Forces and elasticity — AQA GCSE Physics
Test yourself on Forces and elasticity with AQA GCSE practice questions.
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Forces and elasticity explained
One force on a free object simply makes it accelerate; to change its shape you need at least two.
Read the full explanation
A spring hangs from a stand, so the weight of the load pulls down while the stand pulls up, and the spring stretches between them. Bend a ruler and each hand pushes the opposite way; compress a spring and the hand pushes down while the bench pushes up. Because the object stays still, those forces must be balanced, which is why the specification limits this to stationary objects. Deformation is elastic if the object returns to its original length and shape the moment the forces are removed, as a spring balance must so that it reads correctly next time; it is inelastic if part of the change of shape is permanent, as when a spring stretched too far stays longer than it began.
The extension of an elastic object, such as a spring, is directly proportional to the force applied, provided that the limit of proportionality is not exceeded.
Directly proportional has a precise meaning: double the force and the extension doubles, and a graph of force against extension is a straight line through the origin. That is what makes a spring balance useful, because a linear scale turns equal steps of mass into equal steps of length. Extension is not length: it is the stretched length minus the original unstretched length, so a spring 30 mm long that becomes 50 mm long has an extension of 20 mm. The proportionality holds only up to the limit of proportionality. Past that point the line bends away from the straight part. Note that curvature only confirms the limit of proportionality has been exceeded; it does not mean the spring has been inelastically deformed. Inelastic deformation only occurs if the spring is stretched beyond its elastic limit, meaning it no longer returns to its original length when unloaded.
f orce = spring constant × extension F = k e force, F, in newtons, N spring constant, k, in newtons per metre, N/m extension, e, in metres, m This relationship also applies to the compression of an elastic object, where ‘e’ would be the compression of the object.
This equation links the force applied to a spring, the spring constant in newtons per metre and the extension in metres. The spring constant is a property of that particular spring: a large value means a stiff spring that extends very little. Rearranged, k = F/e and e = F/k. A 3.0 N load on a spring of 25 N/m gives e = 3.0/25 = 0.12 m. Almost every lost mark here is a unit. Extensions are usually measured in centimetres or millimetres and must be converted to metres first, and a mass in grams has to be turned into a weight in newtons before it can be used as the force. The same relationship covers compression, where e is how much shorter the spring has become, which is why car suspension springs and bathroom scales obey it too.
A force that stretches (or compresses) a spring does work and elastic potential energy is stored in the spring.
Stretching or compressing a spring means moving its end through a distance against the spring's force, so the applied force does work and that work fills the spring's elastic potential energy store. The store depends on the square of the extension or compression, so doubling the deformation stores four times the energy. That is why pulling a toy launcher back twice as far sends it much further, and why compressing a suspension spring twice as much absorbs four times the energy. When the spring is released the elastic store empties into whatever it pushes: the kinetic store of a launched aeroplane, or the gravitational potential store of a pod rising on bungee cords. In practice not all of it arrives, because air resistance and internal heating take a share.
Provided the spring is not inelastically deformed, the work done on the spring and the elastic potential energy stored are equal.
Below the limit of proportionality a spring is a perfect energy store: every joule of work done stretching it comes back when it is released, so the work done and the energy stored are equal and either can be found from the area under the force-extension line. That area is a triangle, half the force multiplied by the extension, which is the same as half the spring constant multiplied by the extension squared. Once the spring has been stretched far enough to be inelastically deformed the bookkeeping changes: some of the work done goes into permanently rearranging the material rather than into the elastic store, the spring does not return to its original length, and less energy comes back than went in. Mark schemes accept elastic limit as an alternative name for the limit of proportionality.
Students should be able to calculate relevant values of stored energy and energy transfers.
These are the multi-step questions. A typical one gives the energy stored in a stretched cord and asks for the spring constant, so the elastic energy equation is rearranged to make the spring constant the subject and the energy is converted from kilojoules to joules first: 24500 = ½ × k × 35² gives k = 40 N/m. Others chain the equations together - elastic potential to kinetic, so the two expressions are equated and a speed is found; or elastic potential to gravitational potential, giving a height. The two traps are using the total stretched length instead of the extension, and forgetting to square. Keep every value in joules, newtons, metres and kilograms, and write down the transfer you are assuming before you start.
Required practical activity 6: investigate the relationship between force and extension for a spring.
Clamp a spring from a stand with the base weighted or clamped to the bench so it cannot topple. Measure the unstretched length with a millimetre rule held vertically alongside, reading the position of the bottom of the spring at eye level to avoid parallax error. Add weights one newton at a time, measuring the new length after each and subtracting the original unstretched length to calculate the extension. Take readings as the weights are removed to check they match, ensuring no inelastic deformation. Plot force on the vertical axis against extension on the horizontal; the gradient of the straight part is the spring constant. The control variable is the spring itself. The graph curving at the top shows the limit of proportionality.
Your focus
- Name the two forces acting for a stretched spring, a bent ruler and a compressed spring.
- Explain why one force alone cannot change the shape of a stationary object.
- Distinguish elastic from inelastic deformation by what happens when the force is removed.
Show all 21 objectives
- Calculate the extension of a spring from its unstretched and stretched lengths.
- Identify the limit of proportionality on a force-extension graph.
- Distinguish between the limit of proportionality and the elastic limit.
- Calculate the extension of a spring from a given force and spring constant.
- Rearrange the spring equation to find a spring constant from a measured force and extension.
- Apply the same relationship to a compressed spring by treating e as the compression.
- Explain how work done by a stretching or compressing force fills the elastic potential store.
- Predict the change in stored energy when the extension or compression of a spring is doubled.
- Describe the energy transfers when a stretched or compressed spring is released to launch an object.
- Calculate the elastic potential energy stored as the area under a force-extension graph.
