2.3.1 Stretching, bending and compressing

Changing the shape of an object

1. An object undergoes deformation when the forces acting on it change its shape or size.
2. The three basic types of deformation are stretching, compressing and bending.
   1. Stretching pulls an object so that it becomes longer, as when you pull a spring or a rubber band.
   2. Compressing squashes an object so that it becomes shorter or thinner, as when you squeeze a sponge or sit on a cushion.
   3. Bending makes an object curve, as when a diving board sags under a diver or a ruler bows when you press its middle.
3. When an object is stretched, the increase in its length is called its extension.
4. The extension is found by subtracting the original length from the stretched length.
   1. A spring with an original length of $8.0cm$ that is stretched to $11.0cm$ has an extension of $3.0cm$.
5. When an object is compressed, the decrease in its length is called its compression.

Why one force is not enough

1. If only one force acts on an object, the resultant force on it cannot be zero.
2. An unbalanced resultant force makes the whole object accelerate in the direction of that force, so the object moves off instead of changing shape.
   1. If you pull one end of a spring lying loose on a smooth bench, the whole spring slides towards you and does not stretch.
3. To change the shape of an object, more than one force must act on it, and the forces must act on different parts of the object.
4. The forces together stop the object as a whole from moving off, while its different parts are pulled apart, pushed together or made to curve.
   1. To stretch the spring on the bench, you must hold its other end still, so that a second force pulls on it in the opposite direction.
5. The forces that deform an object are often balanced, so the object stays still while its shape changes.

Stretching forces

1. To stretch an object, forces must pull on it in opposite directions, away from each other.
2. A spring hanging from a clamp stand with a mass on its lower end is stretched by two forces.
   1. The weight of the mass pulls the bottom of the spring downwards.
   2. The clamp pulls the top of the spring upwards.
3. The pulling force that is set up inside a stretched spring, rope or cable is called tension.
4. In a tug of war, the two teams pull the rope in opposite directions, so the rope is stretched tight and is under tension.
5. A bungee cord is stretched between the jump platform that holds its top end and the weight of the jumper hanging from its lower end.

Compressing forces

1. To compress an object, forces must push on it in opposite directions, towards each other.
2. When you press down on a sponge resting on a table, your hand pushes down on the top of the sponge while the table pushes up on its bottom, so the sponge is squashed between them.
3. A car's suspension springs are compressed because the car body pushes down on the top of each spring while the axle pushes up on its bottom.
4. The legs of a chair are compressed between the weight of the person sitting on it and the upward push of the floor.
5. The sole of a trainer is compressed between your foot and the ground every time you take a step.

Bending forces

1. To bend an object, forces must act at different places along its length, usually with forces in one direction at the ends and a force in the opposite direction in between, or the other way round.
2. A ruler resting on two books bends when you press down on its middle, because the books push up at each end while your finger pushes down at the centre.
3. A shelf sags under a row of heavy books because the brackets push up near its ends while the weight of the books pushes down along its length.
4. A diving board curves downwards because the clamp and the support hold one end while the weight of the diver pushes down on the free end.
5. When an object bends, the material on the outside of the curve is stretched and the material on the inside of the curve is compressed.
6. A fishing rod bends when a fish pulls on the line at the tip while the angler holds the handle firmly at the other end.

Deformation in everyday objects

1. A trampoline works because the springs around its edge are stretched between the frame and the mat each time you land on it.
2. A climbing rope is stretched between the anchor point above the climber and the weight of the climber below.
3. A tennis ball is squashed against the racket strings during a shot, because the strings push on one side of the ball while the ball presses back on the strings.
4. In every case you can name two objects in contact with the deformed object, and each of them supplies one of the forces that changes its shape.

2.3.2 Elastic and plastic deformation

Elastic deformation

1. An object is elastically deformed when it returns to its original shape and size once the deforming forces are removed.
2. While the forces act, the object is still distorted, but the change in shape is temporary.
3. A rubber band, a steel spring and a trampoline mat all deform elastically when they are stretched by modest forces.
4. A tennis ball deforms elastically when it is squashed against a racket, then springs back to a round shape as it leaves the strings.
5. In an elastic deformation, the forces pull the particles of the material slightly further apart, or push them slightly closer together, without changing how they are arranged.
6. When the deforming forces are removed, the forces between the particles pull them back to their original positions, so the object recovers its shape.

Plastic deformation

1. An object is plastically deformed when it does not return to its original shape and size after the deforming forces are removed.
2. Plastic deformation is also called inelastic deformation, and the change in shape is permanent.
3. A lump of modelling clay, a drinks can crushed underfoot and a paper clip bent open all deform plastically.
4. A car's crumple zone is built to deform plastically in a collision, so the front of the car stays crushed afterwards.
5. In a plastic deformation, layers of particles in the material slide past one another into new positions.
6. When the forces are removed, nothing pulls the particles back to where they started, so the object keeps its new shape.
7. The word plastic describes the type of deformation, not the material, so a steel spring can deform plastically just as a plastic carrier bag can.

