Flashcards · 9630 · AS Topic 2

Mechanics and materials (AS Level) flashcards

53 cards on AS Topic 2, subtopics 3.2.1 to 3.2.10. Pick a subtopic or study them all.

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3.2.1 Scalars and vectors

Scalar vs vector: A scalar has size only (distance, speed, mass, energy). A vector has size and direction (displacement, velocity, acceleration, force, weight).
Resolving a force F at angle θ to the horizontal: Horizontal component = F cos θ. Vertical component = F sin θ.
Resultant of two perpendicular vectors a and b: R = √(a² + b²), at an angle given by tan θ = opposite ÷ adjacent. State the reference direction (e.g. 12 N east + 5 N north = 13 N at 22.6° north of east).
Conditions for three coplanar forces at a point to be in equilibrium: The resultant force is zero: components sum to zero in any direction, and drawn tip to tail the three forces form a closed triangle.
Does equilibrium mean the object is at rest?: No. Equilibrium means zero resultant force, so the object is at rest or moving at constant velocity.

3.2.2 Moments

Moment of a force: Force × perpendicular distance from the point to the line of action of the force. Unit N m.
Principle of moments: For a body in equilibrium, the total clockwise moment about any point equals the total anticlockwise moment about that point.
Couple, and the moment of a couple: A couple is two equal, opposite, coplanar forces acting along different (parallel) lines. Moment = one force × perpendicular distance between the lines. Resultant force is zero, so it turns without accelerating sideways.
Centre of mass: The point through which the whole weight of a body can be taken to act. For a uniform regular solid it is at the geometric centre; it can lie outside the material (e.g. a ring).
Beam problems: why take moments about a point where an unknown force acts?: A force through the chosen point has zero moment about it, so that unknown drops out of the equation.

3.2.3 Motion along a straight line

The four SUVAT equations (uniform acceleration): v = u + at s = ½(u + v)t s = ut + ½at² v² = u² + 2as
Displacement-time and velocity-time graphs: what do gradient and area give?: Displacement-time: gradient = velocity. Velocity-time: gradient = acceleration; area under graph = displacement. Acceleration-time: area = change in velocity.
Displacement, velocity and acceleration: Displacement: distance moved in a specified direction (vector). Velocity: rate of change of displacement. Acceleration: rate of change of velocity.
Required practical: finding g by free fall. Which graph, and how is g found?: From rest, s = ½gt². Plot s against t²: a straight line through the origin with gradient g/2, so g = 2 × gradient.
Velocity-time graph: 0 to 12 m s⁻¹ in 6 s, then 12 m s⁻¹ for 4 s. Total displacement?: Triangle ½ × 6 × 12 = 36 m, plus rectangle 4 × 12 = 48 m: total 84 m.

3.2.4 Projectile motion

Key idea of projectile motion (no air resistance): Horizontal and vertical motions are independent and share only the time. Horizontal: constant velocity. Vertical: uniform acceleration g = 9.81 m s⁻² downwards.
At the top of its path, what are a projectile's vertical velocity, speed and acceleration?: Vertical velocity is zero; speed equals the horizontal component (not zero); acceleration is still 9.81 m s⁻² downwards.
Launch at speed u, angle θ, landing at the same level: time of flight, maximum height, range: u_y = u sin θ, u_x = u cos θ. Time of flight = 2u_y/g. Maximum height = u_y²/2g. Range = u_x × time of flight.
Describe how a falling object reaches terminal speed: At release drag is zero and a = g. As speed rises, drag rises, so resultant force and acceleration fall. When drag equals weight the resultant force is zero and the speed stays constant.
Effects of air resistance on a projectile's path: Horizontal velocity decreases; range and maximum height are smaller; the path is asymmetrical, with a steeper descent than ascent; landing speed is lower.
Direction of drag and of lift: Drag acts opposite to the velocity relative to the fluid and increases with speed. Lift acts perpendicular to the flow of fluid past the object.

3.2.5 Newton's laws of motion

Newton's first law: An object stays at rest or moves at constant velocity unless acted on by a resultant force.
Newton's second law, and its condition: Resultant force = mass × acceleration, F = ma, which applies only when the mass is constant. In general F = rate of change of momentum.
Newton's third law: If body A exerts a force on body B, B exerts an equal and opposite force on A. The pair are the same type of force and act on different bodies.
A book rests on a table. Are its weight and the normal contact force a third-law pair?: No. Both act on the book, so they are balanced forces (first law). The pair to the weight is the book pulling the Earth up.
Tension in a lift cable of mass m accelerating upwards at a: T − mg = ma, so T = m(g + a). Tension exceeds weight for upward acceleration, and is less than weight for downward acceleration.

