2.2.1 Interactions between objects

Forces and interactions

1. A force is a push or a pull that one object exerts on another object.
2. Forces are measured in newtons, which have the symbol $N$.
3. A force can make an object speed up, slow down, change direction or change shape.
4. Forces never come from a single object on its own, because every force is produced by an interaction between two objects.
5. When you kick a football, the force on the ball comes from the interaction between your foot and the ball.
6. When an apple falls from a tree, the force that pulls it down comes from the interaction between the apple and the Earth.
7. Interactions are sorted into two groups depending on whether the objects need to touch.
   1. Non-contact interactions act across a gap, so the objects can be separated.
   2. Contact interactions act only while the surfaces of the objects are touching.

Non-contact forces

1. A non-contact force acts between objects that are not touching, so it can act across empty space.
2. There are three kinds of non-contact interaction that you need to recall: gravitational, electrostatic and magnetic.
3. Gravitational interactions happen between any two objects that have mass.
   1. The gravitational force is always attractive, so it pulls the two objects towards each other.
   2. The gravitational pull of the Earth on an object near its surface is the object's weight, which acts downwards towards the centre of the Earth.
   3. Gravity also holds the Moon in orbit around the Earth and the Earth in orbit around the Sun.
4. Electrostatic interactions happen between objects that carry an electric charge.
   1. Objects with opposite charges, one positive and one negative, attract each other.
   2. Objects with like charges, both positive or both negative, repel each other.
   3. A balloon rubbed on a woollen jumper becomes charged and can then attract small pieces of paper or stick to a wall.
5. Magnetic interactions happen between magnets, or between a magnet and a magnetic material such as iron, steel, nickel or cobalt.
   1. Unlike poles (a north pole and a south pole) attract, while like poles (two north poles or two south poles) repel.
   2. A magnet always attracts a magnetic material that is not itself a magnet, which is how a fridge magnet holds a shopping list in place.
6. Gravitational forces can only attract, while electrostatic and magnetic forces can either attract or repel.
7. Each non-contact force gets weaker as the distance between the objects increases.

Contact forces

1. A contact force acts only when the surfaces of two objects are touching each other.
2. The normal contact force is the push that a surface exerts on an object pressing on it.
   1. The word normal means perpendicular, so this force always acts at right angles to the surface.
   2. A mug resting on a desk is held up by an upward normal contact force from the desk.
   3. A ladder leaning against a wall feels a horizontal normal contact force from the wall, because the wall surface is vertical.
3. Friction is the force that opposes the movement, or the attempted movement, of one surface sliding over another.
   1. Friction acts parallel to the surfaces in contact, in the opposite direction to the sliding.
   2. Even surfaces that feel smooth have tiny bumps that catch on one another, and this is what produces friction.
   3. Friction between your shoes and the pavement lets you walk, and friction between a brake pad and a bicycle wheel slows the bike.
4. Air resistance is a type of friction that acts on an object moving through air, and it gets larger as the object moves faster.
5. A similar resistive force, called drag, acts on objects moving through water or any other liquid.
6. Two further contact forces are common in force problems.
   1. Tension is the pull exerted by a stretched string, rope or cable on the object attached to it.
   2. Upthrust is the upward push of a liquid or gas on an object that is partly or fully immersed in it.

2.2.2 Force pairs in interactions

Every interaction involves two objects

1. An interaction always involves two objects, so it always produces two forces, one acting on each object.
2. If object A pushes or pulls on object B, then object B also pushes or pulls on object A at the same time.
3. These two forces together are called an interaction pair.
4. The two forces in an interaction pair have three features in common.
   1. They are the same type of force, so a gravitational force is always paired with a gravitational force and a normal contact force with a normal contact force.
   2. They act along the same line but point in opposite directions.
   3. They start and stop at the same instant, because they are produced by the same interaction.
5. The two forces act on different objects, which is the most important feature of an interaction pair.
6. To describe a pair fully, use the pattern "the force on object A by object B" for one force and "the force on object B by object A" for the other.

Pairs in contact interactions

1. In a contact interaction, the two surfaces press on each other or rub against each other, so each surface feels a force.
2. When your hand pushes a door, your hand exerts a force on the door and the door exerts a force back on your hand, which is why you can feel the door pressing on your palm.
3. When a mug rests on a desk, the mug pushes down on the desk and the desk pushes up on the mug.
   1. Both of these forces are normal contact forces, acting at right angles to the desk surface.
   2. One force acts on the desk and the other acts on the mug.
4. When you walk, your shoe pushes backwards on the pavement and the pavement pushes forwards on your shoe.
   1. Both of these forces are friction forces between the touching surfaces.
   2. The forward force on your shoe is the force that moves you along.
5. When a hammer hits a nail, the hammer pushes the nail into the wood and the nail pushes back on the hammer, which is why the hammer stops.

Pairs in non-contact interactions

1. Non-contact interactions also produce a force on each of the two objects, even though there is a gap between them.
2. In a gravitational interaction, the Earth pulls a falling apple downwards and the apple pulls the Earth upwards.
   1. The pull of the apple on the Earth has no noticeable effect, because the Earth has an enormous mass.
   2. The force on the Earth is still there, and it acts on the Earth, not on the apple.
3. In a magnetic interaction, a magnet pulls a steel paper clip towards it and the paper clip pulls the magnet towards itself.
   1. If the magnet is placed on a floating cork, the magnet visibly moves towards a paper clip held near it, which shows the paper clip really does exert a force on the magnet.
4. In an electrostatic interaction, two charged balloons hung side by side by threads push each other apart.
   1. Each balloon swings away from the other, which shows that both balloons experience a force.
5. Gravity between the Earth and the Moon is also an interaction pair, with the Earth pulling the Moon and the Moon pulling the Earth.
   1. The Moon's pull on the Earth's oceans helps to cause the tides.

