Work Done by Friction

Introduction

Whenever you push a heavy box across the floor, ride a bicycle, or apply brakes in a car, you experience resistance. This resistance is due to friction. While friction is essential for walking, driving, and gripping objects, it also leads to energy loss.

In physics, this loss is explained through the work done by friction. Unlike conservative forces such as gravity or spring force, friction is a non-conservative force. It does negative work on moving objects and converts useful mechanical energy into heat, sound, and deformation.

In this article, we’ll explore:

• What friction is,
• How friction does work,
• Why energy dissipates due to friction,
• Mathematical expressions,
• Graphical interpretation,
• Examples, numerical problems, and real-life applications.

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What is Friction?

Friction is the resistive force that opposes the relative motion (or tendency of motion) between two surfaces in contact.

Types of friction:

1. Static Friction – Acts when objects are at rest, prevents motion.
2. Kinetic (Sliding) Friction – Acts when surfaces slide past each other.
3. Rolling Friction – Acts when objects roll on a surface (e.g., wheels).
4. Fluid Friction (Drag) – Resistance offered by fluids (air, water) to motion.

In the context of work done, kinetic friction is the most relevant because it directly resists motion.

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Work Done by a Force – Recap

Work is defined as: W=F⋅d⋅cos⁡θW = F \cdot d \cdot \cos \thetaW=F⋅d⋅cosθ

Where:

• FFF = applied force,
• ddd = displacement,
• θ\thetaθ = angle between force and displacement.
• If θ=0∘\theta = 0^\circθ=0∘: Positive work (force aids motion).
• If θ=180∘\theta = 180^\circθ=180∘: Negative work (force opposes motion).

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Work Done by Friction

Friction always acts opposite to displacement.

Thus, Wfriction=Ffr⋅d⋅cos⁡180∘W_{friction} = F_{fr} \cdot d \cdot \cos 180^\circWfriction​=Ffr​⋅d⋅cos180∘ Wfriction=−Ffr⋅dW_{friction} = – F_{fr} \cdot dWfriction​=−Ffr​⋅d

Where:

• Ffr=μkNF_{fr} = \mu_k NFfr​=μk​N (kinetic frictional force)
• μk\mu_kμk​ = coefficient of kinetic friction
• NNN = normal reaction force

So, Wfriction=−μkNdW_{friction} = – \mu_k N dWfriction​=−μk​Nd

👉 Negative sign indicates friction removes energy from the system.

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Nature of Work Done by Friction

1. Always Negative
   • Friction opposes displacement → extracts energy.
2. Path Dependent
   • Friction is non-conservative; work depends on path length, not just initial and final states.
3. Energy Dissipation
   • Work done by friction is converted into heat, sound, or microscopic deformations.

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Energy Dissipation Due to Friction

When a block slides across a rough surface:

• The applied force does positive work, increasing the block’s kinetic energy.
• Friction does negative work, removing energy.
• The lost energy is transformed into thermal energy (heat), sound, and wear.

Equation for Energy Dissipation

ΔEdiss=∣Wfriction∣=μkNd\Delta E_{diss} = |W_{friction}| = \mu_k N dΔEdiss​=∣Wfriction​∣=μk​Nd

This represents the energy lost from the system due to friction.

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Graphical Interpretation

1. Force vs. Displacement Graph
   • Friction force is constant for kinetic motion.
   • Work done = area under curve = Ffr⋅dF_{fr} \cdot dFfr​⋅d.
2. Kinetic Energy vs. Time
   • As an object slides, KE decreases linearly until it stops → energy dissipated.

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Real-Life Examples

1. Braking in Vehicles
   • Friction between brakes and wheels converts kinetic energy into heat.
2. Rubbing Hands Together
   • Friction generates heat.
3. Sliding a Box
   • Work done against friction warms the surfaces.
4. Shoes and Roads
   • Friction allows walking, but energy from muscles partly dissipates as heat.
5. Machines and Engines
   • Bearings and gears lose energy due to friction → efficiency drops.

