Work, Energy & Power – Key Concepts, Formulas & Short Notes for Engineering Entrance Exams

WORK, ENERGY & POWER — SHORT NOTES (Exam Focus)

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1. Work

Work is done when a force causes displacement.

$$W=\vec{F}\cdot \vec{s}=Fs\cos \theta$$

Key Points

• θ = 0° → maximum work (W = Fs)

• θ = 90° → zero work (e.g., centripetal force)

• Work is scalar, unit = Joule (J).

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2. Variable Force

If force is not constant:

$$W=\int Fdx$$

Area under F–x graph = Work.

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3. Work Done by Gravity

$$W_g=mg(h_1-h_2)$$

Depends only on height difference → path independent.

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4. Work Done by Spring Force (Hooke’s Law)

$$F = -kx$$
$$W=-\frac{1}{2}k{x}^2$$

Negative because spring force opposes stretch/compression.

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5. Kinetic Energy (KE)

$$K=\frac{1}{2}m{v}^2$$

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6. Work–Energy Theorem

Net work done on a body equals change in kinetic energy:

$$W_{\text{net}}=\mathrm{\Delta }K=\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2$$

Very important for numericals.

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7. Potential Energy (PE)

Gravitational PE

$$U=mgh$$

Elastic PE

$$U=\frac{1}{2}k{x}^2$$

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8. Mechanical Energy

$$E=K+U$$

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9. Conservation of Mechanical Energy

If only conservative forces act:

$$K_1+U_1=K_2+U_{\mathrm{2}}$$

Used in:

• free fall

• pendulum

• spring motion

• roller coaster

• projectile motion (vertical KE ↔ PE)

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10. Power

Rate of doing work.

$$P = \frac{dW}{dt}$$ $$P=\vec{F}\cdot \vec{v}$$

Unit: Watt (W)
1 HP = 746 W

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11. Conservative & Non-Conservative Forces

Conservative:

• gravity

• spring

• electrostatic

• central forces

Work done is path independent.
Energy is conserved.

Non-Conservative:

• friction

• air resistance

• viscosity

They cause energy loss (heat, sound etc.)

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12. Collision (Basic Intro)

1. Elastic Collision

• Momentum conserved

• KE conserved

• Perfectly elastic: coefficient of restitution ( e = 1 )

2. Inelastic Collision

• Momentum conserved

• KE not conserved

• Perfectly inelastic: objects stick → maximum KE loss, ( e = 0 )

Coefficient of Restitution:

$$e=\frac{\text{relative speed after collision}}{\text{relative speed before collision}}$$

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13. Important Special Results

1. Work done by friction

$$W=-f_{k}s=-{\mu }_{k}Ns$$

2. Power in rotational motion

$$P=\tau \omega$$

3. Work done by centripetal force

$$W=0$$

(Force ⟂ displacement)

4. Escape velocity (advanced but useful)

$$v_e=\sqrt{2gR}$$

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14. Most Expected Numerical Types

1. Work done by variable force (F–x graph)

2. Block sliding on smooth/rough surface

3. Spring compression/extension

4. Energy conservation in vertical motion

5. Power delivered by engine

6. Collision problems (1D elastic)

7. Lift work/energy questions

8. Pulley + energy method

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