Hooke's Law and Springs Virtual Lab

A spring is a device that pushes or pulls back when it is deformed, and returns to its original shape when the deforming force is removed. Car suspensions, kitchen scales, mattresses, pens, watches and the cables of a bridge all rely on this behaviour. Hooke's law is the simplest rule that describes it: for a spring that is not stretched too far, the extension is directly proportional to the force that stretches it. The rule is named after Robert Hooke, who published it in the seventeenth century.

This page works in three layers. Material marked Foundation needs only simple algebra and the idea of a force. Core material matches typical A-level, IB Diploma, AP Physics 1, WASSCE and Cambridge AS/A Level courses and uses graphs, energy and the equations of simple harmonic motion. Extension material goes towards IB HL, AP Physics C and first-year university, and covers the Young modulus, the energy lost in hysteresis, and the mass of the spring itself.

Use the simulation alongside these notes. It hangs masses on a spring, reads the extension from a ruler, lets you combine two springs, and lets you set the mass oscillating. The default spring has a constant of 40 N/m and an elastic limit of 8 N. Before you change anything, predict what will happen, then test your prediction.

Learning outcomes

Foundation

• Describe the extension of a spring as the new length minus the natural length.
• State Hooke's law: for a spring within its limit, force is proportional to extension, F = ke.
• Calculate the weight of a hanging mass from W = mg and use it as the stretching force.
• Explain what an elastic material is and what a permanent set means.
• Read an extension from a ruler and record it to the nearest millimetre.

Core (A-level, IB, AP, WAEC)

• Use F = ke with consistent units, and find k from the gradient of a force-extension graph.
• Distinguish the limit of proportionality from the elastic limit, and describe what happens beyond each.
• Find the effective spring constant of two springs in series and in parallel.
• Calculate elastic energy as the area under a force-extension graph, E = ½Fe = ½ke².
• Explain loading and unloading curves, the permanent set and the energy dissipated in a hysteresis loop.
• Find the period of a mass-spring oscillator, T = 2π√(m/k), and use a graph of T² against m to find k.
• Estimate uncertainties from a line of best fit and from steepest and shallowest lines.

Extension (HL, AP C, university)

• Define stress, strain and the Young modulus, and relate the spring constant of a wire to E, A and L.
• Derive the period of a mass-spring system from Newton's second law, and include the effective mass of the spring, m + m_s/3.
• Describe damping, the amplitude decay and the change in period.
• Explain why the period of a real spring depends on amplitude once the force law is no longer linear.

Natural length and extension

The natural length of a spring is its length when no force acts on it. When a force stretches the spring, its length becomes longer. The extension is the change in length: extension = stretched length − natural length. A spring that is compressed has a compression instead. Extension is measured from the natural length, never from the length of the whole apparatus. In the simulation the ruler is zeroed for you at the pointer of a new, unloaded spring, so the reading is the extension directly. In a real laboratory you must measure the natural length first and subtract it from every later reading.

Hooke's law

The spring pulls back on whatever stretches it with a force of the same size, so F = ke is also the size of the restoring force. In vector form the restoring force is F = −ke, the minus sign showing that the force points opposite to the displacement. At rest, a hanging mass is in equilibrium: the upward spring force equals the downward weight, so ke = mg.

Three important limits

For many metals the two limits are very close together, and some courses treat them as one. In the simulation the limit of proportionality is set at 80% of the elastic limit force, so that you can see a short region where the graph bends but the spring still recovers. The breaking force is twice the elastic limit force.

Elastic and plastic behaviour

An elastic deformation disappears when the force is removed. The atoms are pulled slightly out of their equilibrium positions but stay bonded to their neighbours. A plastic deformation does not disappear: layers of atoms have slipped past one another and stay in the new positions. A spring stretched past its elastic limit is plastically deformed and is unlikely to work properly again.

Spring combinations

In series the springs are joined end to end. The whole load passes through each spring, so each carries the same force F, and the total extension is the sum of the individual extensions. The combination is softer than either spring. In parallel the springs are side by side and share the load. Each has the same extension, the forces add, and the combination is stiffer than either spring. Section C shows the proofs.

