§ 1.1 Physical quantities and measurement techniques
Key ideas
- Measure length with a rule (to the nearest mm); volume of a liquid with a measuring cylinder, read at the bottom of the meniscus at eye level.
- Measure time with clocks or digital timers. For a short repeating interval, time many repeats and divide: time 20 pendulum swings, then divide by 20.
- The same averaging trick works for small lengths: measure the thickness of 100 sheets, then divide by 100.
- A scalar has magnitude only; a vector has magnitude and direction.
- Scalars: distance, speed, time, mass, energy, temperature. Vectors: displacement, velocity, acceleration, force, weight, momentum, gravitational field strength.
- Extended: find the resultant of two perpendicular vectors by calculation (Pythagoras for size, tangent for angle) or by scale drawing.
Fig. 1 · The same 5 m: distance follows the path walked, displacement is the straight line from A to B, stated with a direction.
Watch out: never add the magnitudes of two perpendicular vectors. A 3 N force and a 4 N force at right angles give a 5 N resultant (Pythagoras), not 7 N.
§ 1.2 Motion
Key ideas
- Speed is the distance travelled per unit time; velocity is speed in a stated direction.
- Average speed = total distance ÷ total time, whatever happened along the way.
- Acceleration is the change in velocity per unit time; a negative value is a deceleration.
- Distance-time graph: gradient = speed; a horizontal line means stationary.
- Speed-time graph: gradient = acceleration; the area under the line = distance travelled.
- An object in free fall near the Earth's surface accelerates at a constant g ≈ 9.8 m/s² when air resistance is negligible.
- Extended: with air resistance, a falling object's acceleration shrinks as drag grows, until drag balances weight and it moves at terminal velocity.
Equations
v = s / tspeed = distance ÷ timem/s
a = Δv / tacceleration = change in velocity ÷ time takenm/s²
Fig. 2 · Read the two graphs differently: gradient means speed on the first but acceleration on the second, and only the speed-time graph hides distance in its area.
Watch out: a flat line means stopped on a distance-time graph but constant speed on a speed-time graph. Check the vertical axis before you describe the motion.
§ 1.3 Mass and weight
Key ideas
- Mass is a measure of the quantity of matter in an object; it resists changes to the object's motion and is the same everywhere. Unit: kg.
- Weight is the gravitational force on an object. Unit: newton (N). It changes if the gravitational field changes.
- Gravitational field strength g is the force per unit mass: g = 9.8 N/kg near the Earth's surface, and it is numerically equal to the acceleration of free fall.
- A balance compares masses; a force meter (newtonmeter) measures weight.
Equations
W = m gweight = mass × gravitational field strengthN
Watch out: weight is a force, so it is always in newtons, never kg. A 60 kg student has a mass of 60 kg but a weight of 60 × 9.8 = 588 N.
§ 1.4 Density
Key ideas
- Density is mass per unit volume. Water is 1.0 g/cm³ = 1000 kg/m³.
- Regular solid: measure the dimensions, calculate V, then ρ = m / V. Liquid: use a balance and a measuring cylinder.
- Irregular solid: find its volume by displacement, the rise in level (or the overflow) when it is fully submerged in water.
- An object floats if its density is less than the density of the liquid, and sinks if it is greater. One liquid floats on a denser liquid.
Equations
ρ = m / Vdensity = mass ÷ volumekg/m³
Fig. 3 · Float or sink: compare the block's density with the liquid's, not with water's. The floating block sits with part of its volume above the surface.
Watch out: converting g/cm³ to kg/m³ means multiplying by 1000, not 100. So 0.8 g/cm³ = 800 kg/m³.
§ 1.5 Forces
Key ideas
- A force can change an object's size, shape, speed or direction. The resultant force is the single force equivalent to all the forces acting.
- With zero resultant force an object stays at rest or keeps a constant velocity. Extended: a resultant force gives an acceleration in its own direction, F = ma.
- Friction and drag act against motion; solid friction also transfers energy by heating.
