Work Done

We observe various types of work in our day-to-day life starting from waking up to pushing a lawn roller, and so on. Do you notice something in all the work that you do daily? Also, is there anything that we need to do for doing any work? Well, the thing required is force. To define, if we push a box by some distance ‘d’ by applying force ‘F’, we do some work and the multiplication of Force and ‘d’ is the work done. 

Therefore, for every work we do, we need force or the work is done when a force moves something.

Work Done in Physics

When we give a thrust to a block with some force ‘F’, the body travels with some acceleration or, also, its speed rises or falls liable to the direction of the force. As the speed surges or declines, the kinetic energy of the system alters. We know energy can neither be formed nor be demolished, so the energy must be converted into some other form. In this stance, it is termed as work done. The energy decreases when negative energy is completed, and the energy increases when positive work is completed. Now we will perceive how to determine work done. 

Definition of the Work Done

Work done is elaborated in such a way that it includes both forces exerted on the body and the total displacement of the body. 

This block is preceded by a constant force F. The purpose of this force is to move the body a certain distance d in a straight path in the direction of the force.

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Now, let us do the work done derivation.

What is Work Done for the Motion of a Block?

Consider a block located on a frictionless horizontal surface. A constant force F is acted upon this block. The purpose of this force is to move the body through a certain distance in a straight path in the direction of the force.

Now, the total work done by this force is equal to the product of the magnitude of applied force and the distance traveled by the body. Scientifically Work done formula will be given as, 

W = F * d

In this case, the force exerting on the block is constant, but the direction of force and direction of displacement influenced by this force is different. Here, force F reacts at an angle θ to the displacement d.

W = (|F| cosθ) |d|

We know that work done is defined as the multiplication of magnitude of displacement d and the component of the force that is in the direction of displacement.

Derivation for the Work Done Formula 

We know that the Work done by force (F) is equal to the change in kinetic energy.

\[W=\frac{1}{2mv^{2}}-\frac{1}{2mu^{2}}=\frac{1}{2m}(v^{2}-u^{2})\] ……..(1)

We know that according to third equation of motion:  v2 - u2= 2as …..(2)

Substituting equation (2) in (1) we get:

\[W=\frac{1}{2}m(2as)\]

\[W=m\times a \times s\]

We know from Newton's second law equation, F = ma (substituting now for F).

\[W=F.s\]

Since K.E. is the work done by a force ‘F’, so W = F.s 

Work done by the system

While describing work, we emphasize on the effects that the system does not work on its surroundings. 

Thus, we express work as being positive when the system makes any effort on the surroundings (i.e., energy leaves the system). The work is negative if work is done on the system (i.e., energy added to the system).

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Types of Work Done

• Positive Work: If a force relocates the object in its direction, then the work done is positive. The example of this type of work done is the motion of the ball dropping towards the ground where the displacement of the ball is in the direction of the force of gravity.

• Negative Work: When force & displacement are in reverse directions, then the work is assumed to be negative. 

For instance, when a ball is thrown upwards, the displacement will be in upwards direction; however, the force because of the gravity of the earth will be in the downward direction.

• Zero Work: If the direction of the force and the displacement are perpendicular to each other, the total work done by the force on the object is null.

For example, when we thrust hard against a wall, the force we are applying on the wall does not work, because in this case, the displacement of the wall is d = 0.

Work Done and Energy Relation

To move an object, it should be transferred to energy. Transferring energy can be in the method of force. This quantity of energy transferred by the force to move an object is termed as work done. Therefore, the relation between Work and Energy is related directly. 

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We concluded that the work and the energy are directly proportional to each other. Work done by an object can be scientifically expressed as:

                    W  = \[\frac{1}{2}\]mvf2   -  \[\frac{1}{2}\]mui2

Where,

m = the mass of the object measured using kilograms.

W = the work done by an object measured using Joules.

vf  = the final velocity of an object measured using m/s.

vi = the initial velocity of an object measured using m/s.

Therefore, the work-energy principle states that:

The total work done by all the forces acting on a particle or the work of the resultant force F(in subscript resultant) is equivalent to the change in kinetic energy of a particle.