Oscillation

If a mass on a spring is pulled down and released, the restoring force F = −ke makes it oscillate up and down. The motion is simple harmonic because the restoring force is proportional to the displacement from equilibrium and always points towards it. The time for one complete oscillation, the period T, depends on the mass and the spring constant, but, for a linear spring, not on the amplitude.

C1. Summary of equations

C2. A hanging mass (Core)

A mass m hangs at rest from a spring. The only forces are weight mg down and the spring force ke up. At rest they balance: ke = mg, so the extension is e = mg/k. Doubling the mass doubles the extension, as long as the spring stays within its limit of proportionality.

C3. Springs in series (Core)

Take springs of constants k₁ and k₂ joined end to end, with a load F at the bottom. Every part of the chain carries the same force F, so spring 1 extends by e₁ = F/k₁ and spring 2 by e₂ = F/k₂. The total extension is e = e₁ + e₂ = F(1/k₁ + 1/k₂). Comparing with e = F/k gives the result:

1/k = 1/k₁ + 1/k₂

For two equal springs this gives k = k₁/2. The combination is always softer than the softest spring.

C4. Springs in parallel (Core)

Two springs hang side by side from the same support and carry a light rigid bar with the load F. The bar stays level, so both springs have the same extension e. They carry forces F₁ = k₁e and F₂ = k₂e, and F = F₁ + F₂ = (k₁ + k₂)e. So:

k = k₁ + k₂

For two equal springs this gives k = 2k₁. The combination is stiffer than either spring.

C5. Elastic energy (Core)

To stretch a spring, a force that grows from 0 to F must act through the extension, so the work done is the area under the force-extension graph. Inside the proportional region the graph is a straight line from the origin, and the area is a triangle of base e and height F:

E = ½Fe = ½ke² = F²/(2k)

Notice that the energy is not Fe. The force rises from zero, so the average force is only half the final force. Beyond the limit of proportionality the graph curves and the area must be found by counting squares or the trapezium rule. The elastic energy depends on the extension squared: doubling the extension quadruples the energy.

C6. Period of a mass-spring oscillator (Core and Extension)

Let x be the displacement of the mass below its equilibrium position. At equilibrium, ke₀ = mg. At displacement x the spring force is k(e₀ + x), so the resultant force downward is mg − k(e₀ + x) = −kx. By Newton's second law:

m d²x/dt² = −kx

This is the equation of simple harmonic motion with angular frequency ω = √(k/m). The solution is x = A cos ωt when the mass is released from rest at displacement A. The period is T = 2π/ω:

T = 2π√(m/k)

Gravity does not appear. It only changes the equilibrium position, so the period of a mass on a spring is the same on the Moon as on Earth. The period does not depend on the amplitude (for a linear spring), and it increases as the square root of the mass. Squaring gives T² = (4π²/k) m, a straight line through the origin when plotted against m, with gradient 4π²/k.

C7. Energy in the oscillation (Core)

The total mechanical energy of the undamped oscillator is constant: E = ½kA² = ½mv² + ½kx², measured from equilibrium with gravitational and elastic potential energy combined. The mass is fastest at the equilibrium position, with maximum speed v_max = Aω = A√(k/m), and momentarily at rest at the extremes.

C8. A mass released from the natural length (Core)

A mass attached to a spring and released from rest with the spring at its natural length falls until it stops momentarily. By energy conservation, the gravitational potential energy lost equals the elastic energy gained: mgh = ½kh², where h is the maximum extension. So h = 2mg/k, which is twice the equilibrium extension mg/k. The mass oscillates about the equilibrium position with amplitude mg/k.

C9. Stress, strain and the Young modulus (Extension)

A spring constant depends on the shape of the sample, so it cannot describe a material. The Young modulus removes the shape. Stress is the force per unit cross-sectional area, σ = F/A, in pascals. Strain is the fractional change in length, ε = e/L, with no units. In the proportional region stress is proportional to strain, and the constant is the Young modulus:

E = σ/ε = (F/A) / (e/L) = FL/(Ae)

Rearranging gives F = (EA/L) e, so a uniform wire obeys Hooke's law with spring constant k = EA/L. A longer wire is softer; a thicker wire is stiffer. Steel has a Young modulus of the order of 200 GPa, so a thin steel wire barely stretches.