Elastic or plastic

1. Whether a deformation is elastic or plastic depends on what happens after the forces are removed, not on how far the object stretched while it was loaded.
2. Most objects deform elastically for small forces and plastically for large forces.
3. A steel spring stretched gently springs back to its original length, but the same spring stretched very hard ends up permanently longer, with its coils pulled apart.
   1. A coiled spring toy pulled too far down a staircase stays open in the overstretched section and never coils neatly again.
4. An object can show both types of deformation at once, recovering most of its stretch but keeping a small permanent extension.
5. The permanent extension is the length after all the forces are removed minus the original length.
   1. A spring that is $10.0cm$ long, stretched to $18.0cm$ and then released, might shrink back only to $10.6cm$, leaving a permanent extension of $0.6cm$.

Loading and unloading

1. Loading means adding masses to an object one at a time, so the stretching force increases in equal steps.
2. Unloading means removing the masses one at a time, so the stretching force decreases in the same steps.
3. Comparing the lengths measured during loading and unloading shows which type of deformation has happened.
   1. If the object has the same length at each load on the way down as on the way up, and finishes at its original length, the deformation is elastic.
   2. If the object finishes longer than it started, it has been plastically deformed.
4. A steel spring usually gives almost identical readings during loading and unloading, provided it has not been overloaded.
5. A rubber band behaves differently, because it is longer at each load during unloading than it was during loading, even though it returns to almost its original length at the end.

2.3.3 Force and extension of a spring

Force and extension

1. When a force stretches a spring, the extension of the spring depends on the size of the force.
2. A larger stretching force pulls the coils further apart, so it produces a larger extension.
3. For a spring, equal increases in force produce equal increases in extension, provided the spring is not stretched too far.
   1. If adding $1N$ stretches a spring by $2cm$, adding another $1N$ stretches it by a further $2cm$.
4. This means the extension of a spring is directly proportional to the force applied to it.
   1. Doubling the force doubles the extension.
   2. Halving the force halves the extension.
   3. Applying no force gives no extension.
5. The relationship is written as $extension\propto force$, where the symbol $\propto$ means "is directly proportional to".

Hooke's law

1. The direct proportion between force and extension for a spring is called Hooke's law, after the scientist Robert Hooke, who described it in 1678.
2. Hooke's law holds only up to a point called the limit of proportionality.
3. Below the limit of proportionality, the spring stretches by the same amount for every extra newton of force.
4. Beyond the limit of proportionality, the extension is no longer directly proportional to the force.
5. Many objects obey Hooke's law for small forces, including steel springs and metal wires.
6. The spring inside a newton meter obeys Hooke's law over its whole working range, which is why the markings on its scale are evenly spaced.

Force-extension graphs

1. The relationship between force and extension is shown by plotting a force-extension graph.
2. In an investigation, force is the quantity you change, so it is often plotted on the horizontal axis with extension on the vertical axis.
3. Some graphs have force on the vertical axis and extension on the horizontal axis instead, so always read the axis labels first.
4. For a spring obeying Hooke's law, the graph is a straight line through the origin.
   1. The line is straight because equal steps in force give equal steps in extension.
   2. The line passes through the origin because zero force gives zero extension.
5. On a graph of extension against force, a steeper line belongs to a spring that stretches further for each newton.
6. Values can be read from the straight part of the graph to find the extension for any force within the range plotted.
   1. If the line shows an extension of $6.0cm$ at $3.0N$, the extension at $1.5N$ is $3.0cm$.

Other simple systems

1. Other simple systems can be loaded in the same way as a spring, with the extension measured for each force.
2. A metal wire behaves like a spring for small forces, because its extension is directly proportional to the force, although the extensions are tiny and need very careful measurement.
3. A rubber band and a strip of polythene also extend more as the force increases, but their extension is not directly proportional to the force, so their graphs are not straight lines through the origin.
4. Two identical springs hung one below the other each carry the full load, so each stretches by the same amount and the total extension is double that of one spring.
5. Two identical springs hung side by side share the load equally, so each carries half the force and the extension is half that of one spring.

2.3.4 Linear and non-linear force-extension behaviour

Linear relationships

1. A relationship between force and extension is linear when equal increases in force always produce equal increases in extension.
2. A linear relationship gives a straight-line graph, and the gradient is the same everywhere along the line.
3. A spring loaded below its limit of proportionality has a linear force-extension relationship.
4. A linear relationship is also directly proportional only if the straight line passes through the origin.
   1. A graph of extension against force for a spring is a straight line through the origin, so the relationship is linear and directly proportional.
   2. A graph of total length against force for the same spring is still a straight line, but it crosses the length axis at the original length, so that relationship is linear but not directly proportional.

Non-linear relationships

1. A relationship between force and extension is non-linear when equal increases in force produce unequal increases in extension.
2. A non-linear relationship gives a curved graph, and its gradient changes from one part of the curve to another.
3. A rubber band has a non-linear relationship over most of its stretch.
   1. At first it stretches easily, because the long, tangled chains of molecules in the rubber straighten out with little force.
   2. Once the chains are nearly straight, the band becomes much harder to stretch, so each extra newton produces a smaller extension.
4. A strip of polythene and a strip of liquorice are also non-linear, stretching more and more for each extra newton as they start to thin.
5. A steel spring becomes non-linear when it is loaded beyond its limit of proportionality, because its coils begin to open out and each extra newton stretches it further than before.