Choosing the object you are studying

1. Because the two forces in a pair act on different objects, only one force from each pair acts on any object you choose to study.
2. To find the forces acting on an object, list every interaction that object is involved in and keep only the force that acts on it from each one.
3. For a mug resting on a desk, the mug takes part in two interactions.
   1. The interaction with the Earth gives the mug a downward gravitational force, its weight.
   2. The interaction with the desk gives the mug an upward normal contact force.
4. The partner forces, the mug pulling up on the Earth and the mug pushing down on the desk, act on other objects, so they are left out when you study the mug.

2.2.3 Forces as vectors

Force is a vector

1. Every physical quantity is either a scalar quantity or a vector quantity.
2. A scalar has a size only, such as a mass of $5kg$, a time of $20s$ or an energy of $300J$.
3. A vector has a size and a direction, such as a displacement of $40m$ north or a velocity of $3m/s$ to the left.
4. Force is a vector, because a push or pull always acts in a particular direction.
5. A $20N$ push to the right has a completely different effect from a $20N$ push to the left, even though the magnitude is the same.
6. To describe a force fully, you must give its magnitude in newtons and its direction, such as "$600N$ downwards" or "$50N$ to the left".
7. Direction can be given in words (up, down, left, right, forwards, backwards) or by using positive and negative signs along a chosen line.
   1. If you choose to the right as positive, a $50N$ force to the left is written as $-50N$.

Drawing forces as arrows

1. A vector is shown on a diagram as an arrow.
2. The direction of the arrow shows the direction in which the force acts.
3. The length of the arrow shows the magnitude of the force, so a larger force is drawn with a longer arrow.
4. Two forces of equal size are drawn as arrows of equal length, even if they point in different directions.
5. The arrow is drawn starting from the point where the force acts on the object.
   1. The weight of an object is drawn from the middle of the object, pointing straight down.
   2. A push from a hand or a pull from a rope is drawn from the point where the hand or rope touches the object.
6. An arrow drawn to scale uses a fixed length for each newton, so the size of a force can be read off the diagram.
   1. With a scale of $1cm$ to $10N$, a $35N$ force is drawn as an arrow $3.5cm$ long.
   2. Every arrow on the same diagram must use the same scale, or the lengths cannot be compared.
7. Each arrow is labelled with the name of the force, and with its value when the value is known.

Free body diagrams

1. A free body diagram shows one object on its own, with an arrow for every force that acts on it.
2. The object is drawn as a simple shape, such as a box or a dot, because its detailed appearance does not matter.
3. The surroundings, such as the floor, the rope or the Earth, are not drawn, because only their forces on the object are needed.
4. Forces that the object exerts on other things are not included, because they act on those other objects.
5. To draw a free body diagram, work through the object's interactions one at a time.
   1. Start with weight, because every object near the Earth has a gravitational force pulling it straight down.
   2. Add a force for every surface, string, rope or fluid that touches the object.
   3. Check that each arrow points in the right direction and that larger forces have longer arrows.
   4. Label every arrow with the name of the force.
6. The forces that appear most often on free body diagrams act in predictable directions.
   1. Weight acts vertically downwards.
   2. The normal contact force acts at right angles to the surface, so it acts upwards from a flat floor.
   3. Friction, air resistance and drag act in the opposite direction to the motion, or to the attempted motion.
   4. Tension acts along a string or rope, pulling away from the object.
   5. Thrust or driving force acts in the direction in which an engine, a motor or a person pushes the object.
   6. Upthrust acts vertically upwards on an object in a liquid or gas.
7. In the free body diagram of the car, the vertical arrows for weight and normal contact force are the same length, while the forward thrust arrow is longer than the backward friction arrow.
8. Reading the arrows tells you which forces are larger, which is the first step in working out how the object will move.

2.2.4 Newton's first law

Resultant force

1. Most objects have several forces acting on them at once, and their combined effect decides how the object moves.
2. The resultant force is the single force that would have the same effect on the object as all of the forces acting together.
3. The resultant force is also called the net force.
4. Forces acting along the same line are combined by taking their directions into account.
   1. Forces in the same direction add together.
   2. Forces in opposite directions partly or completely cancel, so the smaller force is taken away from the larger one.
   3. The resultant acts in the direction of the larger total force.
5. When the forces on an object cancel completely, they are balanced forces and the resultant force is zero.
6. When the forces do not cancel, they are unbalanced and there is a non-zero resultant force in one direction.
7. A zero resultant force does not mean that no forces are acting, because several forces can act and still cancel out.

Newton's first law

1. Newton's first law describes what an object does when the resultant force on it is zero.
2. If the resultant force on a stationary object is zero, the object stays at rest.
3. If the resultant force on a moving object is zero, the object keeps moving at the same speed in the same direction, which means at a constant velocity.
4. An object does not need a forward resultant force to keep moving, it only needs a resultant force to change the way it is moving.
5. On Earth, moving objects usually slow down because of friction and air resistance, which is why it can seem as if motion needs a force to keep it going.
6. In deep space, far from any planet or star, a space probe that switches off its engines keeps travelling in a straight line at a constant speed for years.

