Work, Energy and Power: Explanation, Examples, Types & Theorem

Work, Energy, and Power are three fundamental concepts that are interlinked with one another.

• When the point of application on a body is displaced under the effect of a force, then the body is said to be doing work.
• The capacity to do work is called energy.
• Power can be defined as the work done per unit of time. 
• Joule(J) is the SI unit of Work.
• The S.I. unit of power is watts (W).

Key Terms: Work, Energy, Power, Conservative Forces, Non-conservative Forces, Scalar, Time, Displacement Vector, SI unit, Work-energy theorem, Force, Gravitational force

---

What is Work?

[Click Here for Previous Year Questions]

When the point of application on a body is displaced under the effect of a force, then the body is said to be doing work.

• Work is a scalar quantity calculated by multiplying the magnitude of force by the component of displacement along the force direction. 
• In fact, work is the scalar product (dot product) of the force vector and the displacement vector of the point of application.
• Joule(J) is the SI unit of work done.

Thus,

where F and S are the magnitudes of force and displacement vectors and θ is the angle between them.

• For 0 ≤ θ < π/2, positive work done.
• For θ = π/2, zero work done
• For π/2 < θ < 3π/2, negative work done.

---

Examples of Work

[Click Here for Sample Questions]

There are several examples of work. Some include:

1. In a scenario of lifting a body from the ground, the work done by the lifting force is positive (θ = 0°) but the work done by the gravitational force is negative. (θ = π).
2. The work done by the pulling force is positive when a body slides on a fixed rough surface, whereas the work done by the friction force is negative. The usual normal force produces no effect i.e., it is 0.

Total work done by any constant force is path independent, that is, it depends only on initial and final positions of the body.

Work Done by a Variable Force

The Equation, W = F.S = FS cosθ is applicable when F remains constant

But when the force is variable work is obtained by integrating F.dS

Thus, W = ∫F. ds

The spring force, which varies on the extension x, i.e., F α x, is an example of a variable force. 

When the force is time-dependent, we have,

W = ∫F.dx = ∫F.(dx/dt) dt = ∫F.v dt

where F and V are force and velocity vectors at any instant.

Geometrically, the work done is equal to the area between the F(x) curve and the x-axis, between the limits x_i and x_f, e.g., suppose a spring force. 

In the given figure, one end of a spring is attached to a fixed vertical support and the other end to a block that can move on a horizontal frictionless table.

Spring

At x = 0, the elongation in the spring is zero i.e., it is at its natural length. When an amount x displaces the block, a restoring force (F) due to elasticity is applied by the spring on the block.

i.e.  

F = – kx … (i)

where k is the spring's force constant, which depends on the nature of the spring.

Seeing equation (i), we can say that –

1. F is a variable force
2. F-x graph is a straight line which passes through origin with slope – k.
3. Work done by the force F when block is displaced from x = 0 to x is: W = 0x∫ F.dx = 0x∫-k.dx = -1/2kx2 = area under F-x graph.

Work Done by Variable Force Graph

Work Depends on the Frame of Reference

Consider a person pushing a box inside a moving train

• Then the work done in the frame of the train will be F.s.
• While work done with respect to earth will be F. (S + S_o,), where, S_o is the displacement of the train relative to the ground and S is the displacement of the box relative to the train.
• Depending upon the situation, work done by friction may be zero, positive, or negative.
• Work done by the frictional force is zero whenever the force applied to a body is insufficient to overcome the friction.
• Work done by the frictional force is negative whenever this force is large enough to overcome the friction.
• The work done by the friction force on the lower body is positive when force is applied to a body that is placed above another body.

Consider the following scenario: a horizontal rough trolley, on which a block and a guy are standing, is accelerating horizontally. On the trolley, the block is not slipping. The following conclusions can be made from the above scenario:

1. In this case, work done by friction (between trolley and the block) is zero as observed by the man on the trolley.
2. Work done by the force of friction (acting between trolley and the block) is positive as observed by an observer on the ground.
3. Work done by friction is negative as observed by an observer who is moving along the direction of motion of the trolley at a higher speed with the trolley in the same direction.