C10. Effective mass of the spring (Extension)

The coils of a real spring move too. The top coil is fixed and the bottom coil moves with the mass, so on average the spring moves more slowly than the mass. Integrating the kinetic energy of a uniform spring gives an effective extra mass of one third of the spring's mass. The period becomes:

T = 2π√((m + m_s/3) / k)

A graph of T² against m therefore has a positive intercept on the T² axis equal to 4π²(m_s/3)/k, or equivalently an intercept of −m_s/3 on the m axis.

Use g = 9.81 N/kg. Every number in these examples has been checked by computation.

The force-extension graph

• Gradient = F/e = k, the spring constant, in N/m (if extension is in metres).
• Area under the graph = work done on the spring. In the straight part it is the triangle ½Fe.
• Straight through the origin shows direct proportion. A line that does not pass through the origin means a zero error in the extension (for example a wrong natural length) or an initial tension in the spring.
• If force is plotted on the horizontal axis and extension on the vertical axis, the gradient is 1/k, and the area has no direct meaning.

Loading and unloading

If a spring is stretched beyond its elastic limit and then unloaded, the unloading path is not the loading path. It is a line with the original gradient k, displaced to the right, and it reaches zero force at a positive extension. The area between the loading and unloading curves is energy that was dissipated, mostly as heat in the metal. The area under the unloading curve is the elastic energy returned.

Other useful graphs

The T² against m graph is a typical example of linearising a relationship: a quantity that is not directly proportional is plotted so that a straight line results, from which a constant can be found accurately.

The simulation compared with real springs

The virtual spring is a single model that shows the main ideas clearly. A real spring differs from it in several ways.

Rubber and other materials

Not every material obeys Hooke's law. A rubber band has a force-extension graph that is S-shaped, not a straight line, and its loading and unloading curves differ even when it returns to its original length. The area between them is the energy dissipated as heat on each stretch, which is why a rubber band warms up when stretched and released many times. Rubber is elastic but not Hookean. A polythene strip shows a long plastic region, and glass fractures suddenly with almost no plastic region. Hooke's law describes one class of behaviour that is common for metals at small strains; it is not a law of nature in the same sense as Newton's laws.

Large-amplitude oscillations

The period T = 2π√(m/k) holds only while the restoring force is proportional to the displacement. If the motion reaches the curved part of the force-extension graph, the effective stiffness falls and the period becomes longer, and it now depends on the amplitude. The simulation shows this: oscillate a heavy mass whose motion crosses the limit of proportionality and compare the measured period with 2π√(m/k). Oscillators of this kind are called anharmonic.

Atomic origin

Atoms in a solid sit at the minimum of a potential energy curve. For small displacements the bottom of the curve is close to a parabola, and a parabolic potential energy ½kx² gives a linear restoring force. This is why Hooke's law is so widely useful: nearly any system displaced a little from stable equilibrium behaves like a spring. Larger displacements reach the asymmetric parts of the curve, where the law fails.

Research question

A good research question names the variables and the relationship. For example: "How does the extension of a steel spring depend on the load hung from it, and is the relationship consistent with Hooke's law up to the elastic limit?" This is testable, has a theoretical prediction (a straight line through the origin) and lets you find the spring constant from the results. A second question could be: "How does the period of a mass on a spring depend on the mass, and does a graph of T² against m give the same spring constant as the static method?"

Variables

Method for a real laboratory: static loading

1. Clamp the spring vertically. Fix a pointer to the bottom of the spring and a metre rule, held vertically, beside it. Use a set square or a mirror (to avoid parallax) when reading the scale.
2. Read the position of the pointer with only the hanger attached. Treat that as the zero for extension, or measure the natural length with the hanger removed if the hanger is very heavy.
3. Add a 50 g mass, wait until the spring is still, and read the pointer position. Calculate the extension.
4. Continue adding masses to a total that keeps the spring within the straight part. Look for the first signs of curvature in the readings as you go.
5. Unload in the same steps, recording the pointer positions again. This tests whether the spring returned to the same extension each time.
6. Repeat the whole set and take means.

