Why this practical matters

• Hooke’s Law underpins vehicle suspension, weighing scales, force sensors and many engineering structures.
• The practical develops accurate measurement, graphing and gradient skills.
• Students evaluate proportionality, elasticity, hysteresis and the limitations of physical models.
• The investigation provides a strong context for random and systematic uncertainty.

Learning objectives

• Safely set up apparatus to investigate Hooke’s Law.
• Measure spring extension as load is increased.
• Plot force against extension and identify the limit of proportionality.
• Determine the spring constant from the graph gradient.
• Record and compare loading and unloading data.
• Distinguish the limit of proportionality from the elastic limit.
• Estimate the uncertainty in an extension calculated from two ruler readings.

Zero-load convention

Record L₀ with the spring and pointer fitted but with no mass hanger or slotted masses attached. The empty mass hanger is the first non-zero load.

Total suspended mass

Every loaded force calculation must include the mass hanger and all attached slotted masses: m = m_hanger + m_{slotted masses}.

Spring constant

When force is plotted vertically against extension horizontally, the gradient through the linear region gives k. A larger k means a stiffer spring.

Two different limits

Limit of proportionality: the graph stops being linear. Elastic limit: the spring no longer fully returns to its original position. These are not necessarily the same point.

Student equipment

• Clamp stand with a heavy, stable base.
• Bosshead and clamp to support the spring securely.
• Second bosshead and clamp, or suitable ruler clamp, to hold the metre ruler vertically.
• Helical spring inspected before use and suitable for the selected load range.
• Mass hanger of known or measured mass.
• Slotted masses in suitable equal increments.
• Metre ruler with 1 mm divisions.
• Lightweight pointer attached securely to the lower end of the spring.
• Set square, if required, to transfer the pointer position accurately to the scale.
• Bench mat or shallow tray beneath the suspended masses.
• Top-pan balance if hanger or mass values need to be verified.

Safety and variables

• Eye protection: safety goggles.
• Laboratory coat: if required by local procedures.
• Independent variable: applied force, F, calculated from total suspended mass including the hanger.
• Dependent variable: extension, x = L − L₀.
• Controlled conditions: same spring, upper fixing point, pointer arrangement, ruler position, alignment and environmental conditions.
• Standardised procedure: same loading/unloading method, same settling criterion and same eye-level reading technique.

1. Set up the apparatus

Secure the spring to the clamp stand and attach a lightweight horizontal pointer to its lower end. Fix the ruler vertically beside the pointer, close but not touching, parallel to the pointer’s motion and securely clamped. The ruler zero does not need to align with the pointer.

2. Record the no-load reading

With the spring and pointer fitted but no hanger or masses attached, allow the spring to hang freely. Read the pointer at eye level and record L₀ in millimetres. Do not move or re-zero the ruler afterwards.

3. Apply the first load

Confirm the hanger mass, then attach the empty hanger. This is the first non-zero load. Calculate the total suspended mass and F = mg using g = 9.81 N kg⁻¹. Wait for oscillations to stop, record L and calculate x = L − L₀.

4. Increase the applied force

Add slotted masses one at a time using equal increments where practical. For each load, include the hanger in total mass, convert to kilograms, calculate F, wait for the spring to settle, record L, calculate extension and convert extension to metres for graphing.

5. Stay within the safe load range

Never exceed the technician-defined maximum total suspended mass. Stop immediately if the stand moves, a hook or spring appears damaged, the hanger becomes insecure, coils touch, the apparatus rubs or the setup becomes unstable. Graph curvature is not the practical safety limit.

6. Record unloading measurements

Remove slotted masses one increment at a time. At each matching total suspended mass, wait for the spring to settle, record L and calculate extension. Record the hanger-only value before removing the hanger last, then record the final no-load pointer reading.

7. Repeat independently

Repeat the complete loading procedure independently using the same spring, mass increments, maximum load, ruler position and technique. Begin each repeat from the no-hanger condition. Re-reading the same stationary pointer is not a fully independent repeat.