- Hooke's law: up to the limit of proportionality, the extension of a spring is directly proportional to the load. The spring constant is k = F / x.
- Extension is the increase in length from the natural (unstretched) length.
- The moment of a force is force × perpendicular distance from the pivot to the line of action.
- Principle of moments: for a body in equilibrium, total clockwise moment = total anticlockwise moment about the same pivot. Equilibrium also needs zero resultant force.
- An object is stable while the vertical line through its centre of gravity stays inside its base.
Equations
F = m aresultant force = mass × accelerationN
F = k xforce = spring constant × extensionN, k in N/m
moment = F × dforce × perpendicular distance from the pivotN m
Fig. 4 · Hooke's law holds along the straight part only; past the limit of proportionality the graph bends and F = kx no longer applies.
Fig. 5 · A balanced beam: the anticlockwise moment of F1 equals the clockwise moment of F2 about the pivot, so the beam is in equilibrium.
Watch out: x in F = kx is the extension, the stretched length minus the natural length. Substituting the total length of the spring is the classic error.
§ 1.6 Momentum Extended only
Key ideas
- Momentum p = mv is a vector; its direction is the direction of the velocity. Unit: kg m/s.
- Impulse = force × time for which it acts = FΔt = Δp, the change in momentum it causes.
- Conservation of momentum: when no external resultant force acts, the total momentum before an interaction equals the total momentum after it. This applies to collisions and to explosions.
- The resultant force on an object equals its rate of change of momentum. Increasing the impact time (crumple zones, airbags) lowers the force for the same Δp.
Equations
p = m vmomentum = mass × velocitykg m/s
F = Δp / Δtresultant force = change in momentum ÷ time takenN
Fig. 6 · A sticking collision: the trolleys share the momentum but the total, 6 kg m/s, is unchanged.
Watch out: momentum is a vector. In head-on problems choose a positive direction first and give anything moving the other way a minus sign before you add.
§ 1.7 Energy, work and power
Key ideas
- Energy is held in stores (kinetic, gravitational potential, chemical, elastic, nuclear, electrostatic, internal) and moved between them by forces doing work, electrical work, heating, and waves.
- Conservation of energy: energy cannot be created or destroyed, only transferred. A Sankey diagram tracks where every joule goes.
- Work done = energy transferred. A force does work when it moves something along its own direction.
- Wasted energy spreads to the surroundings, usually as internal energy by heating.
- Power is the rate of doing work or transferring energy: 1 W = 1 J/s.
Equations
Ek = ½ m v²kinetic energy of a moving massJ
ΔEp = m g Δhchange in gravitational potential energy over a height change ΔhJ
W = F dwork done = force × distance moved in the force's directionJ
P = E / tpower = energy transferred ÷ time takenW
efficiency = useful / totaluseful energy (or power) out ÷ total in, × 100 for a percentageno unit
Fig. 7 · Every joule is accounted for: 100 J in, 60 J useful, 40 J wasted, so the device is 60% efficient.
Watch out: only the speed is squared in Ek = ½mv². Doubling the mass doubles the kinetic energy, but doubling the speed quadruples it.
§ 1.8 Pressure
Key ideas
- Pressure is the force per unit area: 1 Pa = 1 N/m².
- The same force gives a larger pressure over a smaller area: knife edges and drawing pins concentrate force, snowshoes and wide tyres spread it.
- In a liquid, pressure increases with depth and with the liquid's density, and at a given depth it acts equally in all directions.
- Δh in Δp = ρgΔh is the vertical change in depth below the surface.
Equations
p = F / Apressure = force ÷ area it acts onPa
Δp = ρ g Δhpressure change = density × g × depth changePa
Fig. 8 · The deeper the point below the surface, the greater the pressure from the liquid above it: the arrows grow with depth Δh.
Watch out: ρgΔh is only the extra pressure from the liquid. For the total pressure at depth, add the atmospheric pressure acting on the surface.