# Work Done by Friction: Exhaustive Insights and Facts

Work done by friction comprises the moving body’s displacement opposite to the direction of friction force.

The applied force to the body makes the displacement in its direction. Newton’s third law of motion employs a friction force against the body’s motion as a reaction force. Thus, the body’s displacement opposite to the friction force is called the work done by friction.

The friction force is the opposing contact force exerted by the surface to oppose the motion when two bodies slide on each other. It is a non-conservative force that offers the work done by a force based upon the path along which the force acts. Depending on the chosen path, we can find the different work done by friction.

Suppose you are pushing the table across the room to change its positioning. The below surface first resists the table with a friction force as it exactly counters our applied push force. When we employ more push force that overcomes the friction force, the table begins to move. The maximum force at which the body starts to move and then comes to rest is determined by – the static and kinetic coefficients of friction of the body, respectively.

When two bodies are at rest, then friction between their surfaces is called ‘static friction‘. Whereas, when two bodies relatively move, friction between their surfaces is ‘kinetic friction’, also called “sliding friction‘. As per definition, the displacement due to static friction is zero. Therefore, the work is only done by sliding friction.

The work done depends on the type of energy conversion of the bodies when a force is applied. That means, when we apply force on the body at rest, the conversion of potential to kinetic energy occurs – which accelerates the body to move in the direction of applied force. Likewise, when the surface exerts sliding friction on the moving body, its kinetic energy is again converted into potential energy, which deaccelerates its motion.

The friction converts its work into heat energy as we feel some heat on the surface when bodies slide. When the moving body comes to rest on the horizontal surface, its kinetic energy becomes zero. That means the friction on the body dissipates its kinetic energy, which is estimated as the amount of work done by friction.

## How to Calculate Work Done by Friction?

We can determine the work done by friction through the detailed analysis of the friction force.

To calculate the work done, first, we have to determine the non-conservative force from the surface, the total path length on the surface or displacement, and, more importantly, the angle between friction force and displacement.

It is crucial to identify the force, whether it is a conservative or non-conservative. That is how we will understand that the force will change the body’s total mechanical energy (kinetic + potential) when it does any work. Since friction force acts oppositely to the moving body, it is a non-conservative force that changes the total mechanical energy, involves kinetic to potential energy conversion to resist the motion.

When a net force acts on the body, it changes the kinetic energy.

Work-Energy Theorem says that the work done on the body by a net force equals the difference between their kinetic energy.

[latex]W_{net} = \Delta E_{k} = E_{kf} – E_{ki}[/latex]

If the body gained energy, its work done is positive. If the body lose energy, its work done is negative

The friction force is the only net force acting on a horizontal surface equals the coefficient of kinetic friction μk and normal force N.

Whereas, the normal force perpendicular to the horizontal surface is mgcosθ.

Therefore, the friction force is,

Ffric = μkmg ……………. (*)

## Work Done by Friction Formula

The work done by friction formula is obtained using net forces and also a work-energy theorem.

The normal force and gravitational force, which act perpendicularly, cancel each other since they are opposite. Therefore, the horizontal friction force Ffric is the only net force acting on the body to get work done.

Let’s calculate the net work done by friction force on moving box by a displacement d along a horizontal path.

Wfric = Ffri dcosθ

Substituting equation of friction force (*), we get

Wfric = μkmg.dcosθ

Since there is sliding friction, the angle between displacement and sliding friction is 180°; which gives cosθ = cos180° = -1.

Wfric = – μkmg.d

The above is the equation of work done by friction.

## Is Work Done by Friction Always Negative?

The work done by friction is typically negative because of the 180° angle between friction and displacement.

When we apply force along the rough surface, the friction is employed in the opposite direction to applied force. Hence, the angle between friction and displacement becomes 180° which decreases the kinetic energy; and gives negative work done by friction force.

## Work Done by Friction in Pure Rolling

The work done by friction without sliding is zero in pure rolling.

When a body is in pure rolling with another body, the static force employs perpendicular to the horizontal surface. The rolling body does not undergo the energy conversion as the static force cannot make its displacement. Hence, static friction in pure rolling does not do any work.

The friction force is self-adjusting. It keeps its direction as per the direction of applied force to resist the motion. In pure rolling, the bottom part of the body comes into contact with the ground for a short time, elevating the body up perpendicularly to the direction of friction force. i.e., static friction.

Pure rolling means less translation and more rolling. The static friction does negative work done in translation motion – which deaccelerating translation, and equally positive work done in rotational motion – which means accelerating the rotation. That’s why the net work done by static friction is zero in pure rolling Therefore, to do any work, the body needs to roll with sliding.

## Work Done by Friction is Positive or Negative?

The work done by friction only be positive, negative depending upon choosing the frame of reference.

Since the applied force and the body’s displacement are in the same direction; it increases its energy. So the work done is positive. But the kinetic friction and the body’s displacement are in the opposite direction; it decreases its energy. So the work done is negative.

If the body is sliding, the friction force acting on the body would be sliding friction, and the angle between kinetic friction and displacement is 180° which gives rise to negative work done.

The work done by friction can appear positive if we change a frame of reference since kinetic friction can appear in the direction of the body’s motion. Let’s say a heavy box is put on a rug, and it is picked suddenly. Even the box slides behind but travels forwards relatively with a frame of reference. In such a case, the work done by friction is positive.

## How to find Work Done by Friction without Coefficient?

The work done by friction without coefficient is obtained by conducting an inclined plane experiment.

Let’s set up an inclined ramp first and adjust its inclined angle. The more the inclined angle, the more applied force pushes the object on the ramp down. When we increase the applied force than friction force, we can find the maximum friction force and measure the object’s displacement and the work done by friction without knowing its coefficient.