After data collection

Plot force on the vertical axis against extension in metres on the horizontal axis. Use the gradient of the best-fit line through the linear region to determine k. Keep loading and unloading data separate so hysteresis is not concealed.

1. Total suspended mass

m = m_hanger + m_{slotted masses}

The hanger must be included in every non-zero loaded value.

2. Applied force

F = mg, where g = 9.81 N kg⁻¹.

For a total suspended mass of 0.200 kg: F = 0.200 × 9.81 = 1.96 N.

3. Extension

x = L − L₀

Example: L₀ = 120.0 mm and L = 165.0 mm, so x = 45.0 mm = 0.0450 m.

4. Spring constant

k = ΔF ÷ Δx from the graph gradient.

A single-point estimate using 1.96 N and 0.0450 m gives 43.6 N m⁻¹, but the graph gradient is preferred.

Extension uncertainty

With 1 mm ruler divisions, each reading may be assigned ±0.5 mm. Since x = L − L₀, a conservative extension uncertainty is ±1.0 mm.

Unit conversion

If a graph gradient is obtained in N mm⁻¹, multiply it by 1000 to convert the spring constant to N m⁻¹.

Expected graph

• A straight-line trend in the Hookean region.
• An intercept consistent with zero within experimental uncertainty.
• A systematic departure from the initial straight-line trend at the limit of proportionality.
• Curvature alone does not prove permanent deformation or identify the elastic limit.

Finding the gradient

• Draw a best-fit line through the linear region only.
• Do not automatically force the line through the origin.
• Use two widely separated coordinates on the best-fit line.
• Use a gradient triangle covering at least half the linear region.
• Use k = ΔF ÷ Δx and quote k in N m⁻¹.

Reducing uncertainty

• Use a thin pointer and read at eye level.
• Wait until oscillations and sideways swinging stop.
• Keep the ruler vertical and fixed.
• Repeat the complete loading sequence independently.
• Calculate a mean loading extension at each applied force when repeats are available.
• Choose a suitable spring and pretested safe range that produce larger measurable extensions.

A useful uncertainty point

A fixed ruler-zero offset cancels in L − L₀, provided the ruler does not move between readings.

Movement after L₀ has been recorded does not cancel and introduces a systematic error into subsequent extensions.

Common misconceptions

“Hooke’s Law applies to all extensions.”
It applies only up to the limit of proportionality.

“The force is equal to the mass added.”
Applied force is the weight of the total suspended mass, including the hanger: F = mg.

“A steeper graph means the spring stretches more easily.”
A steeper force–extension graph means a larger k and therefore a stiffer spring.

“Hooke’s Law uses total spring length.”
It uses extension, x = L − L₀.

“Spring constant depends on the mass used.”
Within the linear region and fixed conditions, k is a property of the complete spring.

High-value corrections

• Record L₀ with no hanger attached.
• Include the hanger mass in every non-zero force calculation.
• Convert grams to kilograms before using F = mg.
• Keep the ruler fixed after recording L₀.
• Convert extension to metres before the standard F–x graph or k calculation.
• Plot force, not mass, against extension.
• Do not force the best-fit line through the origin.
• Use a large gradient triangle, not one measurement pair.
• Exclude curved-region data when determining k.
• Do not assume graph curvature means plastic deformation has begun.
• Do not average loading and unloading values together.
• Do not call unloading measurements independent repeats.
• Use ±1.0 mm, not ±0.5 mm, as the conservative uncertainty in x when both ruler readings are ±0.5 mm.
• Never use graph curvature as the safety stopping rule.

Before the lesson

• Check springs are not already overstretched.
• Select a spring and load range giving clear measurable extensions below the pretested mechanical safety limit.
• If the proportional limit is to be investigated, approach it without nearing the mechanical safety limit.
• Ensure clamp stands are stable.
• Avoid springs so stiff that extensions are too small to measure accurately.
• Have spare springs available.
• Determine and label the maximum total suspended mass, including the hanger.