Method for a real laboratory: oscillations

1. Hang a mass of at least 100 g. Pull it down about 2 cm and release. Check that it moves vertically without swinging sideways.
2. Start the stopwatch when the mass passes the equilibrium position moving downward and count 10 complete oscillations, then stop. Divide the time by 10 to get the period. Timing 10 oscillations reduces the effect of reaction time.
3. Repeat for at least five different masses, and repeat each timing at least twice.

Method for the virtual lab

1. In Static loading, choose one spring (the default is 40 N/m with an elastic limit of 8 N). Add 50 g at a time and press Record reading after the pointer has stopped. Watch the status box and the small graph: stay in the green region at first.
2. Go on into the curved region and beyond the elastic limit with a fresh spring if you want to see the permanent set. Then unload in steps and record the readings, including 0 g.
3. Repeat with two springs in series and in parallel, using two equal springs, and compare the extensions at the same load.
4. Switch to Oscillation. Choose a mass, release it, and let the ten oscillations finish before pressing Record period. Repeat for at least five masses.
5. Use the analysis tools in section 5 to find k from the gradient of the force-extension graph and from the T² against m graph, and compare the two values.

Choosing ranges and repeats

Choose a range of loads that gives at least six points in the straight part, and use more points close to the limit so you can locate the bend. Equal load steps make the graph easier to read. The simulation can add a measurement scatter. With the scatter on, repeating a measurement gives slightly different answers, just as in a real laboratory; with it off, repeats are identical and only the rounding to the nearest millimetre remains.

Types of uncertainty

A random uncertainty makes repeated readings scatter about the true value. Reading a ruler by eye, judging when a pointer stops moving and starting a stopwatch late are examples. Repeating and averaging reduces it. A systematic error shifts every reading the same way. A bent ruler, a wrongly measured natural length or a stopwatch that runs fast are examples. Repeating does not reduce it, but a graph with a non-zero intercept can reveal it.

Uncertainty in single measurements

• A millimetre ruler is read to the nearest millimetre. A common estimate of the uncertainty of one reading is ±0.5 mm. An extension is the difference of two readings, so its uncertainty can be as large as ±1 mm if both ends carry an uncertainty.
• Masses on a slotted hanger are usually accurate to 1% or better. Their uncertainty is often much smaller than the uncertainty of the extension, and many courses ignore it.
• The time for 10 oscillations might have an uncertainty of ±0.2 s from reaction time. Dividing by 10 divides the uncertainty by 10, so the period has an uncertainty of ±0.02 s.

Percentage uncertainty and propagation

The percentage uncertainty is (uncertainty / value) × 100. For a quantity found by multiplying or dividing, add the percentage uncertainties; for a power, multiply the percentage uncertainty by the power. For example, in k = F/e with 1% in F and 3% in e, k has an uncertainty of about 4%. A single reading with a small extension has a large percentage uncertainty: 0.5 mm in 10 mm is 5%, but 0.5 mm in 100 mm is only 0.5%. This is why measurements at larger loads are more reliable, and why a graph of many points is better than one reading.

Line of best fit and gradient

1. Plot force on the vertical axis and extension on the horizontal axis. Draw a straight line of best fit with roughly equal numbers of points on each side. In the simulation, the line is a least-squares fit.
2. Choose two points far apart on the line (not data points) and calculate the gradient ΔF/Δe, converting extension to metres.
3. Draw the steepest and shallowest straight lines that still pass through all the uncertainty bars, or through the ends of the data. Their gradients give the uncertainty in k: Δk = (k_max − k_min)/2.
4. Quote k = value ± uncertainty with the unit, to a sensible number of significant figures.

The simulation does this for you in the Hooke's law analysis, and also reports the standard error of the gradient from the least-squares fit. Quote the larger of the two uncertainties. The standard error of the fit is smaller when you have more points and when they scatter less. The steepest and shallowest line method depends mostly on the uncertainty in the end points.

Intercepts

A straight line through the origin is expected for F against e. If the line has an intercept that is much larger than its uncertainty, ask why. A positive force intercept suggests an initial tension in the spring; a positive extension intercept suggests that the natural length was measured wrongly. The tool lets you force the line through the origin, but only do so when the physics says the line should pass through it.