Reading a force-extension graph

1. A spring loaded to large forces gives a graph with a straight section followed by a curved section.
2. The point where the straight line starts to curve is the limit of proportionality.
3. On a graph of extension against force, the line curves upwards beyond this point, showing that each newton now produces a larger extension.
4. On a graph of force against extension, the same behaviour makes the line curve towards the extension axis, so check which quantity is on which axis before describing the shape.
5. To find where a relationship stops being linear from a table of results, work out the extra extension for each equal step in force and look for the step where it changes.

The elastic limit

1. Loading an object beyond its elastic limit causes plastic deformation, so the object is left with a permanent extension when the force is removed.
2. For many springs, the elastic limit lies close to the limit of proportionality, but the two are different points.
3. The limit of proportionality is found from the shape of the graph during loading.
4. The elastic limit can only be found by removing the force and checking whether the object returns to its original length.
5. A curved graph does not by itself show that an object has passed its elastic limit, because a rubber band behaves non-linearly yet still returns almost to its original length.

2.3.5 The spring constant

The spring constant

1. Different springs need different forces to produce the same extension, because some springs are stiffer than others.
2. The spring constant, $k$, is a measure of how stiff a spring is.
3. A spring with a large spring constant is stiff, so it needs a large force for each metre of extension.
   1. A car suspension spring is very stiff, with a spring constant of tens of thousands of newtons per metre.
4. A spring with a small spring constant is easy to stretch, so a small force produces a large extension.
   1. The small spring inside a retractable pen has a spring constant of only a few hundred newtons per metre.
5. The spring constant is measured in newtons per metre, $N/m$.
   1. A spring constant of $50N/m$ means that a force of $50N$ is needed for each metre of extension, which is $0.5N$ for each centimetre.

The spring equation

1. For a spring in the linear part of its behaviour, the force, the spring constant and the extension are linked by an equation you must recall.
2. In words, $force\; exerted\; by\; a\; spring=spring\; constant\times extension$.
3. In symbols, the equation is $F=kx$.
   1. $F$ is the force in newtons, $N$.
   2. $k$ is the spring constant in newtons per metre, $N/m$.
   3. $x$ is the extension in metres, $m$.
4. The equation applies only while the spring obeys Hooke's law, so it cannot be used beyond the limit of proportionality.
5. The force stretching the spring and the force the spring pulls back with are equal in size, so $F$ can stand for either.
6. The same equation works for a spring that is compressed, with $x$ standing for the decrease in length.

Calculating the spring constant

1. Rearranging $F=kx$ gives $k=\frac{F}{x}$, so the spring constant is the force divided by the extension it produces.
2. Rearranging for the extension gives $x=\frac{F}{k}$.
3. Convert the extension into metres before substituting, because the spring constant is in newtons per metre.
   1. Divide a length in centimetres by $100$ to give metres, so $4.0cm=0.040m$.
   2. Divide a length in millimetres by $1000$ to give metres, so $25mm=0.025m$.
4. If a question gives two lengths instead of an extension, subtract the original length from the stretched length first.

The spring constant from a graph

1. On a graph with force on the vertical axis and extension on the horizontal axis, the gradient of the straight section equals the spring constant.
2. The spring constant is found as $k=\frac{change\; in\; force}{change\; in\; extension}$, using two well-separated points on the straight line.
3. A steeper line on this graph belongs to a stiffer spring with a larger spring constant.
4. Many practicals plot extension on the vertical axis against force, and on that graph the gradient is $\frac{1}{k}$, so the spring constant is the change in force divided by the change in extension, not the gradient itself.
5. Draw a large triangle on the straight part of the line to find the gradient, because a large triangle reduces the effect of small reading errors.

2.3.6 Work done in stretching

Work done in stretching

1. To stretch a spring, a force must move the end of the spring through a distance, so work is done on the spring.
2. The work done in stretching a spring is equal to the energy transferred to it.
3. When the spring is stretched elastically, this energy is held in the spring's elastic store.
4. When the spring is released, energy in its elastic store is transferred to other stores, such as the kinetic store of a ball fired from a toy launcher.
   1. A stretched catapult or a drawn bow transfers energy from its elastic store to the kinetic store of the stone or the arrow when it is released.
5. If a spring is stretched beyond its elastic limit, part of the work done permanently rearranges the material and warms it, so not all of the energy can be returned.

Why the force changes during a stretch

1. The force needed to stretch a spring increases as the extension increases.
2. At the start of the stretch the force is zero, and at an extension $x$ it has risen to $kx$.
3. For a spring obeying Hooke's law the force rises steadily, so the average force during the stretch is half of the final force.
4. The work done is therefore not the final force multiplied by the extension, but half of that value.

