Forces

Forces and the load-extension graph

A force is a push or a pull, and it is measured in newtons (N). Forces change the motion of an object, and they can also change its size and shape: a spring stretches, a sponge is squashed, a ruler bends and a bar twists. Whenever an object is stretched or compressed, a pair of forces acts on it in opposite directions.

When you hang weights on a spring it stretches. The extension is the increase in length, so extension = stretched length - original length. If you plot load on the vertical axis against extension on the horizontal axis, a spring gives a straight line through the origin: the extension is proportional to the load. A material that behaves like this is an elastic solid, and the behaviour is called Hooke's law.

The gradient of that straight line is the spring constant, k. It is defined as the force per unit extension: k = F / x, where F is the force in newtons and x is the extension in metres. The unit of k is N/m. A large k means a stiff spring, because a large force is needed to produce a small extension. For example, a force of 3.0 N gives an extension of 0.15 m, so k = 3.0 / 0.15 = 20 N/m.

The straight line does not continue for ever. The limit of proportionality is the point on the graph at which the line stops being straight. Beyond it the extension is no longer proportional to the load: each extra newton produces a bigger extension than the one before, so the graph curves upwards. Loaded this far, a spring may also be permanently stretched and never return to its original length.

• To find k: clamp the spring vertically, measure its original length, add the load, measure the new length, and subtract to find the extension.
• Convert the extension to metres before dividing, and always use the extension, not the total length of the spring.
• Add the loads one at a time, read the ruler at eye level, and stop before the limit of proportionality is passed.

Resultant forces along a straight line

More than one force usually acts on an object at the same time. The resultant force is the single force that has exactly the same effect as all of those forces combined. Finding it is called finding the resultant, and it tells you what the motion of the object will do.

For forces acting along the same straight line, the rule is simple. Forces in the same direction add up, and forces in opposite directions subtract. A 5 N push to the right together with a 3 N push to the right gives a resultant of 8 N to the right. A 5 N push to the right together with a 3 N push to the left gives a resultant of 2 N to the right, because the larger force wins and the resultant acts in the direction of the larger force.

If the forces are equal and opposite they cancel out and the resultant is zero. The forces are balanced, and balanced forces produce no change in the object's motion. If you pull a rope to the left with 200 N while someone else pulls to the right with 200 N, the rope does not move, because the resultant force on it is zero.

A common case is an object resting on a surface. The weight of a book pulls it downwards and the contact force from the table pushes it upwards. These two forces are equal and opposite, so the resultant is zero and the book stays still. Whenever you work out a resultant, decide the direction first, then add or subtract, and always give the direction with the number.

• Same direction: add the forces.
• Opposite directions: subtract the smaller from the larger, and the resultant acts in the direction of the larger force.
• Equal and opposite: the resultant is zero and the forces are balanced.
• Always state the direction as well as the size of a resultant force.

Newton's first law and F = ma

Newton's first law says that an object either remains at rest or continues to move in a straight line at a constant speed, unless it is acted on by a resultant force. A resultant force is the only thing that can change an object's motion. If the resultant force is zero, the velocity does not change.

Velocity is speed in a stated direction, so a change of direction is a change of velocity even when the speed stays the same. A resultant force can therefore change an object's velocity by changing its speed, its direction of motion, or both. A car going round a bend at a steady 30 km/h is changing velocity because its direction is changing.

The size of the effect is given by F = ma: resultant force (N) = mass (kg) x acceleration (m/s2). The acceleration is always in the same direction as the resultant force. This makes sense: push an object forwards and it speeds up forwards, while the braking force on a car acts backwards and gives the car an acceleration backwards, so it slows down. One newton is the force needed to give a 1 kg mass an acceleration of 1 m/s2.

For example, a car of mass 1200 kg accelerates at 2.5 m/s2. The resultant force is F = ma = 1200 x 2.5 = 3000 N. The equation can be rearranged: a = F / m gives the acceleration when you know the force and the mass, and m = F / a gives the mass. Notice that for the same force a larger mass has a smaller acceleration.

• F = ma, where F is in newtons, m in kilograms and a in m/s2.
• The force and the acceleration are always in the same direction.
• Zero resultant force means zero acceleration, so the object stays at rest or keeps a constant velocity.

Friction and drag

Friction is the force between two surfaces in contact that are sliding, or trying to slide, over each other. It acts in the direction that opposes the motion and it produces heating. Rub your hands together and they warm up, because the work done against friction is transferred to the thermal energy store of the surfaces.

Friction is not always unwanted. The friction between your shoes and the ground lets you walk, the friction in a car's brakes is what stops the car, and the friction between the tyres and the road keeps a car from sliding. Where friction is a nuisance it wastes energy and wears surfaces away. It can be reduced by lubricating the surfaces with oil or grease, by making the surfaces smoother, or by using ball bearings so that the surfaces roll instead of sliding.

When an object moves through a liquid or a gas, the friction is called drag. Drag on an object moving through air is called air resistance. Drag acts in the opposite direction to the motion and it increases as the object moves faster, so it is much larger at high speed. This is why a car uses far more fuel on a motorway than in town, and why cars, cyclists and aircraft are given a smooth, streamlined shape to reduce it.