Finding the limit of proportionality

Plot all the loading points and look for the first load at which the points begin to leave the straight line, taking the uncertainty into account. Because of scatter, the position of the bend is uncertain. Record more points near it. The simulation's estimate depends on how many points you recorded: with a few readings it can only say that the bend lies between two loads.

Area under the graph

If the graph is straight, the energy stored is ½Fe. If the graph curves, use the trapezium rule on neighbouring pairs of points: area = ½(F₁ + F₂)(e₂ − e₁). Add the strips together. For a hysteresis loop, subtract the area under the unloading curve from the area under the loading curve. The result is the energy dissipated in the loop.

Graph of T² against m

Plot T² on the vertical axis against m on the horizontal axis. The gradient is 4π²/k, so k = 4π²/gradient. The percentage uncertainty in k equals the percentage uncertainty in the gradient. If the line has a positive intercept on the T² axis, the intercept divided by the gradient is m_s/3, an estimate of one third of the spring's mass.

Comparing with the accepted value

Compare k from the static graph with k from the oscillation graph. If the two values differ by less than the combined uncertainty, they agree. If they do not, search for a systematic error: points beyond the limit of proportionality included in the fit, or a neglected spring mass in the oscillation method.

Apparatus

Reading a scale correctly

• Put the pointer as close to the scale as possible and read at eye level. Looking from above or below moves the apparent position, an error called parallax.
• Read the position of the same part of the pointer each time.
• Wait for the spring to come to rest. A bouncing spring oscillates about the equilibrium and the reading is not meaningful. In the simulation, the pointer settles over a second or so, and recording is refused until it has stopped.
• Estimate the reading to the nearest division, and record every reading in a table with units in the headings.

Reducing errors in the oscillation method

• Use a fiducial marker (a fixed pointer or a mark on a card) at the equilibrium position, and time when the mass passes it, because the speed there is highest and the timing error smallest.
• Time ten complete oscillations, not ten half-oscillations. In the simulation, the stopwatch measures from one downward crossing of the equilibrium to the tenth next.
• Keep the amplitude small, but not so small that it is hard to see. In the simulation you can change the starting displacement; the status panel warns if the motion leaves the straight part of the spring.
• Use a mass large enough that the spring never goes slack at the top of its motion.

Safety in a real laboratory

• Stretched springs and wires store energy. If a spring or wire breaks or slips from its hook, it recoils and can strike the face or eyes. Wear eye protection whenever a spring or wire is loaded close to its limit, and never stretch a spring by hand beyond its range.
• Falling masses. Keep feet clear and place a box or cushion of soft material under hanging masses. A 1 kg mass falling from the bench can injure a foot.
• Stand stability. Clamp the stand to the bench or weight the base, and put the heaviest weight low.
• Wire tests. In a Young modulus experiment the wire can snap. Use a protective screen or goggles and keep the load within the stated range.
• Sharp ends. The ends of springs and wire can be sharp.

What the simulation does

The virtual spring is a one-dimensional model. A single parameter set gives a force-extension curve with a straight section, a short curved section that is still elastic, and a plastic section that ends in a break. Two springs can be combined in series or in parallel. The oscillation is the numerical solution of Newton's second law for the mass on the spring, with an optional damping and an optional spring mass.

Using the simulation responsibly

• Treat the scatter option as part of the experiment: switch it on to practise uncertainty, and off to see the ideal behaviour.
• Do not assume that real springs have the same limits. Use the elastic limit shown in the simulation as an example, and check the data sheet or rating of a real spring.
• The simulation does not replace practical work. Examinations that assess practical skills, such as handling apparatus, reading scales and judging when a spring is at rest, require real equipment.

Before you start a calculation

1. Read the whole question and underline the data: masses, extensions and spring constants, with their units.
2. Convert units straight away: grams to kilograms, millimetres and centimetres to metres.
3. Decide which idea fits. A hanging mass at rest: ke = mg. A graph: gradient and area. Two springs: series or parallel. An oscillation: T = 2π√(m/k).

Writing the answer

• State the formula, substitute the numbers with units, and then give the answer with a unit.
• Give the answer to the same number of significant figures as the least precise data, often two or three.
• Keep unrounded values in your calculator until the last step, to avoid rounding errors.
• If a calculation has several parts, label each part and carry the earlier answer forward. Examiners usually allow marks for correct use of a previous answer, even if that answer was wrong.