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Numerical Examples

Example 1: Sliding Block

A 10 kg block slides 5 m on a horizontal surface with μk=0.2\mu_k = 0.2μk​=0.2. Find work done by friction.

• Normal force:

N=mg=10×9.8=98 NN = mg = 10 \times 9.8 = 98 \, NN=mg=10×9.8=98N

• Friction force:

Ffr=μkN=0.2×98=19.6 NF_{fr} = \mu_k N = 0.2 \times 98 = 19.6 \, NFfr​=μk​N=0.2×98=19.6N

• Work:

W=−Ffr⋅d=−19.6×5=−98 JW = -F_{fr} \cdot d = -19.6 \times 5 = -98 \, JW=−Ffr​⋅d=−19.6×5=−98J

👉 Work done by friction = –98 J.

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Example 2: Car Brakes

A 1000 kg car moving at 20 m/s is stopped by friction. If stopping distance = 50 m, find frictional force.

• Initial KE:

KE=12mv2=0.5×1000×400=200,000 JKE = \tfrac{1}{2}mv^2 = 0.5 \times 1000 \times 400 = 200,000 \, JKE=21​mv2=0.5×1000×400=200,000J

• Work by friction:

W=−Ffr⋅d=−200,000W = -F_{fr} \cdot d = -200,000W=−Ffr​⋅d=−200,000 Ffr=200,00050=4000 NF_{fr} = \frac{200,000}{50} = 4000 \, NFfr​=50200,000​=4000N

👉 Frictional force = 4000 N.

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Example 3: Energy Loss on Incline

A 2 kg block slides down 4 m incline (30∘30^\circ30∘, μk=0.1\mu_k = 0.1μk​=0.1). Find energy lost.

• Normal force:

N=mgcos⁡θ=2×9.8×cos⁡30∘=16.97 NN = mg \cos \theta = 2 \times 9.8 \times \cos 30^\circ = 16.97 \, NN=mgcosθ=2×9.8×cos30∘=16.97N

• Friction:

Ffr=μkN=0.1×16.97=1.697 NF_{fr} = \mu_k N = 0.1 \times 16.97 = 1.697 \, NFfr​=μk​N=0.1×16.97=1.697N

• Work done by friction:

W=−Ffr⋅d=−1.697×4=−6.79 JW = -F_{fr} \cdot d = -1.697 \times 4 = -6.79 \, JW=−Ffr​⋅d=−1.697×4=−6.79J

👉 Energy dissipated = 6.8 J.

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Applications of Work Done by Friction

1. Safety – Friction in brakes allows vehicles to stop.
2. Everyday Life – Walking, writing, gripping objects.
3. Industrial Use – Polishing, grinding, matchstick ignition.
4. Sports – Rubbing hands for warmth, ball control in football.
5. Engineering – Heat generated in machines → need lubrication.

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Reducing Energy Dissipation

Since friction reduces efficiency, engineers try to minimize energy loss by:

• Using lubricants (oil, grease).
• Using ball bearings (convert sliding friction to rolling friction).
• Streamlined designs (reduce air drag).
• Polished surfaces (reduce roughness).

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Misconceptions

1. Friction always bad → False. Friction is essential for walking, driving, gripping.
2. Friction does no work → False. Friction does negative work; energy converts into heat.
3. Energy lost forever → False. It transforms, mainly into heat, but cannot be destroyed.

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Advanced Concept – Microscopic Origin of Friction

At microscopic level:

• Surfaces have irregularities (asperities).
• When they rub, microscopic bonds form and break.
• This generates vibrations (heat, sound), explaining energy dissipation.

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Summary Table

Aspect	Work by Friction
Force Direction	Opposite to displacement
Formula	W=−μkNdW = -\mu_k N dW=−μk​Nd
Nature	Always negative
Energy	Dissipated as heat, sound, deformation
Type of Force	Non-conservative
Applications	Brakes, walking, writing, polishing
Prevention of Loss	Lubrication, ball bearings, streamlining