Uniform velocity

1. An object moving with uniform velocity covers equal distances in equal times without changing direction.
2. An object at uniform velocity has balanced forces acting on it, so the resultant force on it is zero.
3. For a car cruising along a straight motorway at a steady $30m/s$, the forward driving force from the engine is exactly balanced by the backward resistive forces of air resistance and friction.
   1. If the driving force is $2500N$ forwards, the resistive forces add up to $2500N$ backwards.
   2. The car has the same velocity at the end of each second as it had at the start, because nothing is unbalanced.
4. Vertically, the weight of the car is balanced by the normal contact force from the road, so the car neither rises nor sinks.

Changing speed or direction

1. Velocity is a vector, so it changes whenever the speed changes, the direction changes, or both change.
2. A change in velocity can only happen when there is a non-zero resultant force on the object.
3. The way the velocity changes depends on the direction of the resultant force.
   1. A resultant force in the same direction as the motion makes the object speed up.
   2. A resultant force in the opposite direction to the motion makes the object slow down.
   3. A resultant force at an angle to the motion makes the object change direction.
   4. A resultant force on a stationary object makes it start to move in the direction of the resultant.
4. When a bus driver presses the accelerator, the driving force becomes larger than the resistive forces, so the bus speeds up.
5. When the driver brakes, the backward forces become larger than any forward force, so the bus slows down.
6. When a football is kicked sideways as it rolls past a player, the sideways force changes the direction of the ball.

2.2.5 Resolving forces with vector diagrams

Scale vector diagrams

1. A vector diagram shows forces as arrows so that you can find their combined effect by drawing and measuring.
2. Every arrow is drawn to scale, so its length represents the size of the force and its direction matches the direction of the force.
3. Choose a scale that makes the diagram as large as the space allows, such as $1cm$ to $10N$, because a large diagram gives more accurate measurements.
4. Write the scale beside the diagram so that every length can be converted back into a force.
5. Use a sharp pencil and a ruler for lengths, and a protractor for any angle.

Combining forces

1. The resultant force found from a vector diagram is the single force with the same effect as all the forces acting together.
2. To add forces on a vector diagram, use the tip-to-tail method.
   1. Draw the first force arrow to scale.
   2. Draw the second force arrow to scale, starting from the tip of the first arrow.
   3. Draw the resultant as a straight arrow from the tail of the first arrow to the tip of the last arrow.
   4. Measure the length of the resultant and convert it into newtons using the scale, then measure its angle if a direction is needed.
3. For parallel forces, the arrows lie along one line.
   1. Two forces in the same direction give a resultant equal to their sum.
   2. Two forces in opposite directions give a resultant equal to their difference, in the direction of the larger force.
4. For perpendicular forces, the two arrows form two sides of a right-angled triangle and the resultant is the third side.
5. Because the triangle is right-angled, you can check a measured resultant with Pythagoras' theorem, $F_R=\sqrt{F_1^2+F_2^2}$.
6. You can also draw both arrows from the same point, complete a rectangle and draw the diagonal, which gives the same resultant.

Resolving a force

1. Resolving a force means splitting it into two components that are perpendicular to each other.
2. The two components are usually chosen to be horizontal and vertical, or along and at right angles to a slope.
3. The two components together have exactly the same effect as the original force, so they replace it in a calculation rather than being added to it.
4. Resolving is useful because forces along each perpendicular direction can then be combined separately.
5. To resolve a force by scale drawing, follow a fixed set of steps.
   1. Draw the force to scale at the correct angle, starting from one corner of your diagram.
   2. Draw a horizontal line and a vertical line from the start of the arrow.
   3. From the tip of the arrow, draw lines at right angles back to the horizontal and vertical lines to complete a rectangle.
   4. Measure the horizontal and vertical sides of the rectangle and convert each length into newtons with the scale.
6. The horizontal component is the part of the force that moves an object along the ground, and the vertical component is the part that lifts it or presses it down.

Equilibrium

1. An object is in equilibrium when the resultant force on it is zero.
2. An object in equilibrium either stays at rest or moves at a constant velocity.
3. When the forces on an object in equilibrium are drawn tip to tail, the last arrow finishes exactly where the first arrow started, so the arrows form a closed shape.
   1. Two forces in equilibrium form a closed shape only if they are equal in size and opposite in direction.
   2. Three forces in equilibrium form a closed triangle.
4. If a vector diagram does not close, the gap between the start and the finish is the resultant force, so the object is not in equilibrium.
5. To find the force needed for equilibrium, find the resultant of the other forces and add a force of the same size in the opposite direction.
6. Resolving gives a second test for equilibrium, because the horizontal components must cancel and the vertical components must cancel.

2.2.6 Forces acting on an isolated object

Isolated objects and systems

1. To analyse the forces on something, you first draw an imaginary boundary around it and treat it as an isolated object or system.
2. A system can be a single object, such as a ball bearing, or a group of objects that move together, such as a skydiver and her parachute.
3. Only external forces, which are exerted on the system by objects outside the boundary, affect how the whole system moves.
4. Forces between parts of the system are internal forces, and they cancel out in pairs, so they do not change the motion of the system as a whole.
5. The choice of boundary decides which forces count as external.
   1. If the system is the skydiver alone, the tension in the parachute harness is an external force pulling her upwards.
   2. If the system is the skydiver and parachute together, the harness tension is internal and the external forces are the total weight and the air resistance on the whole system.
6. For each external force, you should be able to name the object that exerts it, such as the Earth for weight or the air for air resistance.