The energy equation

1. The energy transferred in stretching a spring is given by an equation that you do not have to memorise, but you must be able to choose it and use it.
2. In words, $energy\; transferred\; in\; stretching=\frac{1}{2}\times spring\; constant\times (extension{)}^2$.
3. In symbols, the equation is $E=\frac{1}{2}k{x}^2$.
   1. $E$ is the energy transferred, equal to the work done, in joules, $J$.
   2. $k$ is the spring constant in newtons per metre, $N/m$.
   3. $x$ is the extension in metres, $m$.
4. The extension is squared, so doubling the extension makes the energy stored four times as large.
5. The equation applies only up to the limit of proportionality, because it assumes that the force rises in proportion to the extension.
6. Rearranging the equation gives $k=\frac{2E}{x^2}$ for the spring constant and $x=\sqrt{\frac{2E}{k}}$ for the extension.

Work done from a force-extension graph

1. On a graph of force against extension, the area under the line equals the work done in stretching.
2. This works because work done is force multiplied by distance, and each thin strip under the graph is a small force multiplied by a small extension.
3. For the straight section of a spring's graph, the area is a triangle, so $work\; done=\frac{1}{2}\times force\times extension$.
4. Substituting $F=kx$ into the triangle area gives $\frac{1}{2}\times kx\times x=\frac{1}{2}k{x}^2$, which is the energy equation.
5. For a curved graph, such as the one for a rubber band, no simple equation applies, so the area is found by counting squares under the curve.
   1. Count each square that is at least half under the curve as a whole square, and ignore squares that are less than half under it.
   2. Find the energy represented by one square by multiplying the force for one square by the extension for one square.
   3. Multiply the number of squares by the energy for one square to give the total work done.
6. The unit of area on this graph is the newton metre, $Nm$, and $1Nm$ is equal to $1J$.

2.3.7 Gravitational fields

Gravitational fields

1. Every object that has mass is surrounded by a gravitational field.
2. This is true of all matter, from a star or a planet down to a person, a pencil or a single atom.
3. Any other object with mass inside the field experiences a gravitational force.
4. The force acts without the objects touching, so gravity is a non-contact force.
5. Gravitational forces are always attractive, pulling objects towards each other and never pushing them apart.
6. The attraction acts on both objects, so the Earth pulls on you and you pull on the Earth with a force of the same size in the opposite direction.
7. A gravitational field is drawn with field lines that point inwards towards the centre of the object, showing the direction of the force on any other mass placed in the field.
8. Where the field lines are closer together, the field is stronger.
9. The lines spread further apart with distance from the object, so the field becomes weaker further away.

Field strength and mass

1. The strength of a gravitational field depends on the mass of the object that creates it.
2. An object with a very large mass has a much stronger gravitational field than an object with a small mass.
3. In physics, massive means having a large mass, not having a large size.
4. The Earth has a mass of about $6\times 10^{24}kg$, which gives it a field strong enough to hold you on the ground and keep the Moon in orbit.
5. The Moon has about one eightieth of the Earth's mass, and the gravitational field strength at its surface is only about $1.6N/kg$, compared with $10N/kg$ at the Earth's surface.
6. Jupiter, the most massive planet in the Solar System, has a gravitational field strength of about $25N/kg$ at the top of its clouds.
7. The Sun holds more than $99\%$ of the mass of the Solar System, and its enormous field keeps every planet in orbit around it.

Attraction between everyday objects

1. Two everyday objects, such as two people standing side by side, also attract each other through their gravitational fields.
2. Their masses are tiny compared with the mass of a planet, so the force between them is extremely small.
   1. Two pupils with masses of $60kg$ standing $1m$ apart attract each other with a force of less than one millionth of a newton.
3. Friction between their shoes and the floor is far larger than this force, so the pupils never notice it and do not move towards each other.
4. The attraction between you and the Earth is large because the Earth's mass is enormous, even though your own mass is small.
5. Gravitational attraction is therefore only noticeable when at least one of the objects is very massive, such as a moon, a planet or a star.

Field strength and distance

1. The gravitational field of an object becomes weaker as you move further away from it.
2. Close to the Earth's surface the field hardly changes over the heights you could climb, so it can be treated as uniform near the ground.
3. Far from the Earth, such as halfway to the Moon, the Earth's field is much weaker, although it never quite falls to zero.

2.3.8 Weight and gravitational field strength

Weight

1. The weight of an object is the gravitational force that acts on it because it is in a gravitational field.
2. Weight is a force, so it is measured in newtons, $N$, and it has a direction as well as a size.
3. Weight always acts towards the centre of the object creating the field, which on Earth means vertically downwards.
4. Mass and weight are different quantities.
   1. Mass is the amount of matter in an object, measured in kilograms, and it is the same wherever the object is.
   2. Weight is a force measured in newtons, and it changes if the object moves to a place where the gravitational field is stronger or weaker.
5. An astronaut with a mass of $70kg$ still has a mass of $70kg$ on the Moon, but weighs far less there because the Moon's gravitational field is weaker.

