Introduction
In physics, forces can be broadly classified as conservative and non-conservative. Conservative forces, such as gravity and spring force, have the special property that the work they do is independent of the path taken and depends only on the initial and final positions of an object.
Non-conservative forces, on the other hand, behave differently. The work done by these forces depends on the path, and they often lead to dissipation of mechanical energy. Examples include friction, air resistance, and tension in ropes during inelastic collisions.
Understanding the work done by non-conservative forces is essential in physics because it explains energy loss, heat generation, and reduced mechanical efficiency in real-world systems. This article explores the concept in detail, with formulas, derivations, numerical examples, and applications.
Definition of Non-Conservative Forces
A non-conservative force is a force for which the work done depends on the path taken between two points and not just on the initial and final positions.
Key characteristics:
- Work depends on the path.
- Mechanical energy is not conserved in the presence of non-conservative forces.
- Energy is dissipated, usually as heat, sound, or deformation.
Examples:
- Frictional forces (sliding, rolling, kinetic friction)
- Air resistance (drag)
- Viscous forces in fluids
- Tension in ropes during inelastic collisions
- Applied forces that cause deformation
Work Done – Recap
In general, the work WWW done by a force F⃗\vec{F}F over a displacement d⃗\vec{d}d is: W=F⃗⋅d⃗=FdcosθW = \vec{F} \cdot \vec{d} = F d \cos \thetaW=F⋅d=Fdcosθ
Where θ\thetaθ is the angle between the force and displacement vectors.
For non-conservative forces, the magnitude and direction of force may vary along the path. The total work done is then calculated as: W=∫pathF⃗⋅ds⃗W = \int_{\text{path}} \vec{F} \cdot d\vec{s}W=∫pathF⋅ds
Here, the integral emphasizes that work depends on the specific path taken, unlike conservative forces.
Non-Conservative Forces vs Conservative Forces
| Feature | Conservative Forces | Non-Conservative Forces |
|---|---|---|
| Work depends on | Only initial & final positions | Entire path taken |
| Energy effect | Mechanical energy conserved | Mechanical energy decreases |
| Examples | Gravity, spring force | Friction, air resistance |
| Work in closed loop | Zero | Non-zero |
| Energy transformation | KE ↔ PE | KE → heat, sound, deformation |
Mathematical Representation
For a system with both conservative and non-conservative forces, the work-energy theorem can be written as: Wtotal=ΔKEW_{\text{total}} = \Delta KEWtotal=ΔKE
Since total work includes contributions from both types of forces: Wtotal=Wconservative+Wnon-conservativeW_{\text{total}} = W_{\text{conservative}} + W_{\text{non-conservative}}Wtotal=Wconservative+Wnon-conservative
- Work done by conservative forces = change in potential energy:
Wconservative=−ΔPEW_{\text{conservative}} = -\Delta PEWconservative=−ΔPE
- Therefore, work done by non-conservative forces is:
Wnon-conservative=ΔKE+ΔPEW_{\text{non-conservative}} = \Delta KE + \Delta PEWnon-conservative=ΔKE+ΔPE
This is a key formula that shows the mechanical energy lost or gained due to non-conservative forces.
Energy Dissipation Due to Non-Conservative Forces
Non-conservative forces transform mechanical energy into other forms, usually:
- Heat – friction between surfaces converts KE into thermal energy.
- Sound – collisions or rubbing generate vibrations.
- Deformation – stretching or compression leads to potential energy in non-elastic deformation.
ΔEdissipated=∣Wnon-conservative∣\Delta E_{\text{dissipated}} = |W_{\text{non-conservative}}|ΔEdissipated=∣Wnon-conservative∣
This is crucial for understanding energy efficiency in machines, vehicles, and daily processes.
Work Done by Friction – A Detailed Example
Consider a block sliding on a rough horizontal surface:
- Mass = mmm
- Coefficient of kinetic friction = μk\mu_kμk
- Displacement = ddd
Frictional force: Ffr=μkN=μkmgF_{\text{fr}} = \mu_k N = \mu_k mgFfr=μkN=μkmg
Work done by friction: Wfr=−Ffr⋅d=−μkmgdW_{\text{fr}} = – F_{\text{fr}} \cdot d = – \mu_k mg dWfr=−Ffr⋅d=−μkmgd
- The negative sign indicates that friction removes energy from the system.
- This energy is dissipated as heat.
