Work Done by a variable force

Introduction

It is captivating to know that the forces that we encounter each day are primarily variable, known as variable force. A force is known to perform work on a specific system if there had been a displacement in the system beyond the application of the force in the direction of the force. In variable force, integration plays a major role in calculating the work executed.

In this article, learn various types of force, work executed by a constant and variable force, and the force-displacement plot.

Types of Forces

1. Force with constant direction and magnitude – A constant force is defined as the force whose direction and amplitude remain constant during the motion of a certain body. We may think of things like weight, kinetic friction force, hydrostatic force, and so on.
2. Variable force is defined as the force whose direction and amplitude vary during the body’s motion and is not constant. As an example, consider spring force, magnetic force, electrostatic force, and so on. 

ΔW = F Δx

Where F is variable force

 Δx is displacement due to force F.

After assuming the displacement approaching zero, the total work executed by force can be stated as: dW = F.dx

Hence, for a variable force, the total work executed can be stated as a definite integral of force over the displacement of any system.

W =F.dx

A force is known to do work while it acts upon a body. Because of this, there is a displacement of the site of application concerning the direction of the force. As a result, a force is effective during a movement.

As we know that the work executed by a static force of magnitude displaying an object by Δx that can be given as:

 W= FΔx

In the case of a variable force, work is estimated with the help of integration. 

For instance, in the spring case, as per Hooke’s law, the force that acts upon any object that is attached to a horizontal spring can be stated as:

Fs = -kx

Where,

x is the displacement of the attached object

k is the spring constant

Here, we know that the force is proportional to the object’s displacement from the equilibrium position. Therefore, the force that acts at every instant during the extension and compression of the spring will always be different. Hence, the infinitesimally few contributions of the work executed throughout every instant are required to be counted to estimate the total work executed.

The work done by variable force formula is : 

W =F(x).dx

Integration is used in the calculation of the work executed by constant forces.

The same integration method used to calculate the work executed by the variable force may also be used to calculate the work executed by the constant force. This indicates that the particular method of determining the work executed by a static strength on a moving body is to integrate the product of distance and force into the equation.

Let us assume that gas is wrapped in the piston. Thermodynamics is concerned with the study of this phenomenon. As a result, the pressure equals the force on the surface area is constant and may be removed from the integral equation. 

W =abPdV

W =PabdV

W = PV

Where P is the pressure 

V is the change in volume.

Conclusion

Work executed by a variable force means when a force displays an object and does work on the object, the force might change its magnitude or direction or both. The forces of such kinds are known as variable forces. Calculating the work executed by variable force is more complex than finding out the work executed by a constant force.