Measuring weight

1. Weight is measured with a newton meter, also called a spring balance or force meter.
2. A newton meter contains a spring that stretches when an object is hung from its hook.
3. The spring obeys Hooke's law, so its extension is directly proportional to the weight on the hook, and the scale can be marked in evenly spaced newtons.
4. Newton meters come in different ranges, such as $0$ to $1N$ for light objects and $0$ to $50N$ for heavy ones.
5. A top-pan balance, kitchen scales and bathroom scales respond to the force an object exerts on them, but they are set up to display mass in grams or kilograms.
   1. These scales divide the force they detect by the Earth's gravitational field strength, so digital kitchen scales taken to the Moon would show a mass about six times too small.

Weight and gravitational field strength

1. The gravitational field strength, $g$, is the force that a gravitational field exerts on each kilogram of mass.
2. Gravitational field strength is measured in newtons per kilogram, $N/kg$.
3. At the Earth's surface, $g$ is about $10N/kg$, so every kilogram of mass has a weight of about $10N$.
4. The weight of an object is its mass multiplied by the gravitational field strength.
5. For an object of fixed mass, weight is directly proportional to the gravitational field strength.
   1. Doubling $g$ doubles the weight, and halving $g$ halves the weight.
   2. A $2.0kg$ bag of potatoes has a weight of $2.0\times 10=20N$ on Earth.
   3. The same bag on the Moon, where $g$ is $1.6N/kg$, has a weight of only $2.0\times 1.6=3.2N$.
6. For a fixed value of $g$, weight is also directly proportional to mass, so a $4.0kg$ bag weighs twice as much as a $2.0kg$ bag in the same place.

2.3.9 Acceleration in free fall

Free fall

1. An object is in free fall when the only force acting on it is its weight.
2. With no other force acting, the weight is an unbalanced force, so the object accelerates towards the ground.
3. True free fall happens where there is no air, such as on the Moon or inside a tube with the air pumped out.
4. Close to the Earth's surface, a small, dense object such as a stone or a ball bearing dropped a short distance is very nearly in free fall, because the air resistance on it is tiny.

The acceleration in free fall

1. The acceleration of an object in free fall near the Earth's surface is about $10{m/s}^2$.
2. A more precise value is $9.8{m/s}^2$, but $10{m/s}^2$ is the value used in most calculations at GCSE.
3. The acceleration is directed vertically downwards, towards the centre of the Earth.
4. An acceleration of $10{m/s}^2$ means that the velocity of a falling object increases by $10m/s$ every second.
   1. A dropped object is falling at $10m/s$ after $1s$, at $20m/s$ after $2s$ and at $30m/s$ after $3s$.
5. The acceleration in free fall has the same value as the gravitational field strength, because $1N/kg$ is the same as $1{m/s}^2$.
   1. A field of $10N/kg$ pulls each kilogram with $10N$, and a resultant force of $10N$ on $1kg$ gives an acceleration of $10{m/s}^2$.
6. The symbol $g$ is used for both the acceleration in free fall and the gravitational field strength.

Mass does not affect free fall

1. In free fall, all objects accelerate at the same rate, whatever their mass.
2. A heavier object has a larger weight pulling it down, but it also has a larger mass, so it needs a larger force to accelerate it.
3. The two effects cancel exactly, because doubling the mass doubles the weight and also doubles the force needed for the same acceleration.
4. In 1971, the astronaut David Scott dropped a hammer and a falcon feather together on the Moon during the Apollo 15 mission, and they hit the ground at the same moment because there is no air on the Moon.
5. On Earth, a feather and a coin released together inside a long tube with the air pumped out also land together.

Air resistance in real falls

1. Most falling objects on Earth are not in true free fall, because air resistance acts upwards on them as they move through the air.
2. Air resistance reduces the resultant force on a falling object, so its acceleration is less than $10{m/s}^2$.
3. Air resistance has the largest effect on objects that are light and have a large surface area, such as a feather, a leaf or a flat sheet of paper.
4. A crumpled sheet of paper falls much faster than a flat sheet of the same mass, because its smaller surface area meets less air resistance.
5. You can treat a falling object as being in free fall whenever a question tells you that air resistance can be ignored.

2.3.10 Relating force, mass and gravitational field strength

The weight equation

1. The weight of an object, its mass and the gravitational field strength are linked by an equation you must recall.
2. In words, $gravitational\; force=mass\times gravitational\; field\; strength$.
3. In symbols, the equation is $W=mg$.
   1. $W$ is the weight, or gravitational force, in newtons, $N$.
   2. $m$ is the mass in kilograms, $kg$.
   3. $g$ is the gravitational field strength in newtons per kilogram, $N/kg$.
4. Use $g=10N/kg$ at the Earth's surface unless a question gives a different value.
5. Convert a mass in grams into kilograms before substituting, by dividing it by $1000$.
   1. A $250g$ apple has a mass of $0.250kg$, so its weight on Earth is $0.250\times 10=2.5N$.

Rearranging the equation

1. Rearranging $W=mg$ gives $m=\frac{W}{g}$, which finds a mass from a weight.
2. Rearranging again gives $g=\frac{W}{m}$, which finds the gravitational field strength from a weight and a mass.
3. The equation $W=mg$ is $F=ma$ applied to an object in free fall, with the weight as the resultant force and $g$ as the acceleration.