Forces on a falling object

1. An object falling through air or a liquid has two main forces acting on it.
   1. Its weight, from the Earth, acts downwards and stays the same throughout the fall.
   2. Drag from the fluid acts upwards, opposite to the motion, and increases as the speed increases.
2. An object in a liquid also feels a small upthrust, which acts upwards alongside the drag.
3. The motion of the falling object follows a fixed sequence of stages.
   1. At the moment of release the speed is zero, so the drag is zero and the resultant force is the weight, which gives the largest acceleration.
   2. As the object speeds up, the drag increases, so the resultant downward force gets smaller and the acceleration decreases.
   3. Eventually the drag grows until it equals the weight, so the forces are balanced and the resultant force is zero.
   4. With a resultant force of zero, the acceleration is zero and the object falls at a constant speed called its terminal velocity.
4. Terminal velocity depends on the size and shape of the object as well as its weight.
   1. A larger surface area facing the direction of motion produces more drag at a given speed, so terminal velocity is reached at a lower speed.
   2. A heavier object needs more drag to balance its weight, so it has a higher terminal velocity than a lighter object of the same shape.
   3. A more viscous fluid, such as cold syrup or glycerol, produces more drag at a given speed than a runny fluid, so the terminal velocity is lower.

The skydiver

1. A skydiver jumping from a plane shows every stage of falling motion.
2. Just after jumping, her weight is much larger than the air resistance, so she accelerates downwards.
3. As she falls faster, the air resistance increases until it equals her weight, and she reaches a terminal velocity of roughly $55m/s$.
4. When she opens her parachute, the surface area increases greatly, so the air resistance suddenly becomes much larger than her weight.
5. The resultant force now acts upwards, so she decelerates while still moving downwards.
6. As she slows down, the air resistance decreases until it equals her weight again.
7. She then falls at a new, much lower terminal velocity of about $5m/s$, which is slow enough for a safe landing.

Vehicles and top speed

1. The same ideas apply to a vehicle moving horizontally, where the driving force takes the place of weight.
2. When a car pulls away, the driving force is larger than the resistive forces of air resistance and friction, so the car accelerates.
3. As the car speeds up, air resistance increases, so the resultant force and the acceleration both decrease.
4. When the resistive forces equal the maximum driving force, the resultant force is zero and the car has reached its top speed.
5. A car with a more streamlined shape has less air resistance at each speed, so it can reach a higher top speed with the same driving force.
6. Adding a roof box increases the air resistance, so the top speed falls and more fuel is used at motorway speeds.

2.2.7 Free body diagrams and resultant force

Reading a free body diagram

1. A free body diagram shows one object with an arrow for each force acting on it, so it lets you see at a glance whether the forces are balanced or unbalanced.
2. Forces are unbalanced when they do not cancel out, so there is a resultant force in one direction.
3. A resultant force makes the object accelerate in the direction of the resultant force.
4. To find the resultant force from a free body diagram, work in two perpendicular directions, usually horizontal and vertical.
   1. Add up the forces in one direction and subtract the forces in the opposite direction.
   2. Do the same for the other direction.
   3. If one direction gives zero, the resultant force lies along the other direction only.
5. The direction of the resultant is the direction of the larger total force, so it is shown by the longer arrows on the diagram.
6. The resultant force is not an extra force on the object, so it is not drawn as another arrow alongside the real forces.

Resultant force along one line

1. When all of the forces act along one line, choose one direction as positive and treat forces in the other direction as negative.
2. The resultant force is the sum of the forces with their signs.
3. A positive answer means the resultant acts in the positive direction, and a negative answer means it acts in the opposite direction.
4. More than two forces can act along the same line, such as a car with a driving force forwards and both air resistance and friction backwards.

Examples of unbalanced forces

1. Free body diagrams can describe many everyday situations where two or more forces give a resultant force.
2. A car speeding up on a level road has a forward driving force that is larger than the total backward resistive forces.
3. A cyclist braking has a backward friction force from the brakes and road that is larger than any forward force, so the resultant force acts backwards and the cyclist slows down.
4. A skydiver straight after jumping has a weight that is much larger than the upward air resistance, so the resultant force acts downwards and she speeds up.
5. A lift starting to rise has an upward tension in its cable that is larger than its weight, so the resultant force acts upwards.
6. A rocket at lift-off has an upward thrust from its engines that is larger than its weight, so it accelerates upwards.
7. A ball thrown upwards has its weight and air resistance both acting downwards while it rises, so the resultant force is downwards and the ball slows down.

2.2.8 Free body diagrams for balanced forces

Balanced forces

1. Forces on an object are balanced when they cancel each other out completely, so the resultant force is zero.
2. An object with balanced forces acting on it is in equilibrium.
3. Balancing is a special case, because the forces must match in both size and direction.
   1. For two forces to balance, they must be equal in size, act in opposite directions and act along the same line.
   2. For several forces to balance, the total force in each direction must equal the total force in the opposite direction.
4. A zero resultant force means the object does not accelerate, so its velocity stays the same.
   1. A stationary object with balanced forces stays stationary.
   2. A moving object with balanced forces keeps moving at a constant speed in a straight line.
5. Balanced forces therefore tell you that the velocity is not changing, not that the object is at rest.

