work-energy theorem

Introduction

According to the work-energy theorem, the change in the kinetic energy of a body is similar to the net work-done by the forces acting on it. The work-energy theorem can also be derived from Issac Newton’s second law.

Work transfers energy from one place to another. In addition to that, in general systems, the particle’s work can change the P.E. of a mechanical device. The term “work” is frequently used in daily life and we understand it as an act of doing something. The work-energy theorem explains the facts behind the physics of no work. Work is claimed to be done when an acting force displaces a particle. According to physics, no work is done without displacement.

In Physics, work may be a result of force and the resulting displacement. As you may know, all moving objects have kinetic energy (KE). Hence, there must be a relation between Work and KE. 

This relation between the KE of an object and work done is termed the work-energy theorem. The work done is in a one-dimensional motion. The equation is as follows:

W = Fdcosθ

Here, W is work done, 

F is the magnitude of the force

d is that the magnitude of the displacement,

and, θ is the angle between the force vector F and displacement vector d.

Work done can have varying values: from positive to zero to negative. Let us look at the instances where positive, negative, and zero work is done.

1. Positive work: When a body moves in the direction of applied force, the work done that is taken into account is positive. For example: When a ball falls down towards the earth surface, having some displacement which is measured, it is an example of this sort of work. The ball comes down in the direction of gravity.
   
   
2. Negative work: The value of work done is negative when the force and displacement on a body act in the opposite direction. For example: If you toss a ball in the upward direction, and measure the displacement which will be upwards against the force of gravity, then the value of the work done will be negative.
3. Zero work: The overall work done by the force on the object is zero if the direction of the applied force and the displacement act perpendicular to one another. The overall work done gives a zero value. Example: If you push forcefully against a wall and this force being exerted on the wall is ineffective, then the wall’s displacement gives a zero value, and hence the work done is zero.

Work Done by a Constant Force

Work is done when a force acts on an object over a distance. This work done on an object is equal to the change in the kinetic energy that the entity experiences. 

The term “work” was proposed in 1826 by a French mathematician named Gaspard-Gustave Coriolis. He came to this term by “weight lifted through a height,” which relied on the utilization of early steam engines to lift buckets of water out of flooded ore mines. 

Work’s SI unit is newton-meter or joule (J). There is a technique that can validate if an expression is correct to see whether a dimensional analysis can be performed or not. 

As we know, work is the change in the kinetic energy of an object and is also equal to the force times the distance. Thus, the units of those two should agree that K.E. and every kind of energy have units of joules (J). Similarly, we can describe the units of force in newtons (N) and distance in meters (m). If the two statements are equivalent, they ought to be like each other. 

It is often asked to calculate the work done by a force on an object. This is often proportional to the force and, therefore, the distance from which the object is displaced. Work is measured by the scalar product of two vectors, namely force and displacement.

Let’s consider force F to be a vector quantity and displacement s as a vector quantity,

Therefore, work = force x displacement

or, W = F. s

W= F.s = Fs cos θ,

[Here, s cos θ is the component of displacement along with the direction of force, F]

Here, θ can be known as the angle between the force and displacement, respectively.

Work Done by a Non-Uniform Force

The forces that we encounter in everyday life are mostly variable in nature. These forces can be defined as non-uniform forces. 

Work is done when a force is applied to a system so that there is a displacement in the system in the direction of the force. But in non-uniform forces, integration is needed to calculate the work done. 

The work done by a uniform force of magnitude F displacing an object by Δx can be expressed as:

       W = F.Δx

In non-uniform forces, the work done is calculated with the assistance of integration. For instance, within the case of a spring, the force acting upon the object attached to a horizontal spring can be expressed as:

Fs = -kx

Here,

k stands for spring constant

x  stands for the displacement of the object attached

If you observe carefully, you can see that this force is proportional to the displacement of the object from the equilibrium position. Thus, the force performing at each instant during the compression and extension of the spring will be different. Therefore, the infinitesimally small contributions of the work done during each instant should be tracked so as to calculate the overall work done.

Work Energy

In order to move an object from rest, energy must be transferred to it. The energy that gets transferred is in the form of force. This energy that gets transferred by force to maneuver an object is referred to as work or work done. Therefore, we can establish a direct relationship between work and energy. 

The difference in the K.E of an object is the work done by the object. In physics, work and energy can be seen as the two sides of the same coin. We can use the proportionality constant to explain the connection between work and energy. Since both are perpendicular to each other, they are directly proportional to each other also. So, the formula for work and energy is as follows: 

W = ½ mvf2 – ½ mvi2

Here, 

W stands for the work done by the thing and is measured in Joules,

M stands for the mass of the concerned object and is measured in kg,

vf stands for the final velocity of the concerned object and is measured in m/s

vi stands for the initial velocity of the concerned object and is measured using m/s.

Therefore, the relation between work and energy (work-energy theorem) states that the net work done on an object by the force is capable of changing its K.E. Thus, another relation between work and K.E. can be written as follows: 

          W = Ki – Kf = ΔK

Where 

Ki stands for the object’s initial K.E,

Kf stands for  the object’s final K.E,

and ΔK is the difference in the final and the initial K.Es.

Conclusion

Imagine a skier moving at a uniform velocity on a flat, frictionless surface. If someone comes up behind him and pushes him, he’ll speed up as a result of his increasing K.E. This sort of work is understood as positive work as the force was applied within the motion of the skier. If however, the person pushed the skier towards himself, the skier would impede as a result of his decreasing K.E. This sort of work is understood as negative work. 

This is just an easy example to make you understand. You can also replace the skier with a block in the same situation. In fact, there aren’t any frictionless surfaces but the friction force opposing the skier’s motion would need to be accounted for.