How the quantities are inter-related

1. In $W=mg$, the weight is directly proportional to the mass when $g$ stays the same.
   1. In the same place, a $6kg$ suitcase weighs three times as much as a $2kg$ suitcase.
2. The weight is also directly proportional to $g$ when the mass stays the same.
   1. Taking an object from Earth, where $g$ is $10N/kg$, to Mars, where $g$ is $3.7N/kg$, multiplies its weight by $0.37$.
3. For a fixed weight, mass and $g$ are inversely proportional, so a place with half the field strength needs twice the mass to give the same weight.
   1. A weight of $1600N$ comes from a mass of $160kg$ on Earth but from a mass of $1000kg$ on the Moon, where $g$ is $1.6N/kg$.
4. The ratio of one object's weights in two places equals the ratio of the field strengths, so $\frac{W_1}{W_2}=\frac{g_1}{g_2}$.
   1. The Moon's field strength is about one sixth of the Earth's, so any object on the Moon weighs about one sixth of its weight on Earth.

Gravitational potential energy

1. The gravitational field strength also appears in the equation for the energy in an object's gravitational store, which you must recall.
2. In words, $gravitational\; potential\; energy=mass\times gravitational\; field\; strength\times height$.
3. In symbols, the equation is $E_p=mgh$.
   1. $E_p$ is the gravitational potential energy in joules, $J$.
   2. $m$ is the mass in kilograms, $g$ is in newtons per kilogram, and $h$ is the height in metres.
4. Lifting an object through a height $h$ at a steady speed needs a force equal to its weight, $mg$, so the work done, and the energy gained by the gravitational store, is $mg\times h$.
5. For the same mass and height, the energy is directly proportional to $g$, so lifting a box $1m$ on the Moon needs about one sixth of the energy needed on Earth.

2.3.11 Forces causing rotation

Forces that cause rotation

1. A force can make an object turn about a fixed point instead of moving the whole object along.
2. The fixed point that an object turns about is called the pivot, or the fulcrum.
3. A force produces a turning effect when its line of action does not pass through the pivot.
4. A force whose line of action passes directly through the pivot has no turning effect, however large it is.
   1. Pushing on a door right at its hinges does not open it, because the force acts through the pivot.
5. The turning effect of a force is larger when the force is larger or acts further from the pivot.
6. The turning effect is greatest when the force acts at right angles to the line joining the point of action to the pivot.

Examples of turning forces

1. A door turns about its hinges, which is why the handle is fitted at the edge furthest from them.
2. A spanner turns a nut about the centre of the nut, and a longer spanner makes a tight nut easier to loosen.
3. A seesaw turns about its central support when children sit on either end.
4. A bicycle pedal turns the crank about the axle, and pushing down when the crank is horizontal gives the largest turning effect.
5. A steering wheel turns about the steering column when you push up on one side and pull down on the other.
6. A tap handle, a screwdriver and a bottle cap are all turned by forces acting some distance from their central axis, which is the pivot.
7. A wheelbarrow turns about the wheel axle when you lift its handles.
8. Scissors and pliers turn about the rivet that joins the two blades or jaws.

Clockwise and anticlockwise turning

1. A turning effect acts in one of two directions about a pivot, clockwise or anticlockwise.
2. To find the direction, imagine the object is free to turn about the pivot and picture which way the force would swing it.
   1. Pushing down on the right-hand end of a horizontal seesaw with a central pivot turns it clockwise.
   2. Pushing down on the left-hand end turns it anticlockwise.
   3. Pushing up on the right-hand end turns it anticlockwise, because reversing the force reverses the direction of turning.
3. When two forces act on an object, one may try to turn it clockwise while the other tries to turn it anticlockwise.
4. The object turns in the direction of the larger turning effect, so the resultant turning effect decides which way it rotates.
5. If the clockwise and anticlockwise turning effects are equal, there is no resultant turning effect, and the object stays balanced.

2.3.12 The moment of a force

The moment of a force

1. The turning effect of a force about a pivot is called the moment of the force.
2. The moment depends on the size of the force and on the perpendicular distance from the pivot to the line of action of the force.
3. The equation for the moment of a force is one you must recall.
4. In words, $moment\; of\; a\; force=force\times distance$, where the distance is measured at right angles to the direction of the force.
5. In symbols, the equation is $M=Fd$.
   1. $M$ is the moment in newton metres, $Nm$.
   2. $F$ is the force in newtons, $N$.
   3. $d$ is the perpendicular distance from the pivot to the line of action of the force, in metres, $m$.
6. Distances given in centimetres must be divided by $100$ before substituting, so that the moment comes out in newton metres.
7. A moment can be increased by using a larger force or by applying the force further from the pivot.
   1. A $20N$ push on the end of a spanner $0.25m$ long gives a moment of $5Nm$, while the same push on a spanner $0.50m$ long gives $10Nm$.
8. Every moment acts in a direction, which is either clockwise or anticlockwise about the pivot.

