Dynamics Chapter 3 – Complete Revision Guide

Definitions · SI units · formulas · derivations · graphs · worked examples

Big idea: Kinematics describes motion; dynamics explains why motion changes. Forces change momentum and energy, while Newton’s laws connect force, mass and acceleration.

Jump to: Definitions · Symbols & units · Formula sheet · Derivations · Graphs · Worked examples

1. Essential Definitions and Concepts

Force: A push or pull that changes, or tends to change, a body’s state of rest, uniform motion, direction or shape. Force is a vector.

Inertia: The tendency of a body to resist any change in its state of rest or uniform straight-line motion. Mass measures inertia.

Momentum: The quantity of motion of a body: p = mv. Its direction is the same as velocity.

Impulse: Force multiplied by the time for which it acts. Impulse equals change in momentum: J = FΔt = Δp.

Balanced forces: Forces whose vector sum is zero. They cannot produce acceleration, although they may deform a body.

Unbalanced force: A non-zero resultant force; it produces acceleration in the direction of the net force.

Normal reaction: The contact force exerted by a surface perpendicular to itself. On a level surface with no other vertical force, R = mg.

Friction: A contact force that opposes actual or impending relative motion between surfaces. Static friction adjusts up to a limiting value; kinetic friction acts during sliding.

Centripetal force: The resultant inward force needed for circular motion. It points toward the centre; it is not a separate new kind of force.

Centrifugal force: The apparent outward force observed in a rotating frame. It is a pseudo-force, not the reaction pair of centripetal force.

Work: Work is done when a force causes displacement: W = Fs cosθ. Work is positive, negative or zero depending on θ.

Energy: The capacity to do work. Kinetic energy belongs to motion; potential energy belongs to position or configuration.

Power: The rate of doing work or transferring energy: P = W/t.

Conservative force: A force for which work depends only on initial and final positions. Gravity and spring force are examples.

Periodic motion: Motion repeated after equal intervals of time.

Oscillatory motion: Repeated to-and-fro motion about a mean position. Every oscillation is periodic, but not every periodic motion is oscillatory.

Simple harmonic motion: Oscillation in which restoring acceleration is proportional to displacement and directed toward the mean position: a = −ω²x.

Renewable energy: Energy replenished naturally on a human timescale, such as solar, wind, tidal and hydro energy.

Non-renewable energy: Finite energy resources formed over very long periods, such as coal, petroleum, natural gas and nuclear fuel.

Newton’s Three Laws

First law (law of inertia): A body remains at rest or in uniform straight-line motion unless acted on by an unbalanced external force.
Second law: The net external force equals the rate of change of momentum and acts in that direction: F = dp/dt; for constant mass, F = ma.
Third law: When body A exerts a force on B, B simultaneously exerts an equal and opposite force on A. The two forces act on different bodies.

2. Symbols, Quantities and SI Units

Quantity	Symbol	SI unit	Dimensions
Mass	m	kilogram (kg)	[M]
Force / weight / friction	F, W, f	newton (N)	[MLT⁻²]
Acceleration	a, g	m s⁻²	[LT⁻²]
Momentum	p	kg m s⁻¹ or N s	[MLT⁻¹]
Impulse	J	N s	[MLT⁻¹]
Work / energy	W, E	joule (J)	[ML²T⁻²]
Power	P	watt (W)	[ML²T⁻³]
Spring constant	k	N m⁻¹	[MT⁻²]
Coefficient of friction	μ	no unit	dimensionless
Time period	T	second (s)	[T]
Frequency	f	hertz (Hz)	[T⁻¹]

Useful conversions: 1 N = 10⁵ dyne; 1 kgf = 9.8 N (approximately 10 N); 1 J = 10⁷ erg; 1 horsepower ≈ 746 W.

3. Complete Formula Sheet

Force and momentum

Weight: W = mg
Momentum: p = mv
Net force: F = ma = Δp/Δt
Impulse: J = FΔt = Δp

Friction

Static: 0 ≤ fₛ ≤ μₛR
Limiting: fₗ = μₛR
Kinetic: fₖ = μₖR
Usually μₛ > μₖ

Work and power

W = Fs cosθ
P = W/t
Instantaneous power: P = Fv (parallel vectors)

Energy

Kinetic: KE = ½mv² = p²/2m
Gravitational PE: U = mgh
Elastic PE: U = ½kx²
Mechanical energy: E = KE + PE

Circular motion

Centripetal acceleration: aᶜ = v²/r = ω²r
Centripetal force: Fᶜ = mv²/r = mω²r

Spring and SHM

Hooke’s law: F = −kx
SHM: a = −ω²x
ω = 2π/T = 2πf

Simple pendulum

T = 2π√(L/g)
g = 4π²L/T²
Valid for small angular displacement

Inclined plane

Along slope: mg sinθ
Into slope: mg cosθ
Normal reaction: R = mg cosθ (simple case)

Conservation laws

Momentum: Σp before = Σp after
Energy: total energy before = total energy after
Work-energy theorem: Wnet = ΔKE

4. Key Derivations

1. Derivation of F = ma

Newton’s second law says F ∝ rate of change of momentum. For constant mass m: F ∝ (mv − mu)/t = m(v − u)/t = ma. In SI units the proportionality constant is 1, so F = ma.

2. Impulse–momentum theorem

F = Δp/Δt. Multiplying by Δt gives FΔt = Δp. Therefore a small force acting for a long time may give the same impulse as a large force acting briefly.