Drawing balanced free body diagrams

1. A free body diagram for balanced forces shows each force on the object as a labelled arrow, with balancing forces drawn as arrows of equal length pointing in opposite directions.
2. Draw the object as a box or a dot, and draw only the forces acting on that object.
3. Consider the vertical forces and the horizontal forces separately.
   1. The upward arrows must have the same total length as the downward arrows.
   2. The arrows pointing one way along the horizontal must have the same total length as the arrows pointing the other way.
4. If a situation has no horizontal forces, such as an object hanging still, the diagram shows only vertical arrows.

Balanced forces on stationary objects

1. A book on a shelf has its weight acting downwards and an equal normal contact force from the shelf acting upwards.
2. A hanging basket outside a café has its weight acting downwards and an equal tension from its chain acting upwards.
3. A boat floating still in a harbour has its weight acting downwards and an equal upthrust from the water acting upwards.
4. Two teams in a tug of war that is not moving pull on the rope with equal forces in opposite directions, so the rope stays still.
5. In each case the arrows on the free body diagram are equal and opposite, so the object stays at rest.

Balanced forces on moving objects

1. A car at a steady speed on a straight, level road has a forward driving force balanced by backward air resistance and friction, and its weight balanced by the normal contact force from the road.
2. A skydiver at terminal velocity has her weight balanced by an equal upward air resistance, so she falls at a constant speed.
3. A lift rising at a steady speed has its weight balanced by an equal upward tension in its cable.
4. A plane cruising at a constant height and speed has its weight balanced by an upward lift force and its thrust balanced by air resistance.
5. In each case the object keeps moving, but its velocity does not change, because the resultant force is zero.

2.2.9 Newton's second law

Newton's second law

1. Newton's second law tells you how much an object accelerates when a resultant force acts on it.
2. The acceleration is directly proportional to the resultant force when the mass stays the same.
   1. Doubling the resultant force on a trolley doubles its acceleration.
3. The acceleration is inversely proportional to the mass when the resultant force stays the same.
   1. Doubling the mass of a trolley halves its acceleration for the same resultant force.
4. The acceleration always acts in the same direction as the resultant force.
5. These ideas combine into one equation, $force=mass\times acceleration$, which you must recall.
6. In symbols the equation is $F=ma$.
   1. $F$ is the resultant force in newtons, $N$.
   2. $m$ is the mass in kilograms, $kg$.
   3. $a$ is the acceleration in metres per second squared, ${m/s}^2$.

The newton

1. The equation $F=ma$ is used to define the unit of force.
2. One newton is the resultant force that gives a mass of $1kg$ an acceleration of $1{m/s}^2$.
3. This means that $1N=1{kg\; m/s}^2$.
4. Large forces are often given in kilonewtons, where $1kN=1000N$.

Second law calculations

1. The equation can be rearranged to find any one of the three quantities.
   1. To find the acceleration, divide both sides by $m$ to give $a=\frac{F}{m}$.
   2. To find the mass, divide both sides by $a$ to give $m=\frac{F}{a}$.
2. Always use the resultant force, so work it out first when more than one force acts along the line of motion.
3. Convert every value into SI units before you substitute.
   1. A mass in grams is divided by $1000$ to give kilograms, so $250g=0.25kg$.
   2. A force in kilonewtons is multiplied by $1000$ to give newtons, so $4.5kN=4500N$.
4. A negative acceleration from your calculation means the object is slowing down, which happens when the resultant force acts against the motion.

2.2.10 Inertia and inertial mass

Inertia

1. Inertia is a measure of how difficult it is to change the velocity of an object.
2. A change in velocity includes starting to move, speeding up, slowing down, stopping and changing direction.
3. Every object has inertia, whether it is moving or at rest.
   1. A stationary object resists being set in motion.
   2. A moving object resists being slowed down, stopped or turned.
4. The greater the mass of an object, the greater its inertia, so the harder it is to change its velocity.
5. An empty supermarket trolley is easy to start, stop and steer, while a fully loaded trolley needs a much bigger push to do the same things.
6. A loaded lorry needs a much larger braking force than a small car to stop in the same distance, because it has far more inertia.
7. Inertia explains why passengers on a bus lurch forwards when it brakes suddenly, because their bodies keep moving forwards until a force from the seat belt or the seat in front changes their velocity.
8. Inertia is a property of the object, not a force, so it is never drawn on a free body diagram.

Inertial mass

1. The inertial mass of an object puts a number on its inertia.
2. Inertial mass is defined as the ratio of the resultant force to the acceleration it produces.
3. In words, $inertial\; mass=\frac{resultant\; force}{acceleration}$.
4. In symbols, $m=\frac{F}{a}$, where $m$ is in $kg$, $F$ is in $N$ and $a$ is in ${m/s}^2$.
5. An object with a large inertial mass gains only a small acceleration from a given resultant force.
   1. A resultant force of $600N$ gives a $60kg$ sprinter an acceleration of $10{m/s}^2$.
   2. The same $600N$ gives a $1200kg$ car an acceleration of only $0.5{m/s}^2$.
6. Measuring the force on an object and its acceleration is one way to find its mass without using a balance.
7. This method works in places where a balance would be useless, such as on board the International Space Station, where objects appear weightless but still have inertia.
   1. Astronauts find their mass by measuring how quickly a known force accelerates them.
8. The force in the equation must be the resultant force, so opposing forces are combined before you divide by the acceleration.

