Work, Energy & Power
Understand the definitions and calculations of work, mechanical energy (kinetic and potential), the work-energy theorem, conservative vs. non-conservative forces, power, and collisions.
1. Work: Scientific Definition & Types
Work is defined as the product of the component of the force in the direction of the displacement and the magnitude of this displacement. Mathematically:
W = F · s · cos(θ)
Where θ is the angle between the Force vector (F) and the Displacement vector (s).
| Type of Work |
Condition (Angle θ) |
Physical Meaning |
Example |
| Positive Work |
0° ≤ θ < 90° (cos θ is positive) |
Force assists the displacement |
A horse pulling a cart, stretching a spring |
| Negative Work |
90° < θ ≤ 180° (cos θ is negative) |
Force opposes the displacement |
Frictional force on a sliding object, brakes stopping a car |
| Zero Work |
θ = 90° (cos θ = 0) |
Force is perpendicular to displacement |
A coolie walking horizontally with a load on his head |
2. Kinetic and Potential Energy
Energy is the quantitative property that must be transferred to a body to perform work on it. Mechanical energy consists of two main types:
| Energy Type |
Definition |
Formula |
Key Exam Characteristics |
| Kinetic Energy (KE) |
Energy possessed by an object due to its motion. |
KE = ½mv² = p² / 2m |
Always positive; depends on frame of reference; related to momentum (p). |
| Potential Energy (PE) |
Energy stored in an object due to its position or configuration. |
PE = mgh (gravitational)
PE = ½kx² (spring) |
Can be positive, negative, or zero; defined only for conservative forces. |
Relation between Kinetic Energy and Momentum:
If momentum (p) of a body is doubled, its kinetic energy increases by 4 times (since KE ∝ p² for constant mass).
3. Work-Energy Theorem & Conservation of Energy
Work-Energy Theorem: The work done by the net force acting on a particle is equal to the change in its kinetic energy.
Wnet = KEf - KEi = ΔKE
This holds true for both constant and variable forces, as well as conservative and non-conservative forces.
Law of Conservation of Energy: Energy can neither be created nor destroyed; it can only be transformed from one form to another. The total energy of an isolated system remains constant.
For a conservative system (like a free-falling body under gravity):
Total Mechanical Energy (E) = KE + PE = Constant
4. Power & Commercial Units of Energy
Power: The rate at which work is done or energy is transferred. Mathematically:
P = W / t = F · v
Where F is the applied force and v is the instantaneous velocity.
- SI Unit: Watt (W) = 1 Joule/second.
- Dimensional Formula: [ML²T⁻³]
- 1 Horsepower (HP) = 746 Watts.
Commercial Unit of Energy:
The standard commercial unit of electrical energy is the Kilowatt-hour (kWh), commonly called a 'Unit'.
1 kWh = 1000 W × 3600 s = 3.6 × 106 J = 3.6 MJ
5. Collisions & Restitution
A collision is a short-duration interaction between two bodies or particles, altering their momentum and energy. Collisions are broadly classified as:
- Elastic Collision: Both total linear momentum and total kinetic energy are conserved. The colliding bodies separate after impact (e.g., collisions between gas molecules).
- Inelastic Collision: Linear momentum is conserved, but kinetic energy is not conserved (it converts into heat, sound, etc.).
- Perfectly Inelastic Collision: The colliding bodies stick together and move with a common velocity after impact (maximum kinetic energy loss).
Coefficient of Restitution (e):
It is the ratio of relative velocity of separation to relative velocity of approach.
e = (v₂ - v₁) / (u₁ - u₂)
- For perfectly elastic collision: e = 1
- For perfectly inelastic collision: e = 0
- For real-world inelastic collision: 0 < e < 1
Historical Evolution of Energy & Work
- 1686 — Concept of Vis Viva: Gottfried Leibniz proposed 'vis viva' (living force), the early formulation of kinetic energy as mv².
- 1807 — Term 'Energy' Introduced: Thomas Young first used the term 'energy' in its modern scientific sense, replacing older colloquial terms.
- 1843 — Mechanical Equivalent of Heat: James Prescott Joule experimentally demonstrated the conservation of energy and quantified the mechanical equivalent of heat.
- 1905 — Mass-Energy Equivalence: Albert Einstein published E = mc², demonstrating that mass and energy are interchangeable and unified.
Key Questions & Answers
- What is the work done when force and displacement are perpendicular?
- **Zero** (W = Fs cos 90° = 0). Example: A satellite orbiting Earth.
- State the Work-Energy Theorem.
- The net work done by all forces on a body equals the **change in its kinetic energy** (W_net = ΔKE).
- What is the difference between conservative and non-conservative forces?
- **Conservative forces**: Work done is path-independent (depends only on initial/final points; e.g., gravity). **Non-conservative forces**: Work done depends on path (e.g., friction).
- Define 1 Horsepower (HP) in Watts.
- **1 HP = 746 Watts** (standard unit used in machinery).
Memory Aids
- Mnemonic 1: Types of Work (Angle Trick): Angle between Force and Displacement determines the work: • **P (Positive)**: θ • **N (Negative)**: θ > 90° (Obtuse angle, e.g., frictional force). • **Z (Zero)**: θ = 90° (Perpendicular, e.g., coolie carrying load on head).
- Mnemonic 2: Conservative Forces: Path-independent (conservative) forces: • **G** — Gravitational Force • **E** — Electrostatic Force • **S** — Spring Force Note: Frictional force is non-conservative (path-dependent).
- Mnemonic 3: Commercial Unit of Energy conversion: Remember that **1 kWh** (1 Unit of electricity) = **3.6 × 10^6 Joules** (or 3.6 Megajoules).
Common Exam Traps
- Trap 1: Believing work is done when you hold a heavy object without moving. If displacement is zero, work done is mathematically zero, despite physiological fatigue.
- Trap 2: Confusing Power with Energy. Energy is the capacity to do work (measured in Joules), whereas Power is the rate of doing work (measured in Watts or J/s).
- Trap 3: Forgetting that potential energy is defined only for conservative forces. You cannot define a potential energy function for friction.
- Trap 4: Assuming kinetic energy is conserved in all collisions. Momentum is conserved in all collisions, but kinetic energy is conserved ONLY in perfectly elastic collisions.