Every time you climb a staircase, kick a football, or turn on a light switch, you are using energy to do work or convert it from one form to another. The concepts of work, energy, and power are not merely physics definitions — they are the language in which the universe describes every physical process, from the falling of an apple to the nuclear fusion burning inside the Sun.
In physics, work is done only when a force causes displacement in its own direction. Energy is the capacity to do work — it can take many forms (kinetic, potential, thermal, electrical, chemical, nuclear) but can never be created or destroyed, only converted. Power is the rate at which work is done or energy is transferred. Together, these three concepts form the foundation of mechanics and thermodynamics.
For CBSE Class 9 Physics Chapter 11 (Work and Energy) and CBSE Class 11 Physics Chapter 6 (Work, Energy and Power), these topics appear in every board examination and are prerequisites for understanding thermodynamics, electricity, and modern physics. This guide covers all subtopics: work done formula and conditions, kinetic energy, potential energy, mechanical energy, the work-energy theorem, the law of conservation of energy, power, efficiency, and five CBSE exam-ready solved problems.
- 1. Work — Definition, Formula, Conditions
- 2. Energy — Definition, Types
- 3. Kinetic Energy — Formula and Derivation
- 4. Potential Energy — Gravitational and Elastic
- 5. Work-Energy Theorem
- 6. Law of Conservation of Energy
- 7. Power and Efficiency
- 8. Work, Energy, Power — Comparison Table
- 9. Solved Numerical Problems
- 10. Real-World Applications
- 11. Frequently Asked Questions (FAQ)
1. Work — Definition, Formula, Conditions
Variables: W = work done (Joules) | F = force applied (Newtons) | s = displacement (metres) | θ = angle between force and displacement
Unit: 1 Joule (J) = 1 Newton × 1 metre = 1 N·m
Definition of 1 Joule: One joule of work is done when a force of 1 N causes a displacement of 1 m in the direction of the force.
Three conditions for work to be done:
(1) A force must be applied to the body.
(2) The body must be displaced (move).
(3) The force must have a component in the direction of displacement (θ must not be 90°).
Special cases of angle θ:
• θ = 0° (force parallel to displacement): W = Fs — maximum positive work (e.g., pushing a box in the direction of motion)
• θ = 90° (force perpendicular to displacement): W = 0 — no work done (e.g., carrying a bag horizontally, gravity on circular orbit)
• θ = 180° (force opposite to displacement): W = −Fs — negative work done (e.g., friction opposes motion, braking force)
When Is Work NOT Done?
2. Energy — Definition and Types
Energy is defined as the capacity or ability to do work. An object that has energy can exert a force on another object and do work on it. Energy is a scalar quantity measured in Joules (J). Like work, it cannot be seen directly but its effects are observable.
3. Kinetic Energy — Formula and Derivation
Derivation using work-energy theorem:
Consider a body of mass m starting from rest (u = 0) and accelerated by a constant force F through a displacement s.
• By Newton’s second law: F = ma
• Work done: W = F × s = mas
• By kinematics: v² = u² + 2as ⇒ v² = 2as (since u = 0) ⇒ as = v²/2
• Substituting: W = m × (v²/2) = ½mv²
This work done = kinetic energy gained: KE = ½mv²
Key properties of KE:
• KE is always positive (or zero). It cannot be negative.
• KE depends on the square of velocity: doubling speed quadruples KE.
• KE is a scalar quantity (no direction).
• A body at rest has KE = 0.
• Relation with momentum p: KE = p²/2m (since p = mv, KE = ½mv² = m²v²/2m = p²/2m)
4. Potential Energy — Gravitational and Elastic
Derivation:
To lift a body of mass m to a height h, we must apply an upward force equal to its weight (F = mg) through a displacement h.
Work done against gravity: W = F × h = mg × h = mgh
This work done is stored as gravitational potential energy: PE = mgh
Variables: m = mass (kg) | g = acceleration due to gravity (9.8 m/s² ≈ 10 m/s²) | h = height above reference level (m)
Note: PE depends on the chosen reference level. It is the change in PE (ΔPE = mgΔh) that matters physically, not the absolute value.
