Motion, forces and energy

📚The physics

Scalars and vectors. A scalar has size only — distance, speed, mass, energy, time. A vector has size and direction — displacement, velocity, acceleration, force, weight, momentum. When vectors act along the same line you add or subtract; at right angles you use a scale drawing or Pythagoras.

Motion. Speed = distance/time; velocity is speed in a stated direction; acceleration = change in velocity / time taken. On a distance–time graph the gradient is the speed, so a horizontal line means stationary. On a speed–time graph the gradient is acceleration and the area under the line is the distance travelled.

All objects in free fall accelerate at about 9.8 m s\(^{-2}\) regardless of mass. With air resistance, a falling object speeds up, drag grows until it equals the weight, the resultant force becomes zero, and the object falls at constant terminal velocity. It does not slow down — it stops speeding up.

Mass and weight are different quantities. Mass is the amount of matter, in kilograms, and is the same everywhere. Weight is the gravitational force on that mass, in newtons, and changes with location: \( W = mg \). On the Moon your mass is unchanged and your weight is about a sixth.

Density \( \rho = m/V \). An object floats if its density is less than that of the liquid. For an irregular solid, find the volume by displacement in a measuring cylinder.

Forces change an object’s speed, direction or shape. If the resultant force is zero the object stays at rest or keeps moving at constant velocity; if it is not zero the object accelerates in the direction of the resultant, with \( F = ma \). Friction and air resistance always oppose motion and transfer energy to the surroundings as heat.

Hooke’s law: extension is proportional to load, up to the limit of proportionality. Beyond that the graph curves and the spring may not return to its original length.

Turning effects. The moment of a force is force × perpendicular distance from the pivot. For an object in equilibrium, the principle of moments says total clockwise moments equal total anticlockwise moments about any pivot. The centre of gravity is where the whole weight appears to act; an object is more stable with a low centre of gravity and a wide base, and topples once its centre of gravity passes outside the base.

Momentum \( p = mv \) is a vector, so direction and sign matter. In a collision, total momentum before equals total momentum after, provided no external force acts.

Energy is never created or destroyed, only transferred between stores.

Note the square on the kinetic energy, so doubling the speed quadruples the energy. Efficiency = useful energy out / total energy in, and can never exceed 100%.

Pressure \( p = F/A \). The same force spread over a smaller area gives a larger pressure — the reason a drawing pin works and snowshoes stop you sinking. In a liquid, pressure increases with depth: \( \Delta p = \rho g \Delta h \), and it acts equally in all directions at a given depth.