Conservative and Non-Conservative Forces

If the work done by a force along a closed path is zero, it is said to be conservative.

• The conservative forces' work is path independent and only depends on the initial and final positions. 
• If the work done by a force along a closed path is not zero, it is said to be non-conservative.
• Non-conservative forces are dissipative, but conservative forces are not. 
• Gravitational force and electrostatic force are examples of conservative forces. 
• Frictional force and viscous force are examples of non-conservative forces.

---

What is Energy?

[Click Here for Previous Year Questions]

Energy can be defined as the ability to do work. Energy can neither be created nor destroyed, and it can only be transformed from one form to another. The unit of Energy is equal to Work, i.e. Joules. There are several forms of energy. All forms of energy are usually either potential or kinetic.

Kinetic Energy

The capacity of a body to do work by virtue of its motion is known as kinetic energy of the body. Kinetic energy is equivalent to work done by an external force on a body of mass ‘m' to bring the body from rest up to its velocity ‘v’ in absence of dissipative forces.

Mathematical expression

Consider the case of a mass ‘m' body that is initially at rest. Let us consider that an external constant force F acts on the body to bring its velocity to ‘v’. If ‘s’ be the displacement, then,

• v² = 2aS
• F = ma

Now, work done by the constant force, 

W = FS = (ma)(v²/2a) = 1/2 mv²

Therefore, according to the definition

Kinetic Energy (KE) = 1/2mv² = m²v² / 2m = p²/2m

Potential Energy

The potential energy of a body is the energy it has as a result of its position. The change in potential energy produced by a conservative force is defined as the negative of the work done by the conservative force.

Hence, 

U_f – U_i = - _rinitial^rfinal∫ F.dr

Where, 

• U_i = Potential energy at the initial reference position,
• U_f  = Potential energy at the final position of the body

Usually, the initial reference position is taken as infinity and the potential energy at infinity is assumed to be 0 as 1/∞ = 0. Then, we get the potential energy of a body as:

U = – r = _∞r^rfinal∫ F.dr

The negative derivative of the potential energy function with respect to the position gives the conservative force acting on the particle. Mathematically,

F = – dU/dr

Potential energy stored in Spring

Suppose a spring is stretched with an elongation of ‘x’ in its original length. Then, 

U = - ₀^x∫ F.dr

⇒ U = kx²/2

Potential Energy Stored when a particle of mass ‘m’ is displaced to a height ‘h’ against gravity is:

U = - ∫F.dx = – ₀^h∫(mg)dx cos 180° ⇒ U = mgh

Where, ‘h’ is height above the reference level.

Potential Stored in a Spring

---

Work-Energy Theorem

[Click Here for Sample Questions]

According to this theorem, work done by all the forces acting on a particle or body is equal to the change in its kinetic energy.

Let us take an example shown in Figure (a), in which a block of mass ‘m’ kept on a rough horizontal surface is acted upon by a constant force ‘F’ parallel to the surface. The corresponding F.B.D. is shown in Fig. (b) which gives F + fk = ma ……(i)

and   Normal Reaction force, N = mg ……(ii)

Initially, while the force F is just applied, the block is at position A and has a velocity Vo. The force acts on it for some interval of time "t' so that the block reaches to position B at a distance x from A.

F + fk + mg + N = Fnet = mdv/dt

dW = Fnet .ds = mdv/dt . vdt = md(v² /2)

Integrating, W = m/2 (v2 - u2)

∑ W = k_final – ki

Work done by net force i.e., Change in kinetic energy = work done by all external forces. 