Command words

Graph questions

• Label both axes with the quantity and unit, choose scales that fill at least half the grid, and plot points to half a small square.
• Draw a single straight line of best fit with a ruler, or a smooth curve, not a join-the-dots path.
• For a gradient, use a large triangle on the line, show its base and height, and give units.
• For an area, show how you counted squares or applied the trapezium rule, and convert the area units to energy (one square of 1 N by 1 cm is 0.01 J).

Typical explanations

• Why do you time 10 oscillations? The reaction-time uncertainty is divided by 10 when the time is divided by 10, so the period has a smaller percentage uncertainty.
• How can you tell the limit of proportionality? The point at which the force-extension graph stops being a straight line through the origin.
• Why does a stretched spring warm up if it is stretched past its elastic limit? Work is done to move atoms to new positions, and the energy is dissipated as heat, so the unloading line lies below the loading line.
• Why is the period independent of g? The restoring force is proportional to the displacement from equilibrium and does not contain g.

About the marks in this laboratory

The tests in this laboratory give marks for learning, not an official grade. Use the review list at the end of the examination, and open the notes it names. Repeat the examination: the questions and the calculation numbers change each time.

Hooke's law and springs appear in the mechanics, materials and oscillations parts of secondary and first-year university physics courses. The list below gives topic names only. Syllabuses change, so check your own current specification and the list of equations your examination gives you, and treat this page as support for it, not a replacement.

Level tags used in these notes

• Foundation: the idea of extension, F = ke, simple weight calculations.
• Core: graphs and gradients, limits, energy, series and parallel, period.
• Extension: Young modulus, hysteresis loop areas, the mass of the spring and the damped oscillator.

Some courses treat springs only as a static Hooke's law topic, while others require full simple harmonic motion with derivations. Check which of the equations in section C you must memorise and which are given.

Stress-strain curves and the yield point

The stress-strain curve of a ductile metal has a linear part, a yield point, a region of plastic flow and work hardening, a maximum stress (the ultimate tensile strength), and then necking and fracture. The Young modulus is the gradient of the linear part. The yield stress marks the start of plastic deformation and is close to what this page calls the elastic limit. A brittle material, such as glass, breaks with almost no plastic region. A ductile one, such as copper, can be drawn into a wire. The area under the whole curve is the energy absorbed per unit volume before fracture, called the toughness.

Strain energy density

The elastic energy stored per unit volume of a stretched wire is ½ × stress × strain. Check this: the energy is ½Fe, the volume is AL, and (½Fe)/(AL) = ½(F/A)(e/L). This quantity lets you compare materials of different shapes. A spring steel can store a large energy per unit mass, which is why it is used for springs.

The torsion spring and the balance wheel

A coiled spring that is twisted instead of stretched exerts a restoring torque τ = −κθ. Newton's second law for rotation gives I d²θ/dt² = −κθ, so the period is T = 2π√(I/κ). It has the same structure as the mass-spring formula, with the moment of inertia I in place of m and the torsion constant κ in place of k.

Damped and driven oscillators

With a damping force proportional to velocity, −bv, the equation is m d²x/dt² + b dx/dt + kx = 0. For light damping the solution is x = A e^−γt cos ω_dt with γ = b/2m and ω_d = √(k/m − γ²). The amplitude decays exponentially and the period is slightly longer than the undamped one. The simulation uses this equation: with light damping you can watch the amplitude shrink while the period barely changes. When b is large enough that γ ≥ √(k/m), the motion no longer oscillates (critical damping). If a periodic force drives the system near its natural frequency ω₀ = √(k/m), the amplitude can become very large: this is resonance.

The mass of the spring, revisited

The factor of one third arises because the kinetic energy of a spring element depends on the square of its velocity, and the velocity grows linearly from zero at the support to v at the mass. Integrating ½(m_s/L)(v x/L)² dx from 0 to L gives ½(m_s/3)v². So the spring behaves as if one third of its mass moved with the load. The result is only approximate when the spring mass is a large fraction of the load.

