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MECHANICS FOR PP4

M1 TOPIC 5 : Energy,Work & Power

Mechanics 1: Energy, Work & Power

A-Level Revision Notes

Based on CAIE 9709 Syllabus

1. Work Done by a Force

Concept: Energy transferred when a force moves its point of application.

W = F × d × cos θ
W = work done (J), F = force (N), d = displacement (m), θ = angle between force & displacement

Force parallel to motion (θ = 0°): W = Fd
Force perpendicular (θ = 90°): W = 0
Force opposite (θ = 180°): W = –Fd (energy lost)

Example: Pulling a suitcase with 20 N at 30° over 5 m:
W = 20 × 5 × cos30° = 86.6 J

2. Kinetic & Potential Energy

KE = ½ m v²
Kinetic Energy: m = mass (kg), v = speed (m/s)

GPE = m g h
Gravitational Potential Energy: g = 9.8 m/s², h = vertical height above reference

GPE can be negative if below reference level.
Always define where h = 0 (often ground or starting point).
Energy is scalar — no direction.

3. Work-Energy & Conservation

Work-Energy Theorem:

Net work done = ΔKE

Conservation of Mechanical Energy (No friction/resistance, only gravity):

KE_initial + GPE_initial = KE_final + GPE_final

With external forces (e.g., pulling, friction):

Work done by external forces = ΔKE + ΔGPE

ON Double Inclines (e.g., 1 mass pulling another):

Energy Method for a Double Inclined Plane System

General Energy Principle:
The loss of gravitational potential energy of the particle moving down its plane is equal to the gain in gravitational potential energy of the particle moving up its plane, plus the work done against friction (if any), plus the total kinetic energy gained by the system.

Definitions

m₁: mass moving down its plane

m₂: mass moving up its plane

α: angle of the downward plane to the horizontal

β: angle of the upward plane to the horizontal

s: distance moved along each plane

v: common speed of the particles

W_f: total work done against friction

g: gravitational field strength

General Energy Equation

m₁ g s sin(α) = m₂ g s sin(β) + W_f + ½(m₁ + m₂)v²

Special Cases

Both planes smooth:

m₁ g s sin(α) = m₂ g s sin(β) + ½(m₁ + m₂)v²

Friction acting on one plane only:

m₁ g s sin(α) = m₂ g s sin(β) + F s + ½(m₁ + m₂)v²

Memory Rule

Down-slope potential energy loss equals up-slope potential energy gain, plus friction work, plus kinetic energy gained.

4. Power

Definition: Rate of doing work.

P = Work done / Time taken
Units: Watts (W) = J/s

For constant force in direction of motion:

P = F v
F = driving force, v = instantaneous velocity

Average power: P_avg = Total work / Total time
Instantaneous power: P = Fv at that moment

📝 Problem-Solving Strategy

Identify forces doing work (driving, friction, gravity).
Choose reference level for GPE (where h = 0).
Apply energy conservation if no resistance/friction.
If resistance present:
Work by driving force – Work against resistance = ΔKE + ΔGPE
For power: Find force and velocity at the required instant using P = Fv.

⚠ Common Exam Traps:

Work done by gravity is already included in ΔGPE — don’t count it twice.
Use vertical height h, not distance along slope, for GPE.
P = Fv only if F is in the direction of motion.
Resistance/drag does negative work (force opposite motion).
On curved slides, normal reaction does no work (perpendicular to motion).

🔍 Past Paper Topics Covered

Work done with angle Q86, Q89, Q97, Q98

KE & GPE changes Q3, Q12, Q24, Q50, Q95

Work-energy with resistance Q2, Q6, Q82, Q91, Q113

Conservation of energy (smooth) Q9, Q21, Q106, Q112

Power P = Fv & acceleration Q1, Q4, Q7, Q23, Q36, Q75, Q105

Hill problems (constant speed/acceleration) Q1(b), Q4(b), Q12(b), Q68, Q107

Pulley + energy Q5(b), Q81, Q74

Variable resistance R = a + bv Q2(b), Q20, Q30, Q51

Energy loss on impact Q13, Q73

ON Double Inclines (e.g., 1 mass pulling another):