Topic 1: Motion, forces and energy

This topic is the foundation of mechanics. You will learn to describe motion with quantities and graphs, link forces to changes in motion, and follow energy as it moves between stores. The same handful of equations reappears throughout the course, so mastering them now pays off everywhere.

Physical quantities and units

Every measurement in physics has a numerical value and a unit, and both must be quoted. The SI base units you meet here are the metre (m) for length, the kilogram (kg) for mass, and the second (s) for time. Other units are built from these, for example speed in m/s and acceleration in m/s^2. Prefixes scale units up or down: kilo (k) means x1000, centi (c) means x0.01, milli (m) means x0.001. So 2.5 km = 2500 m and 30 cm = 0.30 m. A scalar quantity has size only (mass, time, distance, speed, energy), while a vector quantity has both size and direction (force, velocity, acceleration, momentum, weight). Vectors pointing in the same straight line are added by taking one direction as positive and the other as negative.

Speed, velocity, acceleration and motion graphs

Speed is distance travelled per unit time: speed = distance / time. Velocity is speed in a stated direction. Acceleration is the rate of change of velocity: a = (v - u) / t, where u is the starting velocity and v the final velocity. A negative acceleration (deceleration) means slowing down. Worked example: a car speeds up from 8 m/s to 20 m/s in 4 s. a = (20 - 8) / 4 = 3 m/s^2. On a distance-time graph the gradient gives speed: a straight slope is constant speed, a flat line is at rest, and a curve means changing speed. On a speed-time graph the gradient gives acceleration, and the area under the line gives the distance travelled. For an object speeding up from rest to 20 m/s in 4 s, distance = area of triangle = 0.5 x 4 x 20 = 40 m. Near the Earth, objects in free fall accelerate at about g = 9.8 m/s^2 (often taken as 10 m/s^2).

Mass and weight (W = mg)

Mass is the quantity of matter in an object, measured in kilograms (kg); it does not change if you move the object to the Moon. Weight is the gravitational force acting on that mass, measured in newtons (N). They are linked by W = mg, where g is the gravitational field strength (about 9.8 N/kg on Earth, often used as 10 N/kg). Worked example: a 6 kg bag has weight W = 6 x 9.8 = 58.8 N on Earth, but on the Moon where g is roughly 1.6 N/kg its weight is only 6 x 1.6 = 9.6 N - the mass stays 6 kg. The gravitational field strength g can be defined as weight per unit mass: g = W / m. A larger mass also has greater inertia, meaning it resists changes to its motion more strongly.

Density

Density is mass per unit volume: rho = m / V. The standard unit is kg/m^3, though g/cm^3 is common (water is 1000 kg/m^3 = 1.0 g/cm^3). To find the density of a regular solid, measure its mass on a balance and calculate its volume from its dimensions. For an irregular solid, lower it into a measuring cylinder and read the rise in water level (displacement) to get its volume. Worked example: a metal block of mass 270 g occupies 100 cm^3, so rho = 270 / 100 = 2.7 g/cm^3 - which identifies it as aluminium. An object floats in a liquid if its density is less than the liquid's density, and sinks if it is greater.

Forces and Hooke's law

A force is a push or pull, measured in newtons (N), that can change an object's speed, direction or shape. When several forces act, the single force with the same effect is the resultant force. If the resultant force is zero the forces are balanced, so the object stays at rest or moves at constant velocity. An unbalanced (non-zero) resultant force causes acceleration in its direction, given by F = ma. Worked example: a 1500 N resultant force on a 500 kg car gives a = F / m = 1500 / 500 = 3 m/s^2. Hooke's law states that the extension of a spring is directly proportional to the load applied, provided the limit of proportionality is not exceeded: F = k x, where k is the spring constant in N/m and x is the extension. Beyond the limit the spring no longer returns to its original length and the load-extension graph stops being a straight line.

Friction and air resistance

Friction is a force that opposes the relative motion of two surfaces in contact; air resistance (drag) is friction acting through a fluid such as air or water. Friction always acts to slow moving objects and converts kinetic energy into thermal energy, warming the surfaces. It is useful too: without it, walking, braking and gripping would be impossible. When a falling object's air resistance grows until it balances its weight, the resultant force becomes zero and the object stops accelerating - it then falls at a steady terminal velocity. A skydiver reaches terminal velocity, then a smaller, slower terminal velocity once the parachute opens and dramatically increases the air resistance.

