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Mechanics and Materials

Mechanics and Materials

Mechanics is the study of forces, motion, and energy. Combined with the study of material properties, this topic forms a cornerstone of A-Level Physics. It covers kinematics, Newton's laws, momentum, energy, projectile motion, moments, and the mechanical properties of materials.

Kinematics — The SUVAT Equations

For motion with constant acceleration in a straight line, five equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t):

Equation Variables Used
v = u + at v, u, a, t
s = ½(u + v)t s, u, v, t
s = ut + ½at² s, u, a, t
s = vt − ½at² s, v, a, t
v² = u² + 2as v, u, a, s

Key Definition: Displacement is the distance moved in a specified direction (vector). Velocity is the rate of change of displacement (vector). Acceleration is the rate of change of velocity (vector).

Worked Example — SUVAT

A car accelerates uniformly from rest to 30 m s⁻¹ in 12 s. Find (a) the acceleration and (b) the distance travelled.

Part (a): u = 0, v = 30 m s⁻¹, t = 12 s.

a = (v − u)/t = (30 − 0)/12 = 2.5 m s⁻²

Part (b): s = ½(u + v)t = ½(0 + 30) × 12 = 180 m

Exam Tip: Always list the known SUVAT variables first, then choose the equation that contains your unknown and three knowns. This prevents errors from choosing the wrong equation.

Graphs of Motion

  • Displacement-time graph: Gradient = velocity. A straight line means constant velocity; a curve means changing velocity.
  • Velocity-time graph: Gradient = acceleration. Area under the graph = displacement.
  • Acceleration-time graph: Area under the graph = change in velocity.

Projectile Motion

A projectile is any object moving freely under gravity (ignoring air resistance). The key principle is that the horizontal and vertical components of motion are independent:

  • Horizontally: no acceleration, so x = v_x × t (constant velocity)
  • Vertically: acceleration = g (downwards), so use SUVAT with a = −9.81 m s⁻²

Resolving Into Components

For a projectile launched at speed u at angle θ to the horizontal:

  • Horizontal component: u_x = u cos θ
  • Vertical component: u_y = u sin θ

Worked Example — Projectile Motion

A ball is kicked at 20 m s⁻¹ at 30° above the horizontal on level ground. Find (a) the time of flight, (b) the maximum height, and (c) the range.

Part (a): u_y = 20 sin 30° = 10 m s⁻¹. At the top, v_y = 0.

Time to reach top: v = u + at → 0 = 10 − 9.81t → t = 1.02 s.

Total time of flight = 2 × 1.02 = 2.04 s (by symmetry for level ground).

Part (b): s = u_y t + ½at² = 10 × 1.02 + ½(−9.81)(1.02)² = 10.2 − 5.10 = 5.1 m

Part (c): u_x = 20 cos 30° = 17.3 m s⁻¹.

Range = u_x × total time = 17.3 × 2.04 = 35.3 m

Exam Tip: In projectile questions, always resolve the initial velocity into horizontal and vertical components first. Treat each direction independently. Take care with signs — define positive direction clearly.

Newton's Laws of Motion

First Law (Inertia)

An object remains at rest or continues to move at constant velocity unless acted upon by a resultant force.

Second Law

F = ma (for constant mass)

More precisely, F = Δp/Δt, where p = mv is momentum.

Third Law

When two objects interact, they exert equal and opposite forces on each other. These forces act on different objects, are of the same type, and are equal in magnitude but opposite in direction.

Free Body Diagrams

A free body diagram shows all the forces acting on a single object, drawn as arrows from the object. Each arrow is labelled with the type of force (weight, normal contact, friction, tension, etc.) and its magnitude if known.

Described diagram — Free body diagram of a book on a table: A rectangle represents the book. An arrow pointing downward from the centre is labelled "Weight (W = mg)". An arrow of equal length pointing upward from the bottom of the rectangle is labelled "Normal contact force (N)". Since the book is in equilibrium, W = N.