Consider a falling object. Its weight pulls it downwards and drag pushes upwards. At first the drag is small, so there is a resultant force downwards and the object accelerates. As it speeds up, the drag grows, until eventually the drag equals the weight. The resultant force is then zero, so by Newton's first law the object falls at a steady speed and no longer accelerates.

• Solid friction opposes sliding between two surfaces and heats them up.
• Drag is the friction on an object moving through a liquid or a gas; air resistance is drag in a gas.
• Drag increases with speed, and streamlining reduces it.

Motion in a circular path

An object moving in a circle at a constant speed is still accelerating, because its direction is changing all the time. For the object to keep turning, a force must act on it that is perpendicular to its motion and directed towards the centre of the circle. Without that force the object would simply continue in a straight line.

The force needed depends on the mass, the speed and the radius of the circle. If the force is increased while the mass and the radius stay the same, the object moves faster, just as a stone on a string goes round more quickly when you whirl it harder. If the force is increased while the mass and the speed stay the same, the object is pulled into a smaller circle, so the radius decreases. If the mass is increased, a larger force is needed to keep the same speed and radius, because a heavier object is harder to turn.

Think of a car going round a roundabout. The friction between the tyres and the road provides the force towards the centre. If the car goes too fast, the friction cannot supply a large enough force, so the car slides off in a straight line, which is exactly what Newton's first law predicts. A satellite in orbit is kept in its circular path by the gravitational force of the Earth, which pulls it towards the centre.

• The force is perpendicular to the motion and points towards the centre of the circle.
• Increase the force with the mass and radius constant and the speed increases.
• Increase the force with the mass and speed constant and the radius decreases.
• Increase the mass and you need a larger force to keep the same speed and radius.

Moments and the principle of moments

A force can turn an object as well as push or pull it. The moment of a force is a measure of its turning effect about a pivot. You feel it when you push a door near the handle: a small push far from the hinge turns the door easily, while the same push close to the hinge hardly moves it.

The moment is defined by moment = force x perpendicular distance from the pivot. The force is in newtons, the distance is in metres, and the moment is measured in newton metres (N m). The perpendicular distance is the distance from the pivot to the line of action of the force, measured at right angles to that line. Using the distance along the object instead of the perpendicular distance is the commonest mistake in this topic.

The principle of moments states that when an object is in equilibrium, the total clockwise moment equals the total anticlockwise moment. A see-saw balances when the moment of the person on one side equals the moment of the person on the other, so a lighter person sits further from the pivot to make up the difference.

Worked example. A uniform metre rule is pivoted at its centre. A 4.0 N weight hangs at the 20 cm mark and a second weight hangs at the 70 cm mark, and the rule balances. The 4.0 N weight is 30 cm, or 0.30 m, from the pivot, so its moment is 4.0 x 0.30 = 1.20 N m. The second weight is 20 cm, or 0.20 m, from the pivot, so W x 0.20 = 1.20 and W = 1.20 / 0.20 = 6.0 N.

When there is more than one force on each side, add the moments on each side first and then equate the two totals. An object is in equilibrium only when there is no resultant force and no resultant moment acting on it.

• moment = force x perpendicular distance from the pivot, in N m.
• Principle of moments: total clockwise moment = total anticlockwise moment when the object is in equilibrium.
• An object is in equilibrium when there is no resultant force and no resultant moment on it.

Centre of gravity and stability

The centre of gravity of an object is the point at which the whole weight of the object may be taken to act. For a uniform ruler or a regular shape it is at the centre, but for an irregular object it can lie outside the material altogether, as it does for a horseshoe.

To find the centre of gravity of an irregularly shaped plane lamina, make a small hole near one edge and hang the lamina from a pin so that it can swing freely. Hang a plumb line from the same pin and draw a pencil line along the plumb line on the lamina. The centre of gravity lies somewhere on that line, because the lamina hangs with its weight acting through the point of suspension. Repeat with the hole in a second position and draw the second line. The centre of gravity is where the two lines cross.

The position of the centre of gravity decides whether an object is stable. An object is stable if its centre of gravity is low and its base is wide. An object topples when it is tilted so far that the line of action of its weight falls outside the base. Once that happens there is a resultant moment about the edge of the base, and the object turns over.

This is why a double-decker bus is less stable than a racing car even though both have four wheels: the bus has a high centre of gravity, so only a small tilt is needed to push the line of its weight outside the base. A racing car has a low centre of gravity and a wide track, so it can lean much further before it tips. Lowering the centre of gravity, for example by putting heavy luggage at the bottom of a suitcase, makes an object more stable.

• Centre of gravity: the point at which the whole weight of the object may be taken to act.
• Find it by hanging the lamina freely from two different points and drawing the plumb line each time; the centre of gravity is where the lines cross.
• Stability is increased by a low centre of gravity and a wide base.
• An object topples when the line of action of its weight falls outside its base.