- State the condition under which the work done on a spring equals the energy it stores.
- Explain why less energy is recovered from a spring that has been inelastically deformed.
- Rearrange the elastic potential energy equation to find a spring constant from stored energy and extension.
- Calculate the launch speed of an object from the elastic energy stored in a spring.
- Convert kilojoules into joules and lengths into metres before substituting in an energy calculation.
- Describe a method to measure the extension of a spring for a series of known forces.
- Identify the limit of proportionality from the plotted force-extension graph.
- Explain how holding the rule vertically and reading at eye level reduce measurement error.
Forces and elasticity exam tips
Marking Points
- one mark for stating that at least two forces are needed to change the shape of a stationary object
- one mark for naming the pair of forces in a given example, such as the weight and the pull of the support on a hanging spring
- one mark for elastic deformation described as returning to the original length or shape when the force is removed
- one mark for inelastic deformation described as a permanent change of shape that remains after the force is removed
- one mark for a straight line through the origin as the test for direct proportionality
- one mark for extension calculated as stretched length minus original length
- one mark for stating that the relationship holds only up to the limit of proportionality
- one mark for identifying that the graph curving means the limit of proportionality is exceeded, not necessarily the elastic limit
- one mark for the equation force = spring constant × extension, or a correct rearrangement
- one mark for converting the extension to metres, and a mass to a weight in newtons where needed
- one mark for the correct substitution
- one mark for the answer with its unit, N/m for a spring constant or m for an extension
- one mark for stating that the stretching or compressing force does work on the spring
- one mark for that work filling the elastic potential energy store
- one mark for the store depending on the square of the extension or compression, so doubling it stores four times as much
- one mark for the elastic store emptying into the kinetic or gravitational potential store on release
- one mark for equating the work done on the spring with the elastic potential energy stored
- one mark for finding that energy as the area under the force-extension graph
- one mark for stating the condition that the spring has not been inelastically deformed
- one mark for stating that beyond that point some of the work done is not recoverable and the spring stays deformed
- one mark for identifying the extension rather than the total length, such as 35 m of stretch on a cord that began at 20 m
- one mark for converting the energy from kilojoules into joules
- one mark for a correct substitution and a correct rearrangement of the energy equation
- one mark for the answer with its unit, with full credit available for a correct answer shown without working
- one mark for measuring the original unstretched length of the spring before any load is added
- one mark for adding known weights in equal steps and recording the new length each time
- one mark for calculating each extension as new length minus original length
- one mark for plotting force against extension and taking the gradient of the straight section
Examiner Tips
- 💡Name both forces and their directions; one force is never enough for a change of shape.
- 💡Use the words returns to its original length - that is the credited phrase for elastic deformation.
- 💡Subtract the unstretched length before you plot or calculate anything.
- 💡Where a graph curves, name the limit of proportionality. Do not assume curvature immediately means inelastic deformation.
- 💡Underline whether the question has given you a length or an extension before substituting.
- 💡If the question gives a mass, convert it to a weight first; the value of g is printed on the paper.
- 💡If the extension or compression doubles, write four times before reaching for the calculator - it is often all the mark needs.
- 💡Say where the energy goes on release; 'it is released' is not an energy transfer.
- 💡Area under a force-extension graph is energy; the gradient is the spring constant. Check which the question wants.
- 💡Look for the phrase not inelastically deformed in a question - it is telling you the equality applies.
- 💡Write down which stores are involved before touching the calculator; the chain tells you which equations to equate.
- 💡A correct answer without working still scores, but a wrong one without working scores nothing, so show the substitution.
- 💡Say eye level and rule vertical when asked how to reduce measurement error; both are credited.
- 💡Describe repeating the readings as the load is removed, to check the spring has not been inelastically deformed.
Common Mistakes
- describing only the load and forgetting the force from the support or the second hand
- saying a single force stretches a spring, which would only accelerate it
- defining elastic deformation as it stretches rather than as returning to its original length
- treating any large extension as inelastic, when the test is whether the change is permanent
- plotting or using the total length instead of the extension, so the line misses the origin; always subtract the unstretched length
- calling a straight line that does not pass through the origin a proportional relationship; it must pass through the origin
- conflating the limit of proportionality with the elastic limit by assuming a curved graph means permanent deformation; curvature only indicates the limit of proportionality is exceeded
- assuming that doubling the force always doubles the extension, even beyond the straight section; this only applies up to the limit of proportionality
- substituting the total length of the spring instead of its extension
- leaving the extension in centimetres, so the spring constant comes out a hundred times too small
- using a mass in kilograms as the force instead of its weight in newtons
- assuming the equation fails for compression, when e is simply the compression
- saying the energy stored is proportional to the extension or compression rather than to its square
- writing that force is stored in the spring instead of energy
- claiming all the elastic energy becomes kinetic energy, ignoring air resistance and heating; always account for efficiency losses
- forgetting that compressing a spring also stores elastic potential energy; the principle applies to both stretching and compressing
- using the whole rectangle under the graph rather than the triangle, which doubles the energy
- assuming the equality still holds after the limit of proportionality has been passed
- saying the spring loses the energy rather than transferring it to internal energy and permanent deformation
- treating an inelastically deformed spring as broken rather than permanently longer
- using the full stretched length of a bungee cord as the extension, which gives 16.2 rather than 40
- forgetting to square the extension when rearranging for the spring constant
- leaving the energy in kilojoules, so the answer is a thousand times too small
- halving twice, by using the half in the equation and then dividing the answer by two
- holding the rule at an angle, which makes every measured length inaccurate
- recording total lengths and plotting them as extensions without subtracting the initial length
- loading the spring past its limit of proportionality and then including those points in the gradient calculation
- leaving the clamp stand unsecured so that it tips as the load grows