The perpendicular distance

1. The distance in the moment equation is the shortest distance from the pivot to the line of action of the force.
2. When a force acts at right angles to a bar, the perpendicular distance is the distance along the bar from the pivot to the force.
3. When a force acts at a slant, the perpendicular distance is shorter than the distance along the bar, so the moment is smaller.
4. When the line of action passes through the pivot, the perpendicular distance is zero, so the moment is zero.
5. A weight hanging from a horizontal ruler acts vertically downwards, so its perpendicular distance is the horizontal distance from the pivot to the point where it hangs.

The principle of moments

1. An object that is balanced and not turning is in equilibrium.
2. For a balanced object, the total clockwise moment about the pivot equals the total anticlockwise moment about the same pivot.
3. With one force on each side of the pivot, the principle of moments can be written as $F_1{d}_1=F_2{d}_2$.
4. When several forces act on the same side of the pivot, their moments are added to give the total moment on that side.
5. If the total clockwise moment is larger than the total anticlockwise moment, the object turns clockwise, and if it is smaller, the object turns anticlockwise.
6. The principle of moments lets you find an unknown force or an unknown distance for a balanced object.
   1. A light child can balance a heavy adult on a seesaw by sitting further from the pivot, because a smaller force multiplied by a larger distance can give the same moment.

2.3.13 Levers and gears

Levers

1. A lever is a rigid bar that turns about a pivot.
2. The effort is the force you apply to the lever, and the load is the force the lever acts against.
3. The effort produces a moment about the pivot, and the lever transmits this turning effect along the bar to the load.
4. The lever just balances the load, or is on the point of moving it, when the moment of the effort equals the moment of the load about the pivot.
   1. $effort\times distance\; of\; effort\; from\; pivot=load\times distance\; of\; load\; from\; pivot$
5. Everyday levers include a crowbar, a claw hammer pulling out a nail, a bottle opener, a wheelbarrow, scissors, nutcrackers and a screwdriver levering the lid off a tin of paint.
6. Your forearm is a lever too, with the elbow as the pivot and the biceps muscle supplying the effort.

Levers as force multipliers

1. When the effort acts further from the pivot than the load, a small effort can balance a large load, so the lever acts as a force multiplier.
2. The force is multiplied by the ratio of the two distances from the pivot.
   1. $\frac{load}{effort}=\frac{distance\; of\; effort\; from\; pivot}{distance\; of\; load\; from\; pivot}$
   2. A crowbar with the effort $1.2m$ from the pivot and the load $0.10m$ from it multiplies the force by $\frac{1.2}{0.10}=12$.
3. A lever does not supply extra energy, because the effort has to move through a larger distance than the load.
   1. To raise the load at the end of that crowbar by $1cm$, the end you push on must move down by $12cm$.
4. When the effort is closer to the pivot than the load, as in your forearm or a pair of tweezers, the effort must be larger than the load, but the load moves further and faster.

Gears

1. A gear is a wheel with teeth around its edge.
2. When two gears mesh, the teeth of the turning gear push on the teeth of the other gear, so a turning effect is passed from one axle to another.
3. The gear turned by a motor, a handle or your feet is the driving gear, and the gear it turns is the driven gear.
4. Two meshing gears always turn in opposite directions, so a driving gear turning clockwise makes the driven gear turn anticlockwise.
   1. Placing a third gear between them makes the driven gear turn in the same direction as the driving gear.
5. Where the teeth meet, the two gears push on each other with equal and opposite forces, so the force at the rim is the same size for both gears.

Gears as force multipliers

1. The moment on each gear is the force at its teeth multiplied by its radius, the distance from its axle to its teeth.
2. When a small driving gear turns a large driven gear, the same force acts at a larger radius, so the driven gear has a larger moment.
3. The moment is multiplied by the ratio of the radii, which is the same as the ratio of the numbers of teeth.
   1. $\frac{moment\; of\; driven\; gear}{moment\; of\; driving\; gear}=\frac{teeth\; on\; driven\; gear}{teeth\; on\; driving\; gear}$
4. The larger gear turns more slowly, so the gain in turning effect comes at the cost of speed of rotation.
   1. A driving gear with $12$ teeth turning a driven gear with $36$ teeth must make three full turns for every one turn of the driven gear.
5. When a large driving gear turns a small driven gear, the moment is reduced, but the driven gear turns faster.
6. A bicycle uses the same idea, because a large rear sprocket gives a bigger turning effect at the wheel for climbing hills, while a small rear sprocket turns the wheel faster on flat roads.
7. A car gearbox and a hand drill also use gears to switch between a large turning effect and a high speed of rotation.

2.3.14 Pressure in fluids

Pressure in fluids

1. A fluid is any substance that can flow, so all liquids and all gases are fluids.
2. The particles in a fluid are free to move, and they are always moving around randomly.
3. As they move, the particles collide with any surface in contact with the fluid, such as the wall of a container or the skin of a swimmer.
4. Each collision exerts a tiny force on the surface, and the enormous number of collisions every second adds up to a steady force spread across the surface.
5. The force on each unit of area of the surface is the pressure of the fluid.

