Definition of Work
Mechanical Work
Work: energy transfer via force acting through displacement. Requires force component along displacement vector. Scalar quantity. Sign indicates energy direction.
Physical Interpretation
Work quantifies how force changes kinetic or potential energy in a system. No displacement or perpendicular force yields zero work.
Historical Context
Concept developed in 19th century physics by Coriolis and others. Foundation for energy conservation principles.
Mathematical Formula
Basic Equation
Work (W) = Force (F) × Displacement (d) × cos(θ), where θ is angle between force and displacement vectors.
Vector Representation
W = \(\vec{F} \cdot \vec{d}\) (dot product). Only component of force parallel to displacement contributes.
Formula Explanation
Cosine factor projects force onto displacement axis. Zero work if force perpendicular (θ = 90°).
W = F d cos(θ)where:F = magnitude of force (N)d = magnitude of displacement (m)θ = angle between force and displacementUnits of Work
SI Unit
Joule (J): 1 J = 1 Newton × 1 meter = 1 N·m.
Derived Units
1 J = 1 kg·m²/s². Work shares units with energy.
Other Units
Erg (CGS): 1 erg = 10⁻⁷ J. Foot-pound (Imperial): 1 ft·lb ≈ 1.356 J.
| Unit | Symbol | Equivalent in Joules |
|---|---|---|
| Joule | J | 1 J |
| Erg | erg | 1 × 10⁻⁷ J |
| Foot-pound | ft·lb | 1.356 J |
Positive and Negative Work
Positive Work
Force component and displacement same direction (0° ≤ θ < 90°). Energy added to system.
Negative Work
Force opposes displacement (90° < θ ≤ 180°). Energy removed from system.
Zero Work
Force perpendicular to displacement (θ = 90°) or no displacement. No energy transfer.
If:θ < 90°, W > 0 (energy input)θ = 90°, W = 0 (no work)θ > 90°, W < 0 (energy output)Work Done by Variable Force
Non-constant Force
Force magnitude/direction changes during displacement. Requires calculus integration.
Integral Form
W = ∫ \(\vec{F} \cdot d\vec{r}\) over path from initial to final position.
Application
Used for springs, friction, gravitational fields with variable force.
W = ∫ from r₁ to r₂ F(r) · drwhere:F(r) = position-dependent force vectordr = infinitesimal displacement vectorWork-Energy Theorem
Theorem Statement
Net work done on an object equals change in its kinetic energy: W_net = ΔK.
Mathematical Expression
W_net = K_final − K_initial = ½ m v_f² − ½ m v_i².
Significance
Connects force-displacement analysis to energy perspective. Foundation for dynamics.
Conservative and Non-Conservative Forces
Conservative Forces
Work independent of path. Examples: gravity, spring force. Potential energy defined.
Non-Conservative Forces
Work depends on path. Examples: friction, air resistance. Energy dissipated as heat.
Energy Conservation
Conservative forces conserve mechanical energy. Non-conservative forces cause energy loss.
| Force Type | Work Dependence | Energy Implication |
|---|---|---|
| Conservative | Path-independent | Mechanical energy conserved |
| Non-Conservative | Path-dependent | Energy dissipated |
Power and Its Relation to Work
Power Definition
Power: rate of doing work. P = dW/dt.
Average and Instantaneous Power
Average: W/Δt. Instantaneous: derivative of work with respect to time.
Units
Watt (W): 1 W = 1 J/s. Common for engines and machines.
P = dW/dtUnits: Watt (W) = Joule/second (J/s)Work in Rotational Motion
Rotational Work Formula
W = τ θ, where τ is torque, θ angular displacement (radians).
Relation to Angular Variables
Torque analogous to force; angular displacement analogous to linear displacement.
Units
Joule is unit for rotational work. Torque in N·m, angular displacement in radians.
W = τ θwhere:τ = torque (N·m)θ = angular displacement (rad)Worked Examples
Example 1: Horizontal Force
Force 10 N applied horizontally displacing object 5 m. θ=0°.
W = 10 × 5 × cos 0° = 50 J.
Example 2: Inclined Force
Force 20 N at 60° to displacement 3 m.
W = 20 × 3 × cos 60° = 20 × 3 × 0.5 = 30 J.
Example 3: Variable Force (Spring)
Spring constant k=200 N/m, compressed 0.1 m.
W = ½ k x² = 0.5 × 200 × (0.1)² = 1 J.
Common Misconceptions
Work and Force Always Positive
Incorrect: work can be negative or zero depending on force direction.
Work Done Without Displacement
Incorrect: no displacement means no work done regardless of force magnitude.
Work is a Vector Quantity
Incorrect: work is scalar; direction encoded in sign but no vector direction.
References
- Halliday, D., Resnick, R., & Walker, J., Fundamentals of Physics, Wiley, 10th ed., 2013, pp. 120-145.
- Tipler, P. A., & Mosca, G., Physics for Scientists and Engineers, W. H. Freeman, 6th ed., 2007, pp. 200-225.
- Serway, R. A., & Jewett, J. W., Physics for Scientists and Engineers, Cengage Learning, 9th ed., 2014, pp. 150-180.
- Giancoli, D. C., Physics: Principles with Applications, Pearson, 7th ed., 2013, pp. 130-160.
- Young, H. D., & Freedman, R. A., University Physics with Modern Physics, Pearson, 14th ed., 2015, pp. 160-190.