When a spring is stretched or compressed by x from its natural length, it stores elastic PE = ½kx².
Examples: compressed spring, stretched rubber band, bow drawn before shooting arrow, car suspension spring.
5. Work-Energy Theorem
Derivation:
For a body of mass m with initial velocity u and final velocity v under constant net force F through displacement s:
• By kinematics: v² = u² + 2as ⇒ as = (v² − u²)/2
• Net work: W = Fs = mas = m(v² − u²)/2 = ½mv² − ½mu²
• W = KEᵓ − KEᵢ = ΔKE
Applications:
• If net work is positive (W > 0), KE increases (object speeds up).
• If net work is negative (W < 0), KE decreases (object slows down).
• If W = 0, KE is unchanged (constant speed, e.g., uniform circular motion).
• Used to find velocity after a known work is done: v = √(u² + 2W/m)
6. Law of Conservation of Energy
For mechanical energy: Total Mechanical Energy = KE + PE = constant (in absence of friction/non-conservative forces)
At any point: ½mv² + mgh = constant = Total Mechanical Energy (E)
📈 Free-Falling Ball — Energy Conservation at Every Point
This is why a pendulum bob swings back and forth indefinitely in the absence of air resistance — mechanical energy is continuously converted between kinetic (at lowest point) and potential (at highest point) while the total remains constant.
7. Power and Efficiency
Unit: 1 Watt (W) = 1 Joule per second (J/s) = 1 N·m/s
Commercial unit: 1 kilowatt-hour (kWh) = 1000 W × 3600 s = 3.6 × 10&sup6; J = 3.6 MJ (the unit of electrical energy in electricity bills).
Older unit: 1 horsepower (hp) = 746 W ≈ 0.746 kW
Power in terms of velocity: P = W/t = (F·s)/t = F·(s/t) = F·v (where v = s/t = average velocity)
Average power: Pᵃᵛᵉ = Total work done / Total time = Wᵗᵃᵗᵃᵈ / t
Instantaneous power: Pᵢᵗᴸᵗ = dW/dt = F·v (at that instant)
Also: η = (Useful Output Power / Total Input Power) × 100%
Example: A motor consumes 1000 W of electrical power and delivers 750 W of mechanical power. Efficiency = (750/1000) × 100 = 75%.
The remaining 25% (250 W) is lost as heat in the motor windings and friction in bearings.
8. Work, Energy and Power — Comparison Table
| Property | Work (W) | Energy (E) | Power (P) |
|---|---|---|---|
| Definition | Force × displacement (in direction of force) | Capacity to do work | Rate of doing work |
| Formula | W = Fs cosθ | KE = ½mv²; PE = mgh | P = W/t = Fv |
| SI Unit | Joule (J) | Joule (J) | Watt (W = J/s) |
| Type of quantity | Scalar | Scalar | Scalar |
| Can be negative? | Yes (opposing force) | KE: No. PE: Depends on reference | No (rate is positive) |
| Dimensional formula | [ML²T²] | [ML²T²] | [ML²T³] |
| Relation | W = P × t | E = W (work done = energy transferred) | P = E/t = W/t |
| Commercial unit | kilowatt-hour (kWh) | kilowatt-hour (kWh) | kilowatt (kW), horsepower (hp) |
9. Solved Numerical Problems
KE₂ = ½mv² = ½ × 1000 × 40² = 500 × 1600 = 800,000 J = 800 kJ
Note: Doubling the speed quadruples the KE (KE ∝ v²): 800 kJ = 4 × 200 kJ ✓
At the top: KE = 0 (momentarily at rest), PE = mgh
By conservation of energy: KEᵉᵓᵗᵗᵓᵖ = PEᵗᵓₚ
25 = mgh = 0.5 × 10 × h
h = 25/5
Power: P = W/t = 10,000/10 = 1,000 W = 1 kW
By Work-Energy Theorem: W = ΔKE = ½mv² − ½mu²
50 = ½ × 2 × v² − 0
50 = v²
v = √50 = 5√2
10. Real-World Applications
Explore Related Physics Guides & Lab Equipment
11. Frequently Asked Questions (FAQ)
The SI unit of both work and energy is the Joule (J), named after James Prescott Joule.