2.2.11 Momentum in collisions

Momentum

1. Every moving object has momentum, which depends on both its mass and its velocity.
2. Momentum is calculated with $momentum=mass\times velocity$, an equation you must recall.
3. In symbols the equation is $p=mv$.
   1. $p$ is the momentum in kilogram metres per second, $kg\; m/s$.
   2. $m$ is the mass in kilograms, $kg$.
   3. $v$ is the velocity in metres per second, $m/s$.
4. A heavy lorry moving slowly can have the same momentum as a light car moving quickly.
5. A stationary object has zero momentum, because its velocity is zero.
6. Momentum is a vector, because velocity is a vector, so momentum always has a direction.
7. For motion along a line, choose one direction as positive, so an object moving the other way has a negative velocity and a negative momentum.

Momentum in collisions

1. When two objects collide, they push on each other, so the velocity of each object changes and the momentum of each object changes.
2. One object loses momentum and the other gains exactly the same amount, because the forces on the two objects are equal in size and opposite in direction and act for the same time.
3. The total momentum of the two objects therefore stays the same, as long as no external forces such as friction act on them.
4. This is the conservation of momentum: the total momentum before a collision equals the total momentum after the collision.
5. Momentum is conserved whether the objects bounce apart or stick together.
6. When you calculate total momentum, add the momenta with their signs, because momenta in opposite directions partly cancel.
7. Collisions in everyday life show the same pattern.
   1. When a moving snooker ball hits a stationary ball head-on, the first ball can stop dead while the second ball moves off with almost the same velocity.
   2. When a car runs into the back of a stationary car and the two lock together, they move off together more slowly than the first car was moving.
   3. When two dodgem cars collide head-on, each one bounces back, but the total momentum of the pair is unchanged.

2.2.12 Relating force, mass, velocity and acceleration

Linking force and acceleration

1. Force, mass, velocity and acceleration are tied together by two equations that you must recall.
   1. $F=ma$ links the resultant force to the mass and the acceleration.
   2. $a=\frac{v-u}{t}$ links the acceleration to the change in velocity and the time taken.
2. For a fixed mass, the acceleration is directly proportional to the resultant force.
   1. If the resultant force on a shopping trolley is tripled, its acceleration is tripled.
3. For a fixed resultant force, the acceleration is inversely proportional to the mass.
   1. If the same push is given to a trolley with three times the mass, its acceleration is one third as large.
4. Proportional reasoning lets you predict changes without working out every value from the beginning.
   1. If the force is doubled and the mass is also doubled, the acceleration does not change, because $\frac{2F}{2m}=\frac{F}{m}$.
   2. If the force is halved and the mass is doubled, the acceleration falls to one quarter, because $\frac{F/2}{2m}=\frac{1}{4}\times \frac{F}{m}$.

Force and change in velocity

1. Substituting $a=\frac{v-u}{t}$ into $F=ma$ gives $F=\frac{m(v-u)}{t}$.
2. This equation shows how the resultant force depends on the mass, the change in velocity and the time over which the change happens.
   1. A larger mass needs a larger force to produce the same change in velocity in the same time.
   2. A larger change in velocity in the same time needs a larger force.
   3. The same change in velocity spread over a longer time needs a smaller force.
3. The last of these ideas explains why stopping something gently needs a smaller force than stopping it suddenly.
   1. A fielder who pulls their hands back while catching a cricket ball makes the stopping time longer, so the force on their hands is smaller.
   2. Bending your knees when you land from a jump makes the time to stop longer, so the force on your legs is smaller.
4. If the time is not known, the acceleration can be found from the velocities and the distance with $v^2-u^2=2as$, which you need to select and apply rather than recall.
5. Combining $v^2-u^2=2as$ with $F=ma$ shows that, for a given mass and change of speed, a larger force changes the velocity over a shorter distance.

Solving multi-step problems

1. Many questions need two equations, with the answer from the first used in the second.
2. Work out which quantity links the two equations, which is usually the acceleration.
3. Find the acceleration from the motion data first, then use it in $F=ma$, or find the acceleration from $F=ma$ first, then use it to find a velocity, a time or a distance.
4. Keep unrounded values in your calculator between steps and round only the final answer.
5. Check the sign of the acceleration, because a negative value means the resultant force acts against the motion and the object is slowing down.

2.2.13 Work done by a force

Work done by a force

1. Work is done whenever a force makes an object move.
2. The work done depends on the size of the force and how far the object moves along the line of action of the force.
3. The work done is calculated with $work\; done=force\times distance$, an equation you must recall.
4. In symbols the equation is $W=Fs$.
   1. $W$ is the work done in joules, $J$.
   2. $F$ is the force in newtons, $N$.
   3. $s$ is the distance moved along the line of action of the force, in metres, $m$.
5. One joule of work is done when a force of $1N$ moves an object through $1m$ along the line of action of the force.
6. Rearranging the equation gives $F=\frac{W}{s}$ to find a force and $s=\frac{W}{F}$ to find a distance.

