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Work Done By A Variable Force

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What is Work?

Applying force on an object causes that object to move in the direction of the force. This movement in relation to the force is defined as work. You can complete work using a constant force or a variable force. 

What is a Constant Force?

Work done by a force can be divided into work from a constant force and work from a variable force. In the former kind, the magnitude and direction of the force remain unaltered. In such a case, work (W) is equal to the force applied (F) multiplied by displacement (\[\Delta\]x). Therefore,

W = F x \[\Delta\]x

What is Variable Force?

Work done by the variable force is a bit more complex. In such a case, the magnitude and direction of force can change at any time during the work. Most of the work that we complete in our daily life is an example of variable force work. Calculating the same is quite complex and requires integration.

To form a better understanding of the same, let us consider the workings of a spring.

Revising Hooke’s Law

Hooke’s Law states that the spring force for a compressed or stretched spring is equal in magnitude to the force for extension or compression of the spring. However, this spring force has an opposite direction to this extension. 

The figure above relates the force on the spring vs displacement when displacement is 0 for an unstretched spring. Us is the elastic potential energy for a stretched spring. This force-displacement graph for spring can help in assessing force according to Hooke’s Law.

Fs = -kx

Thus, Ws = Fs x vdt

Integration and Formula for Variable Force Work

For work done by a variable force, however, you need to apply integration to arrive at accurate results. Therefore,

\[W_{s} = \int_{0}^{t}F_s .vdt\]

\[W_{s} = \int_{0}^{t} - kx v_{x} dt\]

\[W_{s} = \int_{x^0}^{x} - kx dx\]

\[W_{s} = -\frac{1}{2} k\Delta x^{2}\]

Consequently, by using this approach mentioned above, one can easily derive the work done by variable force.

Quick Exercise – 1

Mass (m) of an object is 2kg. This object undergoes variable force in direction ‘x’. Force variation is a function, Fx = (3 + .2x) N. Determine the work done when object moves from x = 0 to x = 5.


Work done \[ = \int_{x^a}^{x^b} fdx\]

\[W = \int_{0}^{5} (3+.2x) dx\]

W = 17.5 J

Quick Exercise – 2

A bullet weighing 20g is moving at a velocity of 500m/s. This bullet strikes a windowpane and passes through it. Now, its velocity is 400m/s. Calculate work done by a bullet when passing through this obstacle.


We need to determine the change in kinetic energy in this equation. You know that kinetic energy change \[\Delta\] (K.E.) =\[\frac{1}{2} * \left ( mv_{1}^{2} - mv _{2}^{2}\right)\]

Therefore, in this equation, m is 20g or 0.02 kg. Initial velocity (v1) is 500m/s, while final velocity (v2) is 400m/s.

Thus, \[\Delta\] (K.E.) = \[\frac{1}{2} * {0.02 (500)^{2} - 0.02 (400)^{2}}\]

\[\Delta\] (K.E.) = 900 Joules.

Deriving Work Done by a Constant Force with Integration

Just as you can derive the work for a variable force using calculus, you can do the same for work done by a constant force. In such a calculation, pressure remains unchanged, which is why you can take it out of the equation immediately. 

After doing this, you will arrive at an equation, where

Work = \[P\int_{a}^{b} dv\].

\[W = P\Delta V\]

As you can see, the product is the same that we would have evaluated from considering force and distance. Thus, following this integration method for work done by a constant force is redundant.

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This article provides an explanation of work done by a variable force, its formula, and derivations. The frequently asked questions at the end of this article can help you with your doubts if you encounter any while understanding the same.

Variable Force

When a force acts upon a body changes with time(varies) it is known as variable force.  Work done by a variable force is calculated by  Integration.


Integration can be used for calculating both, that is, work done by a variable force and work done by a constant force.

\[W_{s} = \int_{0}^{t}Fs .vdt\]

\[W_{s} = \int_{0}^{t} - kx v_{x} dt\]

\[W_{s} = \int_{x0}^{x} - kx dx\]

\[W_{s} = -\frac{1}{2} k\Delta x^{2}\]


A force does work when it results in movement.

Work done by a constant force of magnitude F on a point moves a displacement

\[ \Delta x\]x  in the direction of the force is the product

\[W=F ⋅ \Delta x\]

In the case of a variable force, integration is essential to calculate the work done. For example, let us consider work done by a spring. Hooke’s law states that the spring or restoring force of a perfectly elastic spring is proportional to its extension but opposite to the direction of extension.

All the small contributions to the work done during small time intervals, for a variable force.

Last updated date: 03rd Jun 2023
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FAQs on Work Done By A Variable Force

1. What is the formula for deriving work from a constant force?

You can multiply constant force with the displacement occurring due to the application of that force to assess the exact work done by a constant force.

2. What are the two requirements for work done?

An external force must be applied on an object due to which, the object must move in the direction of the applicable force.

3. What is the SI unit of work?

The SI unit for measuring work is Joule.

4. What is meant by the work-energy theorem?

In the work-energy theorem, a change in the kinetic energy of the object is caused by the network on an object. Net work done is equal to the change in kinetic energy of an object. It is also known as the work-energy principle.

5. What is the example of variable force?

During the motion of a body when the force whose direction and magnitude changes is known as variable force. Examples of variable force are magnetic force, spring force and electrostatic force. Forces that we encounter in our day-to-day life are mostly variable forces. Work done by a variable force is calculated by dividing displacement into small intervals.

6. What is a constant force?

When the magnitude and direction of an object do not change throughout the motion it is known as a constant force.

Few examples of constant: hydrostatic force kinetic friction force weight etc. In the case of work done by a constant force the force remains constant throughout the displacement and can be calculated by calculating the dot product force and displacement vector.

7. What is the difference between constant and variable force?

When the force acted upon a body is varied (changing with time) it is said to be a variable force. In variable force, the motion of a body is non uniformly accelerated. And in constant force the same amount of force is applied to a body, it does not vary with time. In a constant force, the motion of a body is uniformly accelerated.

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