Topic 5: Forces

A force is a push or a pull that can change an object's shape, speed or direction. This topic builds from describing forces and motion up to Newton's laws and momentum, with several key equations you must apply confidently. Watch for higher-tier (HT) content on fluid pressure and conservation of momentum.

Scalars and vectors

A scalar quantity has only a magnitude (size), such as distance, speed, mass, energy and temperature. A vector quantity has both a magnitude and a direction, such as displacement, velocity, acceleration, force and momentum. Vectors are usually drawn as arrows: the length of the arrow represents the magnitude and the way it points represents the direction. Confusing the two is a common mistake. For example, speed is scalar but velocity is a vector because it specifies a direction, so a car going round a roundabout at a steady 20 m/s has constant speed but changing velocity.

Contact, non-contact forces and weight

Forces are interactions between objects and are themselves vectors. A contact force needs the objects to be touching, for example friction, air resistance, tension and the normal contact (reaction) force. A non-contact force acts at a distance with no touching, for example gravitational force, magnetic force and electrostatic force. Weight is the force of gravity acting on an object's mass and is measured in newtons (N) with a calibrated spring balance (newtonmeter). Calculate weight using W = mg, where m is mass in kg and g is the gravitational field strength (about 9.8 N/kg on Earth). Worked example: a 6 kg bag has weight W = mg = 6 x 9.8 = 58.8 N. Weight acts at the centre of mass, while mass (in kg) stays the same wherever you are.

Resultant forces and free-body diagrams

Most objects have several forces acting on them at once. The resultant force is the single force that has the same effect as all the forces combined. For forces along a straight line, add them when they point the same way and subtract when they oppose. A free-body diagram shows just the object with arrows for each force, drawn from the object and labelled. If the resultant force is zero the forces are balanced and the object stays still or keeps a constant velocity. Worked example: a box pushed with 30 N to the right while friction is 12 N to the left has a resultant force of 30 - 12 = 18 N to the right. HT: two forces at right angles can be combined into a single resultant using a scale drawing (a vector diagram), where the diagonal of the rectangle gives the magnitude and direction.

Work done and energy transfer

Work is done whenever a force makes an object move in the direction of the force, and doing work transfers energy. Calculate work using W = Fs, where W is work in joules (J), F is force in newtons (N) and s is the distance moved along the line of the force in metres (m). One joule equals one newton-metre (1 J = 1 N m). Worked example: dragging a crate 5 m against a friction force of 40 N does W = Fs = 40 x 5 = 200 J of work, which is transferred to the surroundings as heat (raising the temperature of the surfaces).

Forces and elasticity

Stretching, compressing or bending an object needs more than one force. If the object returns to its original shape after the force is removed the deformation is elastic; if it stays deformed it is inelastic (plastic). For a spring within its limit of proportionality, extension is directly proportional to force: F = ke, where F is force in N, k is the spring constant in N/m and e is the extension (or compression) in m. This gives a straight line through the origin on a force-extension graph; beyond the limit of proportionality the line curves. The elastic potential energy stored equals the work done stretching it. Worked example: a spring with k = 25 N/m stretched by an extension of 0.20 m needs a force F = ke = 25 x 0.20 = 5 N.

Moments, levers and gears

A moment is the turning effect of a force about a pivot. Calculate it with M = Fd, where M is the moment in newton-metres (N m), F is the force in N and d is the perpendicular distance from the pivot to the line of action of the force in m. When an object is balanced and not turning, the total clockwise moment equals the total anticlockwise moment (the principle of moments). Levers and gears act as force multipliers. A lever uses a long distance from the pivot to make a small effort produce a large turning effect. Gears transmit rotation: a small gear driving a large gear gives a larger moment but turns more slowly. Worked example: a force of 8 N applied 0.5 m from a pivot gives a moment of M = Fd = 8 x 0.5 = 4 N m.

Pressure in fluids (HT)

Pressure is the force acting per unit area, given by p = F/A, where p is in pascals (Pa), F is force in N and A is area in m squared. In a fluid (a liquid or gas) the pressure at a depth is caused by the weight of the fluid above and acts equally in all directions. The pressure due to a column of liquid is p = h x rho x g, where h is the depth in m, rho is the density in kg/m cubed and g is gravitational field strength. Pressure therefore increases with depth, which is why dams are thicker at the bottom. Upthrust is the upward force from a fluid; an object floats when upthrust equals its weight. The atmosphere also exerts pressure, which decreases with height because there is less air above you.

Describing motion

Distance is how far an object travels (a scalar) while displacement is the straight-line distance and direction from start to finish (a vector). Speed is a scalar; typical values are about 1.5 m/s walking, 3 m/s running and 6 m/s cycling, and the speed of sound in air is roughly 330 m/s. Velocity is speed in a given direction, so an object moving in a circle at constant speed has a continuously changing velocity. Acceleration is the rate of change of velocity, a = (v - u) / t, measured in m/s^2, where u is initial velocity and v is final velocity. HT equation: v^2 - u^2 = 2as. Worked example: a car speeding up from 5 m/s to 20 m/s in 6 s has acceleration a = (20 - 5) / 6 = 2.5 m/s^2.

Distance-time and velocity-time graphs

On a distance-time graph the gradient (slope) gives the speed; a flat horizontal line means the object is stationary, and a steeper line means a faster speed. For an accelerating object the line curves, and the speed at an instant is found from the gradient of the tangent. On a velocity-time graph the gradient gives the acceleration, and a horizontal line means constant velocity. Crucially, the area under a velocity-time graph equals the distance travelled. Worked example: an object at a steady 10 m/s for 8 s covers an area (distance) of 10 x 8 = 80 m on its velocity-time graph.

Newton's three laws

Newton's first law: an object stays at rest or moves at constant velocity unless acted on by a resultant force. The tendency to keep moving the same way is called inertia. Newton's second law: resultant force equals mass times acceleration, F = ma, where F is in N, m in kg and a in m/s^2; inertial mass measures how hard it is to change an object's velocity. Newton's third law: when two objects interact they exert equal and opposite forces on each other. Worked example: a resultant force of 1200 N on a 400 kg trolley gives acceleration a = F/m = 1200 / 400 = 3 m/s^2.

Stopping distance

The stopping distance of a vehicle is the sum of the thinking distance and the braking distance. Thinking distance is how far the vehicle travels during the driver's reaction time, and it increases with speed and with anything that slows reactions, such as tiredness, alcohol, drugs or distractions. Braking distance is how far the vehicle travels while the brakes act; it increases with higher speed and is made worse by poor brakes or tyres and by wet or icy roads. Larger braking forces cause larger decelerations; very large decelerations can be dangerous and may cause the brakes to overheat or skidding.

Momentum and its conservation (HT)

Momentum is a vector defined by p = mv, where p is momentum in kg m/s, m is mass in kg and v is velocity in m/s. In a closed system, with no external resultant force, the total momentum before an event equals the total momentum after it: this is conservation of momentum and it applies to collisions and explosions. Worked example: a 2 kg trolley moving at 3 m/s has momentum p = mv = 2 x 3 = 6 kg m/s; if it sticks to a stationary 1 kg trolley, the combined 3 kg mass moves so total momentum stays 6 kg m/s, giving v = 6 / 3 = 2 m/s. A larger force or longer contact time produces a greater change in momentum (impulse).