Work Done by Air Resistance
Air resistance is another non-conservative force. For a body moving through air: Fdrag=12CdρAv2F_{\text{drag}} = \frac{1}{2} C_d \rho A v^2Fdrag=21CdρAv2
Where:
- CdC_dCd = drag coefficient
- ρ\rhoρ = air density
- AAA = cross-sectional area
- vvv = velocity
Work done by air resistance: Wair=−∫Fdrag dxW_{\text{air}} = – \int F_{\text{drag}} \, dxWair=−∫Fdragdx
- Opposes motion → negative work.
- Converts kinetic energy into heat and turbulence in air.
Numerical Examples
Example 1: Sliding Block
A 5 kg block slides 4 m on a horizontal surface (μk=0.2\mu_k = 0.2μk=0.2). Calculate work done by friction.
- Normal force: N=mg=5×9.8=49 NN = mg = 5 \times 9.8 = 49 \, NN=mg=5×9.8=49N
- Friction force: Ffr=0.2×49=9.8 NF_{\text{fr}} = 0.2 \times 49 = 9.8 \, NFfr=0.2×49=9.8N
- Work: W=−Ffr⋅d=−9.8×4=−39.2 JW = – F_{\text{fr}} \cdot d = – 9.8 \times 4 = -39.2 \, JW=−Ffr⋅d=−9.8×4=−39.2J
Work done by friction = –39.2 J
Example 2: Car Braking
A 1200 kg car moving at 15 m/s is brought to rest by brakes over 50 m. Find average frictional force.
- Initial KE: KE=12mv2=0.5×1200×225=135,000 JKE = \tfrac{1}{2}mv^2 = 0.5 \times 1200 \times 225 = 135,000 \, JKE=21mv2=0.5×1200×225=135,000J
- Work done by friction: W=−135,000 JW = -135,000 \, JW=−135,000J
- Force: F=W/d=135,000/50=2700 NF = W/d = 135,000/50 = 2700 \, NF=W/d=135,000/50=2700N
✅ Average frictional force = 2700 N
Example 3: Energy Loss Due to Air Resistance
A ball moving at 20 m/s through air for 10 m experiences average drag force 2 N. Work done by air resistance: W=−F⋅d=−2×10=−20 JW = – F \cdot d = – 2 \times 10 = -20 \, JW=−F⋅d=−2×10=−20J
Energy dissipated = 20 J
Applications of Work by Non-Conservative Forces
- Engineering & Machines
- Friction in engines, gears, and brakes converts mechanical energy to heat.
- Efficiency analysis requires considering energy lost.
- Transportation
- Air resistance affects cars, planes, and rockets.
- Fuel consumption depends on work against non-conservative forces.
- Sports
- Ball slowing on grass → friction does negative work.
- Cyclists fighting air resistance → energy expenditure.
- Daily Life
- Walking, sliding objects, opening doors – energy dissipates via friction.
- Heating by rubbing hands or matchsticks.
- Industrial Processes
- Grinding, polishing, and machining involve non-conservative work → heat production.
Work Done in Non-Conservative Forces in Oscillations
For damped harmonic motion:
- Air resistance or internal damping causes energy loss.
- Amplitude decreases over time.
- Mechanical energy converted to heat → Wdissipated=ΔKE+ΔPEW_{\text{dissipated}} = \Delta KE + \Delta PEWdissipated=ΔKE+ΔPE
Graphical Interpretation
- Force vs Displacement Graph
- Area under curve = work done by non-conservative force.
- Mechanical Energy vs Time
- Shows gradual decrease in energy due to dissipation.
Misconceptions
- All forces do work → ❌ Not true; only force with component along displacement does work.
- Mechanical energy is always conserved → ❌ Not true in presence of non-conservative forces.
- Energy disappears → ❌ Energy transforms into heat, sound, or deformation.
Advanced Concepts
1. Work in Rotational Systems
- Torque from friction or air resistance:
W=τ⋅θW = \tau \cdot \thetaW=τ⋅θ
- Reduces rotational KE, converted to heat.
2. Non-Conservative Forces in Vehicles
- Drag, rolling resistance, and engine friction determine fuel efficiency.
- Work done = energy lost per distance traveled.
3. Path Dependence
- Non-conservative work is path-dependent: longer path → more energy lost.
4. Relation to Thermodynamics
- Work by non-conservative forces increases internal energy → temperature rise.
- Bridges mechanics and heat transfer.
Summary Table
| Feature | Non-Conservative Forces |
|---|---|
| Work | Path-dependent |
| Energy | Mechanical energy decreases |
| Examples | Friction, air resistance, viscous drag |
| Work Formula | Wnc=ΔKE+ΔPEW_{\text{nc}} = \Delta KE + \Delta PEWnc=ΔKE+ΔPE |
| Energy Transformation | KE → Heat, sound, deformation |
| Application | Brakes, engines, damping, sports |
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