Described diagram — Free body diagram of a block on a rough inclined plane: The block sits on a slope at angle θ. Weight (mg) acts vertically downward. The normal contact force (N) acts perpendicular to the slope surface. Friction (F) acts up the slope (opposing potential motion). Resolving: N = mg cos θ and F = mg sin θ for equilibrium.

Moments and Torque

Key Definition: The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force: Moment = F × d.

The SI unit of moment is N m.

The Principle of Moments

For a body in rotational equilibrium:

Sum of clockwise moments = Sum of anticlockwise moments (about any point)

Torque of a Couple

A couple consists of two equal and opposite parallel forces whose lines of action do not coincide. The torque of a couple is:

Torque = F × d

where d is the perpendicular distance between the two forces. A couple produces pure rotation with no resultant translational force.

Momentum and Impulse

Momentum

Momentum is a vector quantity:

p = mv

Conservation of Momentum

The total momentum of a closed system remains constant provided no external forces act:

Σp_before = Σp_after

Types of Collision

Type Momentum Conserved? Kinetic Energy Conserved?
Elastic Yes Yes
Inelastic Yes No (some KE lost)
Perfectly inelastic Yes No (objects stick together)
Explosion Yes (total = 0 if from rest) No (KE increases)

Impulse

Key Definition: Impulse is the change in momentum of an object: Impulse = FΔt = Δp = mv − mu.

The unit of impulse is N s (equivalent to kg m s⁻¹).

On a force-time graph, the area under the curve equals the impulse (change in momentum).

Applications: Car crumple zones, airbags, and cycle helmets all increase the time of impact, thereby reducing the maximum force experienced.

Terminal Velocity and Drag

When an object falls through a fluid, it experiences a drag force that increases with speed. Initially, the weight is much greater than the drag, so the object accelerates. As speed increases, drag increases until it equals the weight. At this point, the resultant force is zero, acceleration is zero, and the object falls at a constant terminal velocity.

flowchart TD
    A["Object released from rest"] --> B["Weight >> Drag"]
    B --> C["Large resultant force downward"]
    C --> D["Large acceleration (close to g)"]
    D --> E["Speed increases"]
    E --> F["Drag force increases with speed"]
    F --> G{"Does Weight = Drag?"}
    G -->|"No"| H["Resultant force still downward<br/>but decreasing"]
    H --> E
    G -->|"Yes"| I["Resultant force = 0"]
    I --> J["Acceleration = 0"]
    J --> K["Terminal velocity reached<br/>(constant speed)"]

Described diagram — Velocity-time graph for a falling object reaching terminal velocity: The graph starts at v = 0 and rises steeply (large gradient = large acceleration). The curve gradually flattens as drag increases, and the line asymptotically approaches a horizontal value — the terminal velocity. The gradient (acceleration) decreases continuously from g to 0.

Exam Tip: When explaining terminal velocity, always refer to the balance of forces (weight = drag) and state that the resultant force is zero, so by Newton's second law, acceleration is zero.

Density

Key Definition: Density is the mass per unit volume of a material: ρ = m/V. It is measured in kg m⁻³.

Material Approximate Density (kg m⁻³)
Air 1.2
Water 1000
Aluminium 2700
Iron/Steel 7800
Lead 11,300

Work, Energy, and Power

Work Done

W = Fs cos θ

where θ is the angle between the force and the direction of motion.

Kinetic Energy

Eₖ = ½mv²

Gravitational Potential Energy

Eₚ = mgh (near Earth's surface)

Conservation of Energy

Energy cannot be created or destroyed, only transferred from one form to another. In a closed system, the total energy remains constant.

flowchart LR
    A["Gravitational PE<br/>(mgh)"] -->|"Object falls"| B["Kinetic Energy<br/>(½mv²)"]
    B -->|"Work against friction"| C["Thermal Energy<br/>(heat)"]
    B -->|"Object rises"| A
    D["Elastic PE<br/>(½kx²)"] -->|"Spring released"| B
    B -->|"Spring compressed"| D

Power

P = W/t = Fv

Efficiency

Efficiency = (useful output energy / total input energy) × 100% Efficiency = (useful output power / total input power) × 100%

Stress, Strain, and Young's Modulus

Stress

Key Definition: Stress is the force per unit cross-sectional area: σ = F/A (measured in pascals, Pa).

