### Work done by a force

##### March 30, 2017

## Work done by a Force

In Physics Work is done by a force applied only if the object on which the force is applied makes a displacement. Without this displacement, work done is zero irrespective the force you have applied. Work done by a force depends on the 3 parameters, (a) the force applied (b) The displacement and (c) The angle between the line of force applied and the displacement. That’s why when you apply a force by pushing a wall and it never moves, you have not done any work in terms of physics. We will discuss on versions of equations of work done by a force. We’ll study a few cases of work done by a force as well.

#### Equation of Work Done by a Force – simple version

The simple version of the equation of work is as follows, considering that the displacement happens in the direction of the applied force i.e. the angle between the line of force and the displacement of the body is 0 degree

Work = Force * Displacement,

i.e. W=F* S, Here W=Work done, F=applied force on the object and s=the displacement of the object.

#### Equation of Work Done by a Force – Vector Representation

If we consider the Vector representation of the Work, Then we can write it as the dot product of Force and Displacement.

So we can write Work = Force . Displacement

or, W = F . S = F S Cos θ , where θ is the angle between the line of force and the displacement.

When θ = 0 degree, i.e. the Force and the displacement are pointing to the same direction, W = F S , as we have shown in the previous section.

When θ = 180 degree, i.e. the Force and the displacement are pointing to opposite directions, then W = F S cos 180 degree = F S (-1) = – F S. Here the – sign denotes that the displacement happens just in the opposite direction of the applied force.

#### CASE STUDIES – work done

###### Case 1: A person lifts an object from the ground to his head top – work done by a Force

Here Force being applied on the object is the gravity or the weight of the object, against which work needs to be done to lift it from the ground.

So Force (F) = Mg where M = mass of the object, g = acceleration due to gravity and together Mg is the weight of the object. Note that this weight is point towards the center of the earth and the object needs to be lifted in the opposite direction of the force of gravity.

Displacement = height of the person = H

And the angle between the line of force and the displacement is 180 degree.

So Work done (W) = Mg H cos 180 degree = Mg H (-1) = – Mg H.

###### Case 2: A person drops an object from his head top to the ground – Work done by a Force

In this case Force applied on the object is same as the above case i.e Mg. The displacement is as well same as the first case (H). but in this case displacement is happening in the direction of the force, so the angle between force and displacement is zero degree. So we get the Work done as Mg H cos 0 degree =Mg H (1)= MgH

###### Case 3: Summation of case 1 and 2 – Work done by a Force

In this case a person lifts an object first from the ground and then drops it to the ground again. Here we can get the work done of this entire loop in 2 ways:

1) Way 1: As finally displacement(S) of the object is zero, so work done is zero.

2) Way 2: This case is the summation of case 1 and Case 2.

So if we consider the total work done = work done to lift + work done to drop = – MgH + MgH = 0

Case 4: We are pushing a wall but it doesn’t move

###### Case 4: If we push a wall then the following things happen

- We apply force (action) on the wall.
- The wall applies an equal and opposite force on us (reaction)
- If we don’t see it moving, so displacement of the wall along the line of force is zero.
- Work done = force on an object X displacement of the object along the line of force
- In this case, force is non-zero, but displacement along the line of force=0. Hence work done is zero.

We will discuss more later. Thank you.

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