Distance along the line of action

1. Only movement in the direction of the force counts towards the work done by that force.
2. When you lift a bag of shopping onto a table, the lifting force acts upwards, so the distance to use is the vertical height the bag is raised.
3. When you push a wheelie bin along a path, the push acts horizontally, so the distance to use is the horizontal distance moved.
4. If an object does not move, no work is done on it, however large the force.
   1. Pushing hard against a solid wall does no work on the wall, because the wall does not move.
5. A force at right angles to the motion does no work.
   1. When you carry a box along a level corridor, your upward supporting force does no work on the box, because the box moves horizontally, not upwards.
6. When a force opposes the motion, such as friction or air resistance, work is still done, but it is work done against the motion.

Energy transferred by work

1. Doing work is a way of transferring energy, so the work done by a force equals the energy it transfers.
2. A transfer by a force is called a mechanical transfer, which is one of the main ways energy moves between stores.
3. Every time work is done, energy leaves one energy store and enters another.
   1. Lifting a box transfers energy from the chemical store of your muscles to the gravitational store of the box.
   2. A car engine accelerating a car transfers energy from the chemical store of the fuel to the kinetic store of the car.
   3. Stretching a spring transfers energy to the elastic store of the spring.
4. Work done against friction transfers energy to the thermal stores of the surfaces, which is why your hands warm up when you rub them together.
5. When a car brakes, the brakes do work against the motion, so energy is transferred from the kinetic store of the car to the thermal store of the brake discs and pads.
6. To describe a transfer fully, name the force doing the work, the store that loses energy and the store that gains energy.

2.2.14 Work, stored energy and units of energy

Work and stored energy

1. The work done by a force is equal to the energy transferred by that force.
2. This means you can use $W=Fs$ to calculate how much energy goes into or out of an energy store.
3. When a force does work on an object, the energy in one of the object's stores increases by the amount of work done.
   1. A crane that does $40kJ$ of work lifting a steel beam at a steady speed increases the gravitational store of the beam by $40kJ$.
   2. A bow that is drawn back with $60J$ of work stores $60J$ in its elastic store, ready to transfer to the arrow.
4. When an object does work against a force, the energy in one of its stores decreases by the amount of work done.
   1. A bicycle that freewheels to a stop on a level road loses all of the energy in its kinetic store as it does work against friction and air resistance.
5. Energy is never created or destroyed in these transfers, so the energy lost from one store always equals the total energy gained by the other stores.
6. Some of the energy transferred usually ends up in the thermal stores of the surroundings, where it spreads out and is no longer useful.

Calculating energy transfers

1. Energy transfer calculations follow the same steps whichever stores are involved.
   1. Identify the force doing the work and the distance moved along its line of action.
   2. Calculate the work done with $W=Fs$.
   3. Use the work done as the energy transferred, then add it to or take it away from the store involved.
2. A transfer can happen in several stages, and the energy can be followed from one store to the next.
   1. When a ball is lifted and then dropped, the work done lifting it fills its gravitational store, and as it falls this energy moves to its kinetic store.
   2. When the ball hits the ground, the energy moves to the thermal stores of the ball and ground, with some carried away by sound.
3. If a store is given a value, such as the energy in the kinetic store of a moving car, you can use $W=Fs$ in reverse to find a force or a distance.

Newton-metres and joules

1. Multiplying a force in newtons by a distance in metres gives an answer in newton-metres, $N\; m$.
2. Work done is an energy, so its answer can also be given in joules, $J$.
3. The two units are identical in size, so $1N\; m=1J$.
4. Converting between them changes only the unit, not the number, so $750N\; m=750J$.
5. Large amounts of energy are given with prefixes, which do change the number.
   1. $1kJ=1000J$, so $4.2kJ=4200J=4200N\; m$.
   2. $1MJ=1000000J$, so $0.35MJ=350000J$.
6. Convert prefixed values into joules before substituting them into $W=Fs$, so that the force comes out in newtons and the distance in metres.

2.2.15 Power as the rate of energy transfer

Power

1. Power is the rate at which energy is transferred, which means how much energy is transferred each second.
2. When a force does work, the energy transferred equals the work done, so power is also the rate at which work is done.
3. Power is calculated with $power=\frac{work\; done}{time}$, an equation you must recall.
4. In symbols the equation is $P=\frac{W}{t}$.
   1. $P$ is the power in watts, $W$.
   2. $W$ is the work done, or energy transferred, in joules, $J$.
   3. $t$ is the time taken in seconds, $s$.
5. The same equation can be written as $P=\frac{E}{t}$, where $E$ is the energy transferred in joules.
6. Rearranging gives $W=Pt$ to find the work done and $t=\frac{W}{P}$ to find the time.

The watt

1. Power is measured in watts, with the symbol $W$.
2. One watt is a rate of energy transfer of one joule per second, so $1W=1J/s$.
3. Larger powers are given in kilowatts and megawatts.
   1. $1kW=1000W$, which is roughly the power of a strong cyclist sprinting for a few seconds.
   2. $1MW=1000000W$, which is the scale of the power of a train engine or a large wind turbine.
4. Times must be in seconds before you substitute, so a time of $2$ minutes becomes $120s$.

