Mechanics and materials

Explanation

• Scalars have magnitude only; vectors have magnitude and direction. Speed, distance and mass are scalars, whereas velocity, displacement, acceleration, force and weight are vectors.
• Vectors may be added by a scale drawing or, for two perpendicular vectors, by Pythagoras and trigonometry.
• A vector may also be resolved into two perpendicular components; for a force $F$ at angle $\theta$ to the horizontal, these are $F\cos\theta$ horizontally and $F\sin\theta$ vertically.
• For two or three coplanar forces acting at a point, equilibrium means zero resultant force, so the object is at rest or moves with constant velocity.
• Examiners expect both a correct magnitude and an unambiguous direction.

Explanation

• The moment of a force about a point is $M=Fd_{\perp}$, where $d_{\perp}$ is the perpendicular distance from the point to the force’s line of action.
• For rotational equilibrium, the total clockwise moment equals the total anticlockwise moment; full equilibrium also requires zero resultant force.
• A couple consists of equal and opposite coplanar forces on different lines of action, producing a moment equal to one force multiplied by the perpendicular separation of the lines.
• The weight of a uniform regular solid acts through its centre of mass at its geometric centre.
• Examiners expect the chosen pivot, sense of each moment and perpendicular distances to be clear, followed by a separate force balance if an unknown reaction is required.

Explanation

• Velocity is the rate of change of displacement, $v=\Delta s/\Delta t$, and acceleration is the rate of change of velocity, $a=\Delta v/\Delta t$. An instantaneous value is obtained from the tangent gradient, whereas an average value uses the change over an interval.
• On a displacement–time graph the gradient is velocity; on a velocity–time graph the gradient is acceleration and signed area is displacement; on an acceleration–time graph signed area is change in velocity.
• The equations of uniform acceleration apply only when acceleration is constant.
• For free fall near Earth, acceleration has magnitude $g=9.81\,\text{m s}^{-2}$, with its sign set by the chosen positive direction.
• Required practical work may determine $g$ from an appropriate graph and evaluate random and systematic errors.

Explanation

• In a uniform gravitational field with air resistance neglected, horizontal and vertical motions are independent. Horizontal velocity remains constant, while vertical acceleration is $g$ downward.
• Resolve the launch velocity into perpendicular components, apply the uniform-acceleration equations separately, and use the common time to connect the two directions. At maximum height the vertical velocity is zero, but the horizontal velocity is not.
• Friction, lift and drag require qualitative treatment: air resistance increases with speed, reduces range and makes a projectile trajectory asymmetric. Terminal speed occurs when drag balances weight, making the resultant force and acceleration zero.
• For a vehicle, maximum speed occurs when the driving force is balanced by resistive forces.
• Examiners expect the no-air-resistance model to be stated before calculations and drag effects to be described through forces and acceleration.

Explanation

• Newton’s first law states that an object remains at rest or moves with constant velocity unless acted on by a resultant external force.
• For constant mass, the second law is $\sum F=ma$; a free-body diagram should show only forces acting on the chosen object.
• Newton’s third-law pairs are equal and opposite forces of the same interaction acting on different objects, so they do not cancel on one object’s diagram.
• Weight is $mg$, while normal contact force, tension and drag must be obtained from the situation rather than assumed equal to weight.
• Examiners expect a declared positive direction, a signed resultant-force equation and a third-law pair identified by naming both interacting objects.

Explanation

• Linear momentum is the vector $p=mv$. Total momentum is conserved in an isolated system, so the signed total before a one-dimensional collision or explosion equals the signed total afterwards.
• Force is the rate of change of momentum, $F=\Delta(mv)/\Delta t$, and impulse is the change in momentum. For a constant force, impulse is $F\Delta t$; for a varying force, it is the signed area under the force–time graph.
• Elastic collisions conserve both momentum and kinetic energy, whereas inelastic collisions conserve momentum but not kinetic energy.
• Examiners expect a positive direction, signed velocities and a clear distinction between momentum conservation and kinetic-energy conservation.
• Crumple zones and protective packaging increase contact time, reducing mean impact force for the same momentum change; ethical transport design applies this principle to occupant safety.

Explanation

• Work is energy transferred by a force. For a constant force, $W=Fs\cos\theta$, where $\theta$ is the angle between force and displacement.
• For a variable force, work is the area under the force–displacement graph. Power is the rate of doing work or transferring energy, $P=\Delta W/\Delta t$; when a force is parallel to steady velocity, $P=Fv$.
• Efficiency is useful output energy divided by total input energy, or useful output power divided by total input power, and may be quoted as a decimal or percentage.
• A motor-lifting experiment compares measured useful output power with electrical input power and evaluates random and systematic errors.
• Examiners expect the parallel force component, correct graph area and like quantities in an efficiency ratio.

Explanation

• The principle of conservation of energy states that energy cannot be created or destroyed; it is transferred between stores while total energy remains constant in a closed system.
• Near Earth, a change in gravitational potential energy is $\Delta E_p=mg\Delta h$ and kinetic energy is $E_k=\tfrac12mv^2$.
• With negligible resistance, a decrease in gravitational potential energy may equal an increase in kinetic energy.
• If resistive forces act, some mechanical energy is transferred to internal energy of the object and surroundings, so it is not lost.
• Examiners expect a complete energy balance that includes any initial kinetic energy and work done against resistance, together with explicit assumptions such as negligible air resistance.

Explanation

• Density is $\rho=m/V$. Within the limit of proportionality, Hooke’s law gives $F=k\Delta L$, where $k$ is stiffness.
• Elastic strain energy is the area under a force–extension graph and equals $\tfrac12F\Delta L$ only for a straight line through the origin. Tensile stress is $F/A$ and tensile strain is $\Delta L/L$; breaking stress is the stress at fracture.
• Below the elastic limit a material returns to its original dimensions after unloading, while plastic deformation leaves permanent extension. Brittle materials fracture with little plastic deformation.
• Examiners expect force–extension and stress–strain graphs to show the limit of proportionality, elastic limit, energy area and fracture behaviour, and may link spring energy to kinetic or gravitational energy.
• Ethical transport design weighs deformation energy and occupant protection against material and environmental costs.

Explanation

• Tensile stress is $\sigma=F/A$ in pascals and tensile strain is $\varepsilon=\Delta L/L$, which is dimensionless. Within the linear elastic region, Young modulus is $E=\sigma/\varepsilon=FL/(A\Delta L)$ and is the gradient of a stress–strain graph.
• A simple determination uses a long wire of measured original length and diameter, adds known loads below the limit of proportionality and measures each extension.
• The diameter should be measured with a micrometer at several positions and perpendicular orientations because area depends on diameter squared.
• Examiners expect SI conversions, a valid graph-based method, and practical controls such as a reference wire or correction for support movement.
• Required practical 4 assesses this measurement and its uncertainties.