Questions to explore

• Two identical springs carry the same mass, once in series and once in parallel. Show that the period in series is twice the period in parallel, then check it in the simulation.
• Find how the period changes when the oscillation crosses the limit of proportionality in the simulation.
• Estimate the energy dissipated per cycle of a rubber band from a loading and unloading graph.

• Extension = stretched length − natural length.
• Hooke's law: F = ke, with k in N/m, valid up to the limit of proportionality.
• A hanging mass at rest: ke = mg, so e = mg/k.
• The gradient of a force-extension graph is k. The area under it is the work done on the spring.
• The limit of proportionality is where the graph stops being straight. The elastic limit is where recovery stops being complete. The two can differ.
• Beyond the elastic limit there is a permanent set. Unloading follows a line parallel to the original line. The area between loading and unloading curves is energy dissipated as heat.
• Series: 1/k = 1/k₁ + 1/k₂ (softer). Parallel: k = k₁ + k₂ (stiffer).
• Elastic energy: E = ½Fe = ½ke². It goes as the square of the extension.
• Period of a mass on a spring: T = 2π√(m/k). It does not depend on g, or on the amplitude for a linear spring.
• T² against m is a straight line with gradient 4π²/k. A positive intercept points to the mass of the spring: T² = (4π²/k)(m + m_s/3).
• Young modulus E = stress/strain = FL/(Ae). A wire has k = EA/L.
• Time 10 oscillations, record readings to a consistent precision, and quote uncertainties from the gradient with steepest and shallowest lines.

Amplitude

The greatest displacement of an oscillating object from its equilibrium position.

Breaking point

The force or stress at which a sample fractures.

Damping

A force that removes energy from an oscillation and makes its amplitude fall.

Elastic limit

The largest force from which a sample returns to its original length when unloaded.

Elastic potential energy

Energy stored in a stretched or compressed elastic object.

Equilibrium position

The position at which the net force on an object is zero.

Extension

The increase in length of a spring or wire from its natural length.

Frequency

The number of oscillations per second, in hertz (Hz). It equals 1/T.

Hooke's law

Within the limit of proportionality, extension is proportional to the force producing it.

Hysteresis

The difference between the loading and unloading curves. The area between them is the energy dissipated.

Limit of proportionality

The largest force for which force and extension remain directly proportional.

Natural length

The length of a spring when no force acts on it.

Parallax

An apparent shift in the position of an object read against a scale, caused by viewing from the wrong angle.

Period

The time for one complete oscillation, in seconds.

Permanent set

The extension that remains when a sample that has been loaded beyond its elastic limit is unloaded.

Plastic deformation

A deformation that remains after the force is removed.

Restoring force

A force that acts to bring an object back to its equilibrium position.

Simple harmonic motion

Oscillation in which the restoring force is proportional to the displacement and directed towards equilibrium.

Spring constant

The force per unit extension, k = F/e, in N/m. It measures the stiffness of a spring.

Strain

Extension divided by original length. It has no unit.

Stress

Force divided by cross-sectional area, in pascals.

Yield point

The stress at which a material begins to deform plastically.

Young modulus

Stress divided by strain in the proportional region, in pascals. It is a property of a material, not of a sample.

These are well-known texts and organisations that cover the topic. Titles and publisher home pages are given for orientation. This laboratory is an independent original resource and is not endorsed by any of them.

• Fundamentals of Physics, by Halliday, Resnick and Walker. Publisher: Wiley (wiley.com). Chapters on energy, oscillations and elasticity.
• University Physics, by Young and Freedman. Publisher: Pearson (pearson.com). Chapters on Hooke's law, elasticity and periodic motion.
• Physics for Scientists and Engineers, by Serway and Jewett. Publisher: Cengage (cengage.com). Chapters on spring force, potential energy and oscillations.
• The Feynman Lectures on Physics, by Feynman, Leighton and Sands. Publisher: Basic Books (basicbooks.com). An accessible and deep account of oscillators and elasticity.
• Guide to the Expression of Uncertainty in Measurement (GUM). Joint Committee for Guides in Metrology, hosted by the International Bureau of Weights and Measures (bipm.org). The reference on measurement uncertainty.
• The Institute of Physics (iop.org). Teaching and learning resources and guidance on practical physics.
• Your examination board's own specification, data booklet and practical handbook. They are the final word on what is assessed in your course.