Comparing power

1. A more powerful machine transfers energy more quickly than a less powerful one.
2. Two things decide the power: the energy transferred and the time taken.
   1. If two machines transfer the same energy, the one that takes less time has the greater power.
   2. If two machines run for the same time, the one that transfers more energy has the greater power.
3. Two pupils of equal weight who climb the same staircase do the same work against their weight, but the pupil who runs up does it in less time and so develops a greater power.
4. A crane lifting a load to the top of a building does the same work whether it lifts slowly or quickly, but the faster crane needs a more powerful motor.
5. Power describes how quickly energy is transferred, not how much, so a low-power machine running for a long time can transfer more energy than a high-power machine running briefly.

2.2.16 Newton's third law

Newton's third law

1. Newton's third law states that when two objects interact, they exert forces on each other that are equal in size and opposite in direction.
2. The two forces form an interaction pair, often called a third law pair.
3. Every third law pair has the same set of features.
   1. The forces are equal in size.
   2. The forces act in opposite directions along the same line.
   3. The forces act on different objects.
   4. The forces are the same type of force, such as two gravitational forces or two normal contact forces.
   5. The forces start and stop at the same time.
4. The law applies to every interaction, whether the objects are touching or not, and whether they are moving or still.
5. A short way to state a pair is "if A exerts a force on B, then B exerts an equal and opposite force on A".

Third law pairs and balanced forces

1. Third law pairs never cancel each other out, because the two forces act on different objects.
2. Forces can only cancel when they act on the same object, and then they are balanced forces.
3. A mug resting on a desk shows the difference clearly.
   1. The weight of the mug (the Earth pulling the mug down) and the normal contact force (the desk pushing the mug up) both act on the mug, so they are a pair of balanced forces, not a third law pair.
   2. The third law partner of the mug's weight is the mug pulling the Earth upwards.
   3. The third law partner of the desk pushing the mug up is the mug pushing the desk down.
4. Two forces that are equal and opposite are therefore not automatically a third law pair, so you must check which object each force acts on.

Situations in equilibrium

1. In equilibrium situations, the object you are studying has balanced forces on it, but third law pairs are still present between it and every object it touches or attracts.
2. When you lean on a wall without moving, your hand pushes the wall and the wall pushes your hand back with an equal force, which you feel in your palm.
3. When a picture hangs on a hook, the picture pulls down on the hook and the hook pulls up on the picture with an equal force.
4. When two teams in a tug of war are not moving, each team pulls the rope and the rope pulls back on each team with an equal force.

Situations that are not in equilibrium

1. Third law pairs also act when objects are speeding up, slowing down or changing direction.
2. Both objects feel the same size of force, but the effects of the forces can be very different because the objects can have very different masses.
3. When you walk, your foot pushes backwards on the ground and the ground pushes forwards on your foot with an equal force, which accelerates you forwards.
4. When a swimmer pushes water backwards with her arms, the water pushes her forwards with an equal force.
5. When a rocket fires its engines, it pushes hot exhaust gases downwards and the gases push the rocket upwards with an equal force.
6. When a gun is fired, it pushes the bullet forwards and the bullet pushes the gun backwards with an equal force, so the gun recoils.
   1. The bullet has a tiny mass, so it gets a huge acceleration.
   2. The gun has a much larger mass, so it gets a much smaller acceleration.
7. When a car hits a fly, the force on the fly is exactly equal to the force on the car, but the fly is squashed because its mass is so small.

2.2.17 Circular motion and changing velocity

Speed and velocity

1. Speed is a scalar, so it tells you only how fast an object is moving.
2. Velocity is a vector, so it tells you how fast an object is moving and in which direction.
3. Two cars travelling at $13m/s$, one heading north and one heading east, have the same speed but different velocities.
4. The velocity of an object changes if its speed changes, if its direction changes, or if both change.
5. This means an object can keep a constant speed while its velocity changes, as long as its direction is changing.

Moving in a circle

1. An object moving in a circle is always changing the direction in which it travels.
2. At any instant, the object moves along the tangent to the circle, which is at right angles to the radius at that point.
3. As the object moves round, the tangent points in a different direction at every point on the circle.
4. If the object moves at a constant speed, the velocity arrows all have the same length, but each one points in a different direction.
5. The velocity is therefore continuously changing, even though the speed is not.
6. A car driving round a roundabout at a steady $10m/s$ shows this clearly.
   1. At one point the car is heading east at $10m/s$.
   2. A quarter of the way round, the car is heading north at $10m/s$.
   3. Halfway round, the car is heading west at $10m/s$, which is the opposite velocity to the one it started with.

Acceleration in a circle

1. Acceleration is the rate of change of velocity, not only the rate of change of speed.
2. An object moving in a circle at a constant speed has a changing velocity, so it is accelerating all the time.
3. By Newton's first law, a changing velocity needs a resultant force, so there must be a resultant force on any object moving in a circle.
4. This resultant force acts towards the centre of the circle, which is why the direction of motion keeps turning inwards.
5. Different forces provide this inward resultant force in different situations.
   1. For a conker whirled on a string, the tension in the string pulls the conker towards the centre.
   2. For a car on a roundabout, friction between the tyres and the road pushes the car towards the centre.
   3. For a rider on a fairground ride, the normal contact force from the seat or the wall pushes the rider towards the centre.
6. If the inward force suddenly disappears, the object stops turning and carries on in a straight line along the tangent, as Newton's first law predicts.
   1. If the string of a whirling conker snaps, the conker flies off along the tangent, not outwards along the radius.