus,

Poutput = W/t
= 24000 J/6 s
= 4000 W

and

Pinput = Etotal/t
= 30000 J/6 s
= 5000 W

Therefore,

e = Poutput/Pinput × 100%
= 4000 W/5000 W × 100%
= 0.8 × 100%
= 80%

Method 2 - Using the equation (5), i.e. calculating the efficiency directly, without dividing anything by time. Thus,

e = W/Etotal × 100%
= 4000 W/5000 W × 100%
= 0.8 × 100%
= 80%

Interpretation of result: Having an efficiency of 80 percent, means that only 80% of the energy provided by the source is converted into work done by the machine. The rest (20 %) has turned into other forms of energy such as heat, light, sound energy etc. This part is considered as wasted energy although it may be useful for the operation processes of machine such as to turn on the buttons, to make the gears rotate, etc. The fact that it is not used to do work, i.e. to lift the object, makes us consider this part of total energy as wasted.

### Another Formula for Calculating the Mechanical Power

Let's transform the equation of (output) power as follows:

P = W/t
= F × ∆x/t
= F × ∆x/t

Thus,

### Equation 6

P = F × v

where v is the velocity of the object in the direction of force F.

The equation (6) enables us formulate another alternating definition for 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."

### Example 3

An object starts moving from rest and it accelerates constantly reaching a velocity of 8 m/s after 10 s. The output power delivered by the source is 120 W and the efficiency of the process is 40%.

1. What is the mass of the object?
2. What is the total energy supplied by the source for this process?

### Solution 3

Clues:

v0 = 0
v = 8 m/s
t = 10 s
Poutput = 120 W
e = 40 %

1. m = ?
2. Etotal = ?
1. The first thing to do is finding the average velocity of the process as its value will be used in the equation (6) in order to work out the average force of the source later. Thus,
< v > = v0 + v/2
= 0 + 8 m/s/2
= 4 m/s

Now, let's calculate the average force produced by the source. We have

Poutput = < F > × < v >
< F > = Poutput/< v >
120 W/4 m/s
=30 N

Now, using the kinematic equation

a = v - v0/t

we calculate the acceleration, and then use this value to find the object's mass by applying the Newton's Second Law of Motion. Thus,

a = 8 m/s - 0 m/s/10 s
= 0.8 m/s2

Hence, given that

a = < F >/m

then,

m = < F >/a
= 30 N/0.8 m/s2
= 37.5 kg
1. Using the equation
Poutput = W/t

we calculate the work done by the source on the object. Thus,

W = Poutput × t
= 120 W × 10 s
= 1200 J

Therefore, giving that

e = W/Etotal × 100%

we obtain after the substitutions

40% = 1200 J/Etotal × 100%
Etotal = 1200 J × 100%/40%
= 1200 J × 2.5
= 3000 J

Remark! Constructors of electrical appliances often write the power of appliance (in watts) on its back or lateral side. Power is stamped on a white plate amongst other parameters such as operating voltage, frequency, noise level etc. This value represents the output power delivered by the appliance. Since Energy = Power × Time, people (wrongfully) try to calculate the electrical energy consumed by the appliance by multiplying the power shown at the plate by the time (in seconds). Then, using the conversion factor 1 kw - h = 3 600 000 J (we discuss this in detail in our Physics tutorials on Electricity), they find a value in kw-h which in fact doesn't correspond to the value measured by the electric board meter. This occurs because the board meter counts the total energy consumed by the appliance while the abovementioned calculations give the output (useful) energy (i.e. the part of total energy that is used for doing work).

## Summary

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

P = W/t

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,

[1 W] = [1 J/s]

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

Pinput = Etotal/t

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

Poutput < Pinput

and

W < Etotal

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

Etotal = W + Ew

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:

e = Poutput/Pinput × 100%

Or

e = W/Etotal × 100%

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

Another Formula for Calculating the Mechanical Power is

P = F × v

where v is the velocity of the object in the direction of force F. 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.

## Power and Efficiency Revision Questions

1. The engine of a car produces 12000 J of energy every second. From this amount, 4500 J convert into heat, 100 J into sound energy and 200 J into light energy. What is the efficiency of the engine?

(Hint: The rest is the mechanical energy of the car)

1. 40 %
2. 60 %
3. 93.75 %
4. 72 %

2. An electric motor shows the value 2000 W stamped on its back. Its owner turns off all the other electric appliances except the motor and finds out the motor has drawn 150 000 J electrical energy from the main source in one minute. What is the efficiency of this electric motor?

1. 80 %
2. 20 %
3. 75 %
4. 1.3 %

3. A crane can lift a 1.2 t heavy object at h = 10 m in 20 s. Its input power is 10 kW. What is the crane's efficiency? (Take g = 10 N/kg. Also, 1 t = 1000 kg and 1 kW = 1000 W).

1. 50 %
2. 60 %
3. 80 %
4. 83 %