1 Joule is defined as: The amount of work done when a force of 1 Newton causes a displacement of 1 metre in the direction of the force.
1 J = 1 N·m = 1 kg·m²/s²
Other units of energy:
• 1 kilowatt-hour (kWh) = 3.6 × 10&sup6; J = 3.6 MJ (unit of electrical energy in bills)
• 1 calorie (cal) = 4.18 J (used in nutrition)
• 1 electron-volt (eV) = 1.6 × 10²³ J (used in atomic physics)
Work done by a force is negative when the force acts opposite to the direction of displacement (θ = 180°, cos 180° = −1), giving W = −Fs.
Examples of negative work:
(1) Friction on a moving body: A book slides across a table. Friction acts backward (opposite to motion), doing negative work: W = −f·s. This negative work reduces the book’s kinetic energy until it stops.
(2) Braking force on a car: Brakes apply a backward force on a forward-moving car. The braking force does negative work (W = −Fs), reducing the car’s KE from ½mv² to zero.
(3) Gravity on a rising ball: A ball thrown upward: gravity acts downward while displacement is upward. Wᵭᵅᵃᵗᵀ = −mgh (negative), which reduces the ball’s KE as it rises.
Law of Conservation of Energy: Energy can neither be created nor destroyed. It can only be converted from one form to another. The total energy of an isolated system remains constant.
Example — Free-falling ball: A ball of mass m held at height h has PE = mgh, KE = 0. Total mechanical energy = mgh.
As it falls, PE decreases and KE increases. At height h/2: PE = mgh/2, KE = mgh/2. Total = mgh ✓
Just before hitting the ground (height = 0): PE = 0, KE = mgh = ½mv² ⇒ v = √(2gh). Total = mgh ✓
At every point, KE + PE = mgh = constant. After hitting the ground, KE converts to heat and sound — energy is still conserved, just in different forms.
Practical consequence: No machine can ever deliver more energy than it receives (efficiency ≤ 100%). If a machine claims to produce more energy than it consumes, it violates conservation of energy and must be fraudulent.
Work-Energy Theorem: The net work done on a body equals the change in its kinetic energy.
Wᵗᵃᵗ = KEᵓᵢᵗᵃᵈ − KEᵢᵗᵢᵗᵢᵃᵈ = ½mv² − ½mu²
Derivation (brief): From Newton’s second law and kinematics: F = ma and v² = u² + 2as ⇒ W = Fs = mas = m(v² − u²)/2 = ΔKE.
Usefulness:
(1) Calculates final speed after a given work input without knowing acceleration: v = √(u² + 2W/m).
(2) Finds work done by individual forces (e.g., friction) by comparing energy changes.
(3) Explains stopping distance of vehicles: W = −fs (friction does negative work), ½mv² = fs ⇒ s = mv²/(2f). Doubling speed requires 4 times longer stopping distance — a critical road safety insight.
Work (W): Work is the product of force and displacement in the direction of force (W = Fs cosθ). It measures the total amount of energy transferred. Doing more work means more energy was used, but it says nothing about how quickly it was done. Unit: Joule (J).
Power (P): Power is the rate at which work is done (P = W/t). It measures how quickly work is done or energy is transferred. Two machines doing the same amount of work can have different powers if they take different times. Unit: Watt (W = J/s).
Example: A person carries a 50 kg box up a 10 m staircase in 20 seconds; a crane does the same in 5 seconds. Both do W = mgh = 50×10×10 = 5000 J of work. But the crane’s power P = 5000/5 = 1000 W = 1 kW; the person’s power P = 5000/20 = 250 W. Same work, four times more power for the crane (faster rate).
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