3.2.6 Momentum

Momentum: equation and unit: p = mv, a vector in the direction of the velocity. Unit kg m s⁻¹ (the same as N s).
Principle of conservation of linear momentum: The total momentum of a closed system stays constant provided no resultant external force acts on it.
Impulse: The change in momentum: FΔt = Δ(mv) for a constant force. It equals the area under a force-time graph. Unit N s.
Elastic vs inelastic collision: Momentum is conserved in both. Elastic: total kinetic energy is also conserved. Inelastic: some kinetic energy is transferred to other forms. Check by calculating KE before and after.
Why does an airbag or crumple zone reduce injury?: It increases the time taken to stop. The change in momentum is the same, so the rate of change of momentum, and hence the force, is smaller.
Water hits a wall at 0.80 kg s⁻¹ and 6.0 m s⁻¹ and stops. Force on the wall?: F = (Δm/Δt) × v = 0.80 × 6.0 = 4.8 N.

3.2.7 Work, energy and power

Work done by a force: Force × displacement in the direction of the force: W = Fs cos θ, where θ is the angle between F and s. Unit J. Also the area under a force-displacement graph.
Power: definition and equations: The rate of doing work (rate of energy transfer). P = ΔW/Δt, and P = Fv for a force F along velocity v. Unit W.
Efficiency: Useful output power ÷ input power (or useful energy out ÷ total energy in). No unit; never more than 100%.
An 18 kW engine drives a car at a steady 30 m s⁻¹. What is the resistive force?: At steady speed driving force = resistive force. F = P/v = 18 000 ÷ 30 = 600 N.
Work done by a force at 90° to the displacement: Zero, since cos 90° = 0 (e.g. the normal force on a block sliding along a level floor).

3.2.8 Conservation of energy

Principle of conservation of energy: Energy cannot be created or destroyed. In an isolated system the total energy stays the same while it is transferred between stores.
Equations for gravitational potential energy and kinetic energy: ΔEp = mgΔh (vertical height change, uniform field). Ek = ½mv².
Speed of an object dropped from rest through height Δh (no air resistance): mgΔh = ½mv², so v = √(2gΔh). The mass cancels. For 1.8 m: v = √(2 × 9.81 × 1.8) = 5.9 m s⁻¹.
Energy equation when resistive forces act: Initial energy + work done on the system = final energy + work done against resistive forces.
Why does each bounce of a ball reach a lower height?: At each impact some kinetic energy is dissipated as thermal energy and sound in the ball and floor, and energy is also transferred by air resistance.

3.2.9 Bulk properties of solids

Hooke's law: The extension is proportional to the applied force, up to the limit of proportionality: F = kΔL, where k is the stiffness (spring constant) in N m⁻¹.
Elastic limit: The point beyond which the material does not return to its original length when the force is removed.
Tensile stress and tensile strain: Stress = F/A, force per unit cross-sectional area (Pa). Strain = ΔL/L, extension ÷ original length (no unit).
Elastic strain energy stored in a stretched spring or wire: The area under the force-extension graph. Within Hooke's law: E = ½FΔL = ½kΔL².
Elastic vs plastic deformation; ductile vs brittle: Elastic: returns to its original length. Plastic: permanent extension remains. Ductile: large plastic region before fracture. Brittle: fractures with little or no plastic deformation.
Density: Mass per unit volume: ρ = m/V. Unit kg m⁻³.

3.2.10 The Young modulus

Young modulus: definition and equation: Tensile stress ÷ tensile strain, in the region where they are proportional. E = (F/A)/(ΔL/L) = FL/(AΔL). Unit Pa.
How is E found from a stress-strain graph?: It is the gradient of the straight (initial) part of the stress-strain graph.
Required practical: Young modulus of a wire. Outline and analysis: Long thin clamped wire with a marker against a fixed scale. Measure L and the diameter at several places with a micrometer. Add loads and record extensions. Gradient of extension against force = L/(EA), so E = L/(A × gradient).
Which measurement dominates the uncertainty in the Young modulus practical, and why?: The diameter: it is small, and A = πd²/4 depends on d², so its percentage uncertainty is doubled.
Stiffness k vs Young modulus E: k depends on the size and shape of the sample (k = EA/L). E is a property of the material.