Definition of Work
Mechanical Work
Work: scalar quantity. Defined as force applied over displacement in the direction of force. Units: joule (J).
Mathematical Expression
Work done, W = F · d · cos(θ), where F = force magnitude, d = displacement magnitude, θ = angle between force and displacement vectors.
Sign of Work
Positive work: force component and displacement in same direction. Negative work: opposite directions. Zero work: force perpendicular to displacement.
Types of Work
Positive Work
Force aids displacement. Example: lifting an object upwards.
Negative Work
Force opposes displacement. Example: friction slowing a moving object.
Zero Work
Force perpendicular to displacement. Example: centripetal force in uniform circular motion.
Variable Force Work
Work done by force varying with position calculated by integral W = ∫ F·dx.
Energy Concepts
Definition of Energy
Energy: capacity to perform work. Scalar quantity, unit: joule.
Forms of Mechanical Energy
Two primary forms: kinetic energy (energy of motion) and potential energy (energy of position/configuration).
Energy Transformation
Energy converts between kinetic and potential during motion, maintaining total mechanical energy in ideal systems.
Kinetic Energy
Definition
Kinetic energy (KE): energy possessed due to motion. Dependent on mass and velocity.
Formula
KE = ½ m v²Properties
Scalar quantity. Always positive or zero. Increases with velocity squared. Basis for work-energy theorem.
Potential Energy
Definition
Potential energy (PE): energy stored due to position or configuration in a force field.
Gravitational Potential Energy
PE = m g hm = mass, g = acceleration due to gravity, h = height above reference point.
Elastic Potential Energy
PE = ½ k x²k = spring constant, x = displacement from equilibrium.
Work-Energy Theorem
Statement
Net work done on an object equals change in its kinetic energy.
Mathematical Formulation
W_net = ΔKE = KE_final - KE_initialImplications
Relates forces and motion via energy. Simplifies dynamics analysis. Valid for any net force.
Conservative and Non-conservative Forces
Conservative Forces
Work independent of path. Examples: gravity, spring force. Potential energy definable.
Non-conservative Forces
Work depends on path. Examples: friction, air resistance. Dissipate mechanical energy as heat.
Energy Implications
Conservative forces conserve mechanical energy. Non-conservative forces reduce total mechanical energy.
Power and Efficiency
Definition of Power
Power: rate of doing work or energy transfer. Unit: watt (W).
Formula
P = W / tW = work done, t = time interval.
Efficiency
Efficiency = (useful output work / input work) × 100%. Indicates energy conversion effectiveness.
Work Calculation Methods
Constant Force
Work calculated using scalar product W = F d cos θ.
Variable Force
Work via integral: W = ∫ F · dx along displacement path.
Graphical Interpretation
Area under force vs displacement curve equals work done.
| Force Type | Work Calculation |
|---|---|
| Constant Force | W = F d cosθ |
| Variable Force | W = ∫ F dx |
Energy Conservation Principles
Law of Conservation of Energy
Total energy in isolated system constant. Energy cannot be created or destroyed, only transformed.
Mechanical Energy Conservation
In absence of non-conservative forces: KE + PE = constant.
Energy Losses
Non-conservative forces cause conversion to thermal/internal energy, reducing mechanical energy.
| System Type | Energy Behavior |
|---|---|
| Conservative system | ME conserved, KE and PE interconvert |
| Non-conservative system | ME decreases, energy dissipated as heat |
Applications of Work Energy
Engineering Mechanics
Design of machines using work and energy for efficiency optimization.
Projectile Motion
Energy methods simplify calculations of velocities and heights.
Roller Coasters and Amusement Rides
Energy conservation principles predict speeds, forces, and safety margins.
Biomechanics
Work-energy analysis explains muscle effort and movement energetics.
Renewable Energy Systems
Optimization of power output using work and energy concepts.
Problems and Examples
Example 1: Work Done by Constant Force
A 10 N force moves an object 5 m at 30° angle. Calculate work done.
W = F d cosθ = 10 × 5 × cos30° = 10 × 5 × 0.866 = 43.3 JExample 2: Kinetic Energy Change
Mass 2 kg accelerates from 3 m/s to 7 m/s. Find change in kinetic energy.
ΔKE = ½ m (v₂² - v₁²) = 0.5 × 2 × (49 - 9) = 40 JExample 3: Potential Energy in Spring
Spring constant k = 200 N/m compressed 0.1 m. Calculate stored energy.
PE = ½ k x² = 0.5 × 200 × 0.01 = 1 JExample 4: Power Output
Work done 500 J in 10 seconds. Find power.
P = W / t = 500 / 10 = 50 WExample 5: Energy Conservation
Object dropped from 20 m. Find speed at 5 m above ground neglecting friction.
mgh_initial = mgh_final + ½ m v²=> v = sqrt[2g(h_initial - h_final)] = sqrt[2 × 9.8 × (20 -5)] = 17.15 m/sReferences
- Halliday, D., Resnick, R., & Walker, J. Fundamentals of Physics. Wiley, Vol. 1, 2017, pp. 140-180.
- Tipler, P. A., & Mosca, G. Physics for Scientists and Engineers. W. H. Freeman, Vol. 1, 2007, pp. 120-165.
- Serway, R. A., & Jewett, J. W. Physics for Scientists and Engineers with Modern Physics. Cengage Learning, Vol. 1, 2014, pp. 160-210.
- Young, H. D., Freedman, R. A. University Physics with Modern Physics. Pearson, 14th Edition, 2015, pp. 180-230.
- Giancoli, D. C. Physics: Principles with Applications. Pearson, 7th Edition, 2013, pp. 150-195.