Strain

Key Definition: Strain is the fractional change in length: ε = ΔL/L (dimensionless).

Young's Modulus

E = σ/ε = (F/A) / (ΔL/L) (measured in Pa)

It is a measure of the stiffness of a material. A high Young's modulus indicates a stiff material (e.g. steel ~200 GPa), while a low value indicates a flexible material (e.g. rubber ~0.01 GPa).

Worked Example — Young's Modulus

A steel wire of length 2.00 m and diameter 0.80 mm supports a load of 50 N. The wire extends by 0.25 mm. Calculate the Young modulus.

Cross-sectional area: A = π(d/2)² = π(0.40 × 10⁻³)² = 5.03 × 10⁻⁷ m²

Stress: σ = F/A = 50 / 5.03 × 10⁻⁷ = 9.94 × 10⁷ Pa

Strain: ε = ΔL/L = 0.25 × 10⁻³ / 2.00 = 1.25 × 10⁻⁴

Young modulus: E = σ/ε = 9.94 × 10⁷ / 1.25 × 10⁻⁴ = 7.95 × 10¹¹ Pa ≈ 800 GPa

(Note: this is higher than the typical value for steel because the diameter is very small — in reality, the wire would likely be thicker.)

Stress-Strain Curves

The stress-strain graph reveals important properties:

  • Elastic region — the material returns to its original shape when the force is removed (Hooke's law applies: F = kx)
  • Elastic limit — beyond this point, permanent deformation occurs
  • Yield point — the stress at which significant plastic deformation begins
  • Ultimate tensile stress (UTS) — the maximum stress the material can withstand
  • Breaking point — the stress at which the material fractures

Described diagram — Stress-strain curve for a ductile metal (e.g. copper): The graph starts as a straight line from the origin (linear elastic region). At the elastic limit, it begins to curve. There is a noticeable yield point where the curve flattens slightly. The curve then rises to a maximum (UTS) before dropping as the material necks and eventually fractures at the breaking point. The area under the curve up to the elastic limit represents recoverable elastic strain energy; the total area up to fracture represents the total energy absorbed (toughness).

Described diagram — Stress-strain curve for a brittle material (e.g. glass): The graph is a straight line from the origin (obeying Hooke's law throughout) and then snaps suddenly at the breaking point with virtually no plastic deformation. The fracture stress and UTS are essentially the same.

Described diagram — Stress-strain curve for a polymer (e.g. rubber): The graph shows a non-linear curve with very large strains (the material stretches enormously). The loading and unloading curves do not coincide — the unloading curve lies below the loading curve. The area between the two curves represents energy dissipated as heat (hysteresis).

Material Behaviour Summary

Property Description Example
Brittle Fractures with little/no plastic deformation Glass, ceramics
Ductile Large plastic deformation before fracture; can be drawn into wires Copper, gold
Hard Resistant to surface indentation Diamond
Stiff High Young modulus; resists deformation Steel
Tough Absorbs a lot of energy before fracturing (large area under curve) Mild steel
Elastic Returns to original shape when force removed Rubber (within limits)
Plastic Permanently deformed; does not return to original shape Clay

The area under a force-extension graph represents the work done (energy stored or dissipated). For a material obeying Hooke's law, the elastic strain energy is ½FΔL = ½kx².

Exam Tip: When asked to compare materials from their stress-strain graphs, comment on: gradient (stiffness/Young modulus), area under curve (toughness), whether there is plastic deformation (ductile vs brittle), and the breaking stress. Use correct technical vocabulary.

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