Forces change how things move, and Newton's three laws describe how. The topic also involves describing motion precisely, including the distinction between speed and velocity, the two kinds of motion graph and what stopping a car actually involves.
A scalar has size only: mass, speed, distance, energy, temperature. A vector has size and direction: force, velocity, displacement, acceleration, momentum, weight.
Vectors are drawn as arrows whose length shows the magnitude and whose direction shows the direction. Distance and speed are scalars; displacement and velocity are their vector partners.
A force is a push or a pull arising from an interaction between two objects, measured in newtons.
Contact forces need the objects to touch: friction, air resistance, tension, the normal contact force. Non-contact forces act across a gap: gravitational, electrostatic and magnetic.
Weight is the force of gravity on an object: \(W = mg\), where \(g\) is the gravitational field strength, about 9.8 N/kg on Earth.
Mass and weight are different. Mass is a scalar in kilograms and does not change with location; weight is a vector in newtons and does change, because \(g\) differs from one body to another. Weight acts from the object's centre of mass and is measured with a calibrated spring balance.
Several forces on an object can be replaced by a single resultant that has the same effect. Along one line, add forces in the same direction and subtract those in opposite directions.
If the resultant is zero the forces are balanced and the motion does not change. If it is not zero the object accelerates in the direction of the resultant.
Work is done whenever a force moves an object along the line of the force: \(W = Fs\), in joules. One joule is one newton-metre.
Work done against friction transfers energy to thermal stores, so brakes and rubbed hands get warm.
Stretching, bending or compressing an object needs more than one force, because a single force would just move the object. A deformation is elastic if the object returns to its original shape, and inelastic if it does not.
Up to the limit of proportionality, extension is directly proportional to force: \(F = ke\), where \(k\) is the spring constant. Beyond that limit the force–extension graph curves. Proportionality and elasticity are separate limits: passing the limit of proportionality only ends the straight line, while permanent deformation begins once the elastic limit is passed. The work done in stretching is stored elastically: \(E_e = \tfrac12 k e^2\), which is the area under the linear part of the graph.
A force can turn an object about a pivot. The moment is \(M = Fd\), where \(d\) is the perpendicular distance from the pivot to the line of action of the force.
If an object is balanced, the total clockwise moment equals the total anticlockwise moment. A lever is a force multiplier: a long handle gives a large moment from a small force. Gears do the same job for rotation. A large gear driven by a small one turns more slowly but with a greater moment.
Pressure is force per unit area acting at right angles to a surface: \(p = \frac{F}{A}\), in pascals.
In a liquid, pressure increases with depth because there is more liquid above pressing down, and it increases with the density of the liquid: \(p = h\rho g\). Because pressure acts in all directions, the pressure on the bottom of a submerged object is greater than on the top.
That difference in pressure between top and bottom produces a resultant upward force called upthrust. Its size equals the weight of the fluid the object displaces.
If the upthrust equals the object’s weight there is no resultant vertical force, so the object floats at rest or stays at the depth it has reached; if the weight is greater it sinks. Whether an object floats therefore comes down to its density compared with the fluid's, which is how a steel ship, mostly air inside, floats.
The atmosphere is a fluid too, and atmospheric pressure is caused by the weight of the air above a surface. At sea level it is about 100 kPa.
As altitude increases there is less air above, so atmospheric pressure falls. The air also becomes less dense as altitude increases.
Speed is a scalar; velocity is speed in a stated direction. Average speed is \(v = \frac{s}{t}\).
Typical values: walking about 1.5 m/s, running about 3 m/s, cycling about 6 m/s, and the speed of sound in air about 330 m/s.
On a distance–time graph the gradient is the speed. A horizontal line means stationary, a straight sloping line means constant speed, and a steeper line means faster.
A curve means the speed is changing; to find the speed at an instant, draw a tangent and take its gradient.
Acceleration is the rate of change of velocity: \(a = \frac{\Delta v}{t}\), in m/s². An object slows down when its acceleration is in the opposite direction to its velocity. A negative value on its own only means the acceleration points whichever way has been taken as negative.
There is also \(v^2 - u^2 = 2as\), used when the time is unknown. Near the Earth's surface any object in free fall accelerates at about 9.8 m/s² regardless of its mass. Falling through a fluid, an object reaches terminal velocity when drag has grown to equal its weight, so the resultant force is zero and it falls at a steady speed.
On a velocity–time graph the gradient is the acceleration and the area under the line is the displacement, which is also the distance travelled whenever the velocity never becomes negative. A horizontal line means constant velocity.
The two graph types are easily confused: on a distance–time graph the gradient is speed, and the area under it has no meaning.
If the resultant force on an object is zero, a stationary object stays stationary and a moving object keeps moving at the same speed in the same direction.
So a car travelling at a steady speed on a straight road has balanced forces, with the driving force equal to the resistive forces. A change in speed or direction always means a non-zero resultant force.
Acceleration is proportional to the resultant force and inversely proportional to the mass: \(F = ma\).
For the same force, a heavier object accelerates less. For the same mass, a bigger force gives a bigger acceleration.
When two objects interact, the forces they exert on each other are equal in size and opposite in direction.
The two forces of an interaction pair always act on different objects, which is why they never cancel out. A swimmer pushes the water backwards and the water pushes the swimmer forwards.
Inertia is the tendency of an object to keep its state of rest or uniform motion. Inertial mass measures how difficult it is to change an object's velocity.
It is defined as the ratio of force to acceleration, \(m = \frac{F}{a}\), so a large inertial mass means a small acceleration for a given force.
Stopping distance = thinking distance + braking distance. Thinking distance is how far the vehicle travels during the driver's reaction time; braking distance is how far it travels while the brakes are applied.
Both increase with speed, so stopping distance rises sharply as speed rises.
A typical human reaction time is between 0.2 and 0.9 s. It can be measured by catching a dropped ruler, or with a computer-based test.
Tiredness, alcohol, drugs and distractions all increase reaction time, and therefore thinking distance, but they do not affect braking distance.
Braking distance is increased by anything that reduces the grip or the braking force: wet or icy roads, leaves or gravel on the surface, worn tyres and worn brake pads.
These are the conditions a driver should respond to by leaving a bigger gap.
Braking transfers the vehicle's kinetic energy to thermal stores in the brakes, so the work done by the braking force equals the kinetic energy: \(Fs = \tfrac12 m v^2\).
Because kinetic energy depends on the square of the speed, doubling the speed roughly quadruples the braking distance, for the same vehicle and the same braking force. Large decelerations mean large forces on the vehicle and its occupants, and risk brakes overheating or loss of control.
Momentum is a property of moving objects: \(p = mv\), measured in kg m/s. It is a vector, so direction matters and one direction must be taken as positive.
In a closed system, the total momentum before an event equals the total momentum after it. This holds for collisions and for explosions.
A stationary object has zero momentum, so when something explodes from rest the fragments must carry equal and opposite momenta.
A resultant force changes an object’s momentum, and the force is the rate of change of momentum. For constant mass \(F = \frac{m\Delta v}{\Delta t}\), which gives the average resultant force over the interval.
Because the time is on the bottom, extending the time over which momentum changes reduces the force. Seatbelts stretch, cars have crumple zones and a gymnast bends their knees on landing for this reason.
An object moving in a circle at constant speed is still accelerating, because its direction is changing and velocity is a vector.
That acceleration needs a resultant force directed towards the centre of the circle, called the centripetal force. Gravity provides it for a planet, tension for a whirled conker, and friction for a car going round a bend.