Considering 1/mvo² = kf (initial kinetic energy)

and 1/2mv² = kf , (final kinetic energy)

Now, 

W = kf - ki … (iii)

Now, equation (iii) can be explained as:

• the net work done by all the forces on a system gives the change in the kinetic energy (KE) of the system.
• This is called the work-energy theorem. 
• Thus, the change in kinetic energy of the body equals the total work done by all the forces (conservative and non-conservative).

Conservation of Energy and Conservation of Mechanical Energy

Energy conservation entails the preservation of all kinds of energy at the same time.

• The overall energy of an isolated system remains constant after accounting for all forms of energy. 
• Only two types of energy are accounted for by mechanical energy: kinetic energy (KE) and potential energy (U).

If only conservative forces act on a system, then the total mechanical energy of the system remains constant. i.e., 

K + U = constant

Therefore, 

ΔK + ΔU = 0 or, ΔK = - ΔU 

---

Types of Energy

[Click Here for Previous Questions]

Some other forms of energy are also present in nature. They are:

• Internal Energy: A body's internal energy is controlled by its temperature. Molecules are thought to make up a body. Internal energy is the sum of the kinetic and potential energies of all the molecules that make up the body. If the temperature of a body increases, this change causes an increase in the kinetic and potential energy and hence, in the internal energy.
• Heat Energy:  A body has heat energy because of the irregular motion of its molecules.
• Chemical Energy: Due to the chemical bonding of its atoms, a body possesses chemical energy.
• Electrical Energy: Work is required to transport an electric charge from one place to another in an electric field or to move a current-carrying conductor in a magnetic field transversely. This work done is known as the electrical energy of the system.
• Nuclear Energy: When a heavy nucleus is attacked by a neutron, it splits apart into lighter nuclei, releasing a lot of energy. Nuclear energy is the name for this type of energy.

---

What is Power?

[Click Here for Sample Questions]

The rate at which work is done is defined as power i.e., Work/time. S.I. unit of power is watts (W). Mathematically, the formula of power is,

As, 

⇒ dW = F.dx

Therefore, P = dW/dt = F.dx/dt ⇒ P = F.v

If the force is variable, the average power is calculated as:

P_average = ΔW/Δt =  ₀^t∫ Pdt / ₀^t∫ dt

Power can also be expressed as the rate of change of kinetic energy (KE). Assume a body of mass of ‘m' is travelling at a velocity of ‘v.'’. Then, its kinetic energy is,

KE = ½ mv²

Now,   dKE / dt = ½ d(mv2) /dt

= mv.(dv/dt)

= F_external \(\times\) v

Therefore, 

P = dKE / dt

Read Also:

---

Previous Year Questions

1. A child is sitting on a swing. Its minimum and maximum heights … [NEET 2001]
2. A force F = 20 + 10y action a particle in y-direction … [NEET 2019]
3. A mass mm is attached to a thin wire and whirled in a vertical circle … [NEET 2019]
4. A mass of 0.5kg moving with a speed of … [NEET 2004]
5. A metal ball of mass 2 kg moving with speed … [NEET 1997]
6. A particle of mass 10g10g moves along a circle of radius … [NEET 2016]
7. A particle of mass 5 m at rest suddenly breaks … [NEET 2019]
8. A particle of mass m is driven by a machine that … [NEET 2015]
9. A uniform force of … [NEET 2013]
10. An engine pumps water through a hose pipe … [NEET 2010]
11. An explosion blows a rock into three parts … [NEET 2009]
12. Body A of mass 4m moving with speed u collides … [NEET 2019]
13. The coefficient of restitution … [NEET 1988]

---

Things to Remember

1. When the point of application on a body has been displaced under the effect of a force, then the body is known to perform work.
2. The capacity to perform work is known as energy. 
3. Power can be expressed as the work done per unit of time. 
4. There are several types of energy, Mechanical energy, Potential energy, Kinetic energy, Internal energy, Heat energy, Chemical energy and more.
5. Formula of Power is, P = dW/dt