3. Work–energy theorem

For a constant force parallel to motion, W = Fs = mas. From v² − u² = 2as, s = (v² − u²)/(2a). Hence W = ma(v² − u²)/(2a) = ½mv² − ½mu² = ΔKE.

4. Kinetic energy

Work needed to accelerate a body from rest to speed v equals its gain in KE. Put u = 0 in the work–energy result: KE = ½mv².

5. Gravitational potential energy

Raise mass m slowly through height h. Upward force balances weight mg, and displacement is h. Work done = force × displacement = mgh. This work is stored as PE = mgh.

6. Elastic potential energy

For a spring, force rises from 0 to kx. Average force = (0 + kx)/2 = kx/2. Work = average force × extension = (kx/2)x, so U = ½kx².

7. Conservation of mechanical energy in free fall

At height H, total energy = mgH. After falling distance x: v² = 2gx, so KE = ½m(2gx) = mgx and PE = mg(H−x). Sum = mgx + mg(H−x) = mgH, a constant.

8. Simple pendulum formula

For a small angle, the restoring component produces SHM. Its angular frequency is ω = √(g/L). Since T = 2π/ω, T = 2π√(L/g). Therefore g = 4π²L/T².

9. Inclined-plane components

Weight mg is resolved perpendicular and parallel to the slope. In the right triangle, the component adjacent to θ is mg cosθ and the opposite component is mg sinθ. On a smooth slope, ma = mg sinθ, hence a = g sinθ.

10. Conservation of momentum from Newton’s third law

During collision F₁₂ = −F₂₁. Acting for the same time: Δp₁ = −Δp₂. Therefore Δ(p₁+p₂)=0 and m₁u₁+m₂u₂ = m₁v₁+m₂v₂.

5. Graphs and How to Read Them

Force–extension graph (Hooke’s law)

The straight line through the origin means F ∝ x. The slope is spring constant k. Area under the graph is elastic PE = ½kx².

Force–time graph (impulse)

The area under an F–t graph equals impulse and hence change in momentum Δp. A tall short pulse can equal a low broad pulse.

Energy during free fall

KE rises while PE falls by the same amount; total mechanical energy remains constant when air resistance is ignored.

Displacement–time graph for SHM

The wave repeats after one period T. At extreme displacement speed is zero; at the mean position speed is maximum.

6. Worked Examples

Example 1: Impulse

A 0.2 kg ball changes velocity from +15 m/s to −10 m/s in 0.05 s. Δp = m(v−u) = 0.2(−10−15) = −5 kg m/s. Impulse = −5 N s. Average force = Δp/Δt = −5/0.05 = −100 N. The minus sign shows the force is opposite to the initial motion.

Example 2: Collision

A 3 kg trolley at 4 m/s strikes a stationary 1 kg trolley and they stick. Momentum before = 3×4 = 12 kg m/s. After: (3+1)v = 4v. Thus v = 3 m/s. Momentum is conserved; kinetic energy is not conserved in this perfectly inelastic collision.

Example 3: Work at an angle

A 20 N force moves a box 5 m at 60° to displacement. W = Fs cosθ = 20×5×cos60° = 100×0.5 = 50 J. The perpendicular component does no work.

Example 4: Power

A motor lifts 200 kg through 5 m in 10 s. Work = mgh = 200×9.8×5 = 9800 J. Power = W/t = 9800/10 = 980 W.

Example 5: Friction

A 10 kg block rests on a horizontal surface with μ = 0.3. R = mg = 98 N. Limiting friction = μR = 0.3×98 = 29.4 N. A 20 N horizontal push is balanced by 20 N static friction, so the block remains at rest.

Example 6: Inclined plane

A 5 kg block is on a smooth 30° slope. Down-slope force = mg sin30° = 5×9.8×0.5 = 24.5 N. Acceleration = F/m = 24.5/5 = 4.9 m/s². Normal reaction = mg cos30° ≈ 42.4 N.

Example 7: Energy conservation

A 2 kg stone falls from 20 m. Initial PE = 2×9.8×20 = 392 J. After falling 15 m it is 5 m high: PE = 98 J and KE = 392−98 = 294 J. Just before ground, KE = 392 J.

Example 8: Pendulum experiment

For L = 1.00 m, suppose 20 oscillations take 40.1 s. T = 40.1/20 = 2.005 s. g = 4π²L/T² = 39.478/(2.005)² ≈ 9.82 m/s². Timing many oscillations reduces reaction-time error.

7. Exam Checklist and Common Mistakes

Draw a free-body diagram and choose a positive direction before using signs.
Write units with every numerical answer; momentum and impulse share dimensions but describe different ideas.
Action and reaction are equal and opposite but act on different bodies, so they do not cancel each other on one body.
Centripetal force points inward; centrifugal force is apparent in a rotating frame.
Work is zero if displacement is zero or the force is perpendicular to displacement.
Static friction is not always μR; it adjusts from zero up to its limiting value.
Use conservation of mechanical energy only when non-conservative work such as friction is absent or separately included.
For collisions, conserve vector momentum. Kinetic energy is conserved only in elastic collisions.
For pendulums, measure length from the point of suspension to the centre of the bob and use small swings.

Fast revision order: definitions → Newton’s laws → momentum and impulse → work, energy and power → friction → circular motion → springs, SHM and pendulum → conservation laws.

Physics

Dynamics Chapter 3 – Complete Definitions, Formulas, Derivations and Graphs