Moments and centre of gravity

A moment is the turning effect of a force about a pivot: moment = force x perpendicular distance from the pivot. The unit is the newton metre (N m). The principle of moments states that for an object in equilibrium, the total clockwise moment about any pivot equals the total anticlockwise moment. Worked example: a child of weight 300 N sits 1.2 m from a seesaw pivot. To balance, an adult on the other side must produce 300 x 1.2 = 360 N m; if the adult sits 0.8 m from the pivot, the needed force is 360 / 0.8 = 450 N. The centre of gravity is the single point at which the whole weight of an object appears to act. An object is more stable when it has a low centre of gravity and a wide base, and it topples once its centre of gravity passes outside the base.

Momentum (Supplement)

Momentum is mass in motion: p = mv, measured in kg m/s, and it is a vector. Worked example: a 2 kg ball moving at 5 m/s has momentum p = 2 x 5 = 10 kg m/s. Force can be defined as the rate of change of momentum: F = (mv - mu) / t. This explains why a longer impact time means a smaller force - crumple zones and crash mats extend the collision time and so reduce the force on people. The principle of conservation of momentum states that in a closed system with no external forces, total momentum before a collision equals total momentum after. Worked example: a 4 kg trolley at 3 m/s strikes and sticks to a stationary 2 kg trolley. Total momentum before = 4 x 3 = 12 kg m/s; after, the combined 6 kg moves at v = 12 / 6 = 2 m/s.

Energy stores, transfers and conservation

Energy is measured in joules (J) and is held in stores such as kinetic, gravitational potential, elastic (strain), chemical, nuclear, internal (thermal) and electrostatic. Energy is transferred between stores mechanically (by forces), electrically, by heating, or by waves such as light and sound. The principle of conservation of energy states that energy cannot be created or destroyed, only transferred from one store to another; the total amount in a closed system stays constant. For a swinging pendulum, energy moves repeatedly between gravitational potential at the top of the swing and kinetic at the bottom, while friction gradually transfers some to the thermal store of the surroundings.

Work, power and efficiency

Work is done when a force moves its point of application along its line of action, and work done equals energy transferred: W = F d, in joules. Worked example: pushing a box with 40 N over 3 m does W = 40 x 3 = 120 J of work. Power is the rate of doing work or transferring energy: P = W / t, measured in watts (W), where 1 W = 1 J/s. If the 120 J above is done in 4 s, P = 120 / 4 = 30 W. Efficiency compares useful output energy with total input energy: efficiency = (useful energy output / total energy input) x 100%. A motor supplied with 200 J that gives 150 J of useful kinetic energy is 75% efficient, the remaining 50 J being wasted, usually as thermal energy.

Kinetic and gravitational potential energy

Kinetic energy is the energy of a moving object: KE = 0.5 m v^2. Note that velocity is squared, so doubling speed gives four times the kinetic energy. Worked example: a 1000 kg car at 20 m/s has KE = 0.5 x 1000 x 20^2 = 200000 J. Gravitational potential energy is the energy stored by raising an object in a gravitational field: change in GPE = m g h, where h is the change in height. Lifting a 5 kg box 2 m (taking g = 10 N/kg) stores GPE = 5 x 10 x 2 = 100 J. When the box falls, this GPE transfers to KE: ignoring air resistance, m g h = 0.5 m v^2, which lets you find the landing speed.

Pressure (p = F/A and in liquids)

Pressure is the force acting per unit area at right angles to a surface: p = F / A, measured in pascals (Pa), where 1 Pa = 1 N/m^2. The same force spread over a smaller area gives a higher pressure, which is why sharp knives cut and snowshoes stop you sinking. Worked example: a 600 N person standing on shoes of total area 0.02 m^2 exerts p = 600 / 0.02 = 30000 Pa. In a liquid, pressure increases with depth and acts equally in all directions. The pressure difference due to a column of liquid is given by the change in pressure = rho g h, where rho is the liquid density and h the depth. Worked example: at 3 m depth in water (rho = 1000 kg/m^3, g = 10 N/kg), the extra pressure is 1000 x 10 x 3 = 30000 Pa.