Forces at right angles to surfaces

1. The pressure in a fluid causes a net force at right angles to any surface it touches.
2. The force is at right angles, or normal, to the surface whichever way the surface faces, whether it is horizontal, vertical or sloping.
3. The particles strike the surface from many directions, and the sideways parts of their pushes cancel out, leaving only a push straight into the surface.
4. At any point in a fluid, the pressure acts equally in all directions, so a tiny surface placed at that point feels the same force per unit area however it is turned.
5. A plastic bottle full of water with small holes punched around its side squirts water out of every hole at right angles to the wall at that hole.
6. The air inside a balloon pushes outwards at right angles to every part of the rubber, which is why a balloon inflates into a smooth, rounded shape.
7. The water around a submarine pushes inwards at right angles to every part of its hull, so the hull must be strong on its sides and underneath as well as on top.

The net force from fluid pressure

1. A surface often has fluid on both sides, and the fluid on each side pushes on it at right angles.
2. If the pressure is the same on both sides, the two forces are equal and opposite, so there is no net force on the surface.
3. If the pressure is different on the two sides, there is a net force at right angles to the surface, directed from the higher-pressure side towards the lower-pressure side.
   1. The air in a pumped-up car tyre is at a higher pressure than the air outside, so there is a net outward force on the tyre wall that keeps the tyre firm.
   2. An aircraft cabin is kept at a higher pressure than the thin air outside at cruising height, so there is a net outward force on every window and door.
   3. Sucking air out of a plastic drinks bottle lowers the pressure inside it, so the net inward force from the air outside crushes the bottle.

2.3.15 Pressure, force and contact area

Pressure, force and area

1. Pressure describes how concentrated a force is, by comparing the force with the area it acts on.
2. The equation linking pressure, force and area is one you must recall.
3. In words, $pressure=\frac{force\; normal\; to\; a\; surface}{area\; of\; that\; surface}$.
4. In symbols, the equation is $p=\frac{F}{A}$.
   1. $p$ is the pressure in pascals, $Pa$.
   2. $F$ is the force acting at right angles to the surface, in newtons, $N$.
   3. $A$ is the area of the surface in square metres, $m^2$.
5. The unit of pressure is the pascal, and $1Pa$ is equal to $1{N/m}^2$.
6. Large pressures are often given in kilopascals, where $1kPa=1000Pa$.
7. Rearranging the equation gives $F=pA$ to find a force and $A=\frac{F}{p}$ to find an area.

Converting areas

1. Areas must be in square metres before they are substituted into the pressure equation.
2. One metre is $100cm$, so one square metre is $100\times 100=10000{cm}^2$.
   1. Divide an area in ${cm}^2$ by $10000$ to convert it into $m^2$, so $25{cm}^2=0.0025m^2$.
3. One metre is $1000mm$, so one square metre is $1000000{mm}^2$.
   1. Divide an area in ${mm}^2$ by $1000000$ to convert it into $m^2$, so $500{mm}^2=0.0005m^2$.
4. Dividing an area in ${cm}^2$ by $100$ instead of $10000$ makes the calculated pressure $100$ times too small.

How contact area affects pressure

1. For the same force, a smaller area gives a larger pressure, and a larger area gives a smaller pressure.
2. When the force stays the same, pressure is inversely proportional to area, so halving the contact area doubles the pressure.
3. Some objects are designed with a small contact area to produce a large pressure.
   1. A sharp knife has a very thin edge, so the pressure under it is large enough to cut through food.
   2. Drawing pins, nails and needles have fine points, so they push into materials easily.
   3. Football boot studs concentrate a player's weight onto small areas, so they sink into the turf and grip.
4. Other objects are designed with a large contact area to produce a small pressure.
   1. Snowshoes spread a walker's weight over a large area, so they do not sink into soft snow.
   2. Tractors have wide tyres and some diggers have caterpillar tracks, so they do not sink into muddy ground.
   3. The wide straps of a school rucksack spread its weight across your shoulders, so they dig in less than thin straps would.

Hydraulic systems

1. A hydraulic system uses a liquid, usually oil, trapped in pipes and cylinders to transmit a force from one place to another.
2. Liquids are almost incompressible, so pushing on the liquid at one point does not squash it but passes the push through it.
3. A pressure applied to an enclosed liquid is transmitted equally to every part of the liquid.
4. A simple hydraulic system has a narrow piston where the effort is applied and a wide piston that pushes on the load.
   1. A small force $F_1$ on the narrow piston of area $A_1$ produces a pressure $p=\frac{F_1}{A_1}$ in the liquid.
   2. The same pressure acts on the wide piston of area $A_2$, producing a force $F_2=p{A}_2$.
   3. The wide piston has the larger area, so the output force is larger than the input force, and the system is a force multiplier.
5. The force is multiplied by the ratio of the piston areas, $\frac{F_2}{F_1}=\frac{A_2}{A_1}$.
6. A hydraulic system does not create energy, because the narrow piston must move a long way to push the wide piston a short distance.
7. Car brakes are hydraulic, because a light push on the brake pedal moves a narrow piston, and the pressure is transmitted through the brake fluid to wider pistons that press the brake pads onto the discs.
8. A hydraulic car jack, a dentist's chair and the arm of a digger all use the same principle to move heavy loads with a small effort.
9. Gases are not used in these systems, because a gas is easily compressed, so pushing on it would squash it instead of passing on the pressure.