Work Energy Theorem And Its Application

Introduction

We already discussed in the previous article (link here) that there is some relation between work done and energy. Now we will see the theorem that relates them. According to this theorem, the net work done on a body is equal to the change in kinetic energy of the body. This is known as Work-Energy Theorem. It can be represented as:

Kf – Ki = W

Where Kf = Final kinetic energy

Ki = Initial kinetic energy

W = net work done

So the above equation follows the law of conservation of energy, according to which we can only transfer energy from one form to another. Also, here the work done is the work done by all forces acting on the body like gravity, friction, external force etc. For example, consider the following figure.

Work Energy Theorem

According to Work energy theorem,

Work done by all the forces = Change in Kinetic Energy

Wg + WN + Wf  =Kf – Ki

Where Wg = work done by gravity

WN = work done by a normal force

Wf = work done by friction

Kf = final kinetic energy

Ki = initial kinetic energy

Work done by a constant force

A constant force will produce constant acceleration. Let the acceleration be ‘a’.

From the equation of motion,

v2 = u2 + 2as

2as = v2 – u2

Multiplying both sides with mass ‘m’

\(\begin{array}{l}\textup{(ma).s = } \frac {(mv^2 – mu^2)}{2}\end{array} \)

\(\begin{array}{l}\textup{F.s = } \frac {(mv^2 – mu^2)}{2} \end{array} \)

Comparing the above equation, we get,

Work done by force (F) = F.s

Where ‘s’ is the displacement of the body.

Work done by Non-Uniform Force

Now the equation,

W = F.ds

This is only valid when force remains constant throughout the displacement. Suppose we have a force represented below,

Work Energy Theorem

For these kinds of forces, we can assume that force remains constant for a very small displacement and then integrate that from the initial position to the final position.

\(\begin{array}{l}\textup{W = } \int^{x_f}_{x_i} F(x) dx \end{array} \)

This is work done by a variable force. A graphical approach to this would be finding the area between F(x) and x from xi  to xf.

Work Energy Theorem

The shaded portion represents the work done by force F(x).

Frequently Asked Questions – FAQs

We will discuss some of the most common questions on this topic.

Frequently Asked Questions – FAQs

Q1

What is kinetic energy?

Kinetic energy is the energy possessed by an object due to its motion or movement.
Q2

What is the formula of the work energy theorem?

The work-energy theorem is given by the formula: Kf – Ki = W

Q3

State the work energy theorem?

The work-energy theorem states that the work done by the net force on a body is equal to the change in kinetic energy.

Q4

What is the equation to find the relationship between acceleration and the net force?

The relationship between acceleration and the net force is given by the equation: F=ma.
Q5

State the law of conservation of energy.

In a closed system, i.e., a system that is isolated from its surroundings, the total energy of the system is conserved.
Q6

What is the statement of the work energy theorem?

According to the work-energy theorem, the work done by the net force on a body equals a change in kinetic energy.
\(\begin{array}{l}\int~\overrightarrow{F}.d\overrightarrow{r} = K_f~-~K_i\end{array} \)

\(\begin{array}{l}\int~\overrightarrow{F}.d\overrightarrow{r} = K_f~-~K_i\end{array} \)

Q7

How can we prove the work-energy theorem?

We know,
\(\begin{array}{l}K =  \frac{1}{2}mv^2\end{array} \)

Differentiating with respect to time we get,

\(\begin{array}{l}\frac{dK}{dt} = mv. \frac{dv}{dt}\end{array} \)

If the resultant force makes some angle with the velocity, then,

\(\begin{array}{l}\frac{dK}{dt} = Fv~cos\theta\end{array} \)

\(\begin{array}{l}\Rightarrow~dK = \overrightarrow{F}.d\overrightarrow{r}\end{array} \)

Q8

Steps to approach problems on work energy theorem?

The following steps should be considered:

Step-1: Draw the FBD of the object, thus identifying the forces operating on the object.

Step-2: Find the initial and final kinetic energy.

Step-3: Equating the values according to the theorem.

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