Formula for Work Done by Friction: W = −μmgd, with Examples

The formula for work done by friction on a sliding body is $W_f = -\mu_k N d$ : the kinetic friction force $\mu_k N$ multiplied by the distance d, with a minus sign because friction acts against the motion. On a level floor the normal force equals the weight, so the formula becomes $W_f = -\mu_k m g d$ .

The minus sign means friction removes mechanical energy from the body and turns it into thermal energy in the surfaces. A box shoved across a floor slows down for exactly this reason. This page shows how to find that work, how to use it in energy problems, and how it links to the rest of the work and energy formulas.

Quick facts

Formula
$W_f = -\mu_k N d$
Level ground
$W_f = -\mu_k mgd$
On an incline
$W_f = -\mu_k mg\cos\theta\, d$
SI unit
joule (J)
Dimensional formula
[M¹ L² T⁻²]
Sign
Negative on the sliding body
Path
Depends on path length, so friction is non conservative

The formula for work done by friction, symbol by symbol

What each part of W = −μₖNd means
Symbol	Meaning	Unit
W_f	work done on the body by friction	J
μₖ	coefficient of kinetic friction, a ratio for the two surfaces	no unit
N	normal (perpendicular) contact force	N
d	distance the body slides, measured along the path	m

The formula comes from $W = Fd\cos\theta$ with $F = f_k = \mu_k N$ and $\theta = 180^\circ$ , since friction points backwards. The cosine of 180° is −1, which is where the minus sign comes from. The friction force page and the coefficient of kinetic friction page cover the force itself.

Step by step method

Sketch the body and work out N. With no extra vertical pushes on flat ground, N = mg. On a slope at angle θ, N = mg cos θ.
Find the kinetic friction force, $f_k = \mu_k N$ .
Multiply by the distance slid. Use the full path length, not the displacement, if the body changes direction.
Attach the minus sign for the work done on the body.
Put it into the energy equation: $KE_i + PE_i + W_f = KE_f + PE_f$ .

A box pushed forward 4 m and back 4 m ends where it started, but friction has done work over 8 m. That is why friction work depends on the path.

Worked examples

Example 1: a box on a floor

A 5.0 kg box slides 3.0 m over a flat floor, with μₖ = 0.20 between box and floor. Find the work done by friction.

Given: m = 5.0 kg, μₖ = 0.20, d = 3.0 m, g = 9.8 m/s². Normal force N = (5.0)(9.8) = 49 N.

$W_f = -(0.20)(49)(3.0) = -29.4\ \text{J}$

Answer: about −29 J. The box loses 29 J of kinetic energy to heat.

Example 2: stopping distance of a hockey puck

A 170 g puck is struck and glides off at 6.0 m/s on ice with μₖ = 0.10. How far does it slide before stopping?

Convert: m = 0.170 kg. All its kinetic energy is removed by friction: $\frac{1}{2}mv^2 = \mu_k m g d$ .

The mass cancels: $d = \frac{v^2}{2\mu_k g} = \frac{(6.0)^2}{2(0.10)(9.8)} = \frac{36}{1.96} = 18.4\ \text{m}$ .

Friction work: $W_f = -\tfrac{1}{2}(0.170)(6.0)^2 = -3.06\ \text{J}$ . Answer: the puck slides about 18 m, and friction does about −3.1 J of work.

Example 3: a crate on a slope

A 12 kg crate slides 2.5 m down a ramp at 30° with μₖ = 0.25. Find the work done by friction.

$N = mg\cos 30^\circ = (12)(9.8)(0.866) = 101.8\ \text{N}$

$W_f = -(0.25)(101.8)(2.5) = -63.6\ \text{J}$ . Answer: about −64 J.

Sign, units and special cases

The work done by friction on the sliding body is negative. The total work done by kinetic friction on both surfaces together is also negative, and its size equals the thermal energy produced, $Q = \mu_k N d$ , where d is the distance one surface slides over the other. The SI unit is the joule and the dimensions are [M¹ L² T⁻²], like all work. Static friction is different: when there is no slipping, it can do zero work on a body, as with the tyres of a car rolling without skidding. A conveyor belt that carries a box forward is a case where friction does positive work on the box.

Common mistakes

Using mg as the normal force on a ramp. Tilted surfaces give N = mg cos θ, and an extra push or pull at an angle also changes N.
Using the static coefficient for a sliding body. Once it moves, use μₖ.
Forgetting the sign. Friction work added to the energy balance as a positive number makes the final speed too large.
Using displacement for a back and forth path. Use the total distance.

For the full energy picture, see the work energy theorem and kinetic energy.

Frequently asked questions

What is the formula for work done by friction?

W_f = −μₖNd, the kinetic friction force times the distance slid, with a minus sign. On a level surface, N = mg, so W_f = −μₖmgd.

Why is work done by friction negative?

Friction acts opposite to the direction of sliding, so the angle between force and displacement is 180° and cos 180° = −1. Negative work means the body loses kinetic energy.

Is work done by friction path dependent?

Yes. It depends on the total distance slid, so a longer route between the same two points means more energy lost. That makes friction a non conservative force.

Where does the energy removed by friction go?

It becomes thermal energy in the body and the surface, with a little sound. The total energy is still conserved.

Can friction ever do positive work?

Yes. Friction from a moving conveyor belt or the floor of an accelerating truck can push an object forward and do positive work on it.