Technician tips

• Pretest every spring with the actual hanger and mass set.
• Choose a spring producing several centimetres of measurable extension without permanent deformation.
• Secure rulers and align them with the pointer.
• Use a lightweight pointer attached to the lower spring end.
• Check hooks are closed, the load hangs above the base or protective tray and the stand remains stable.
• Consider one conditioning loading–unloading cycle if springs show initial settling.
• Record the actual hanger mass and ensure results sheets use it.
• Confirm g = 9.81 N kg⁻¹.
• Reject springs with corrosion, open hooks or permanent extension.

Common exam questions

1. State Hooke’s Law.
2. Why plot force against extension rather than total length?
3. How is the spring constant determined from a graph?
4. What is the limit of proportionality?
5. Give one source of random uncertainty.
6. How can parallax be reduced?
7. Why might a spring fail to return to its original length after a large load?
8. Calculate k for a 2.0 N force producing 4.0 cm extension.
9. Why must the hanger mass be included?
10. If L and L₀ are each ±0.5 mm, what is the conservative uncertainty in x?
11. Explain the difference between the limit of proportionality and elastic limit.
12. Why should a best-fit line not automatically be forced through the origin?
13. What could slightly different loading and unloading extensions indicate?

Key answer points

• Force is directly proportional to extension while the limit of proportionality is not exceeded.
• The graph gradient through the linear region gives k = ΔF/Δx.
• The limit of proportionality is where force and extension cease to be directly proportional.
• Parallax is reduced with a pointer and eye-level reading.
• 2.0 N ÷ 0.040 m = 50 N m⁻¹.
• The hanger contributes weight; omitting it systematically underestimates every non-zero applied force.
• The conservative extension uncertainty is ±1.0 mm.
• The elastic limit concerns recovery; the limit of proportionality concerns linearity.
• A forced origin can bias the gradient and conceal baseline or systematic problems.
• Different loading/unloading values may indicate hysteresis; a shifted final no-load reading can indicate permanent deformation.

Stronger improvement suggestions

• Use a digital displacement sensor, video analysis with a fixed scale, cathetometer or travelling microscope for higher resolution.
• Collect more data points using smaller mass intervals near the expected proportional limit.
• Repeat the complete loading procedure independently and calculate mean loading extension at each force.
• Measure or verify hanger mass and include it in every total-mass value.
• Draw steepest and shallowest acceptable gradient lines to estimate uncertainty in k.

Thinking like a physicist

• Ask whether the evidence supports the model rather than stating the model is simply “true”.
• Use the graph to reveal the full dataset, scatter, anomalies, linear range and intercept.
• Distinguish precision from accuracy and random from systematic effects.
• Before attributing curvature to the material, rule out slack, rubbing, coil contact and apparatus movement.
• Recognise that most physical models apply only under specified conditions.

Why is force plotted against extension?

With F on the vertical axis and x on the horizontal axis, the gradient equals the spring constant, k.

Why is L₀ measured without the hanger?

The guide defines the no-hanger condition as zero load. The hanger itself has weight, so hanger-only is the first non-zero applied force.

Why use a pointer?

A thin horizontal pointer gives a clear reference against the ruler and helps reduce parallax and ambiguity.

Are unloading readings repeat readings?

No. Unloading measurements test recovery and hysteresis. Independent repeats require the load to be removed and reapplied, or the complete loading–unloading cycle to be repeated.

Why does the graph curve at high force?

The spring has passed its limit of proportionality, so force and extension are no longer directly proportional. Curvature alone does not prove permanent deformation.

Core practical equipment

• Clamp stands
• Springs
• Slotted masses
• Metre rulers

Supporting equipment

• Bossheads
• Clamps
• Safety goggles
• Bench mats

Laboratory planning

• School science equipment
• Physics equipment
• Contact Better Equipped