describe an experiment to determine the Young modulus of a metal in the form of a wire

Stress & Strain – Determining Young’s Modulus (Cambridge 9702)

1. Key Definitions (Syllabus 6.1)

• Load (F) – external force applied to the wire (N). In the lab it is produced by hanging calibrated masses: F = mg.
• Extension (ΔL) – increase in length of the wire under load (m). The opposite effect is compression.
• Stress (σ) – normal force per unit area, σ = F / A [Pa].
• Strain (ε) – normal extension per unit original length, ε = ΔL / L₀ [–] (dimensionless).
• Limit of proportionality – the greatest stress for which σ and ε remain directly proportional (the linear part of the σ‑ε curve).
• Elastic limit – the greatest stress at which the material returns completely to its original dimensions when the load is removed.
• Hooke’s law – within the linear region, σ = E ε or F = k ΔL, where k is the spring constant.
• Spring constant (k) – k = F / ΔL [N m⁻¹]. For a uniform wire
   k = (E A) / L₀ ⟹ E = k L₀ / A.
• Young’s modulus (E) – a measure of stiffness; the slope of the straight‑line portion of a σ‑ε graph (Pa).

2. Objective

To carry out the **static‑loading method** required by the Cambridge syllabus and determine the Young’s modulus of a metal wire from measurements of load, extension and cross‑sectional area.

3. Apparatus

• Metal wire (uniform diameter, length ≈ 1 m)
• Fixed clamp or support
• Hook or low‑friction pulley
• Set of calibrated masses (or a digital force sensor)
• Micrometer or vernier calliper (to measure wire diameter)
• Vernier height gauge, travelling microscope or digital displacement sensor (to read ΔL)
• Stopwatch (optional, to confirm equilibrium after each load)
• Thermometer (to record ambient temperature, ≈ 20 °C)
• Safety goggles and lab coat
• Data sheet / spreadsheet

4. Safety

• Secure the clamp firmly – a loose wire can snap and cause injury.
• Wear safety goggles at all times.
• Do not exceed the elastic limit; plastic deformation can cause sudden breakage.
• Handle masses carefully to avoid dropping them.

5. Experimental Procedure (Static Loading)

1. Measure the wire diameter. Take at least three readings at different points along the wire, record the mean d (mm), and calculate the cross‑sectional area
    A = π d² / 4 [m²].
2. Set up the apparatus. Clamp one end of the wire securely, attach the other end to the hook/pulley, and ensure the wire hangs vertically without touching any support.
3. Record the initial length. Measure the distance between the two supports (unloaded wire) – this is L₀ (m).
4. Apply loads gradually. Add masses one at a time. After each addition wait until the reading is steady for about 5 s (use the stopwatch if desired).
5. Read the extension. For each load record the extension ΔL (mm) from the height gauge or microscope. Also note the total mass m on the hook.
6. Stay within the elastic region. Stop adding masses when ΔL reaches roughly 0.5 %–1 % of L₀ (well below the limit of proportionality). This guarantees Hooke’s law is valid.
7. Hysteresis check (quantitative). After the maximum load has been recorded, remove the masses in reverse order, recording ΔL each time.
   Calculate the percentage difference between loading and unloading for each point:
    % difference = |ΔL_load – ΔL_unload| / ΔL_load × 100.
   If any difference exceeds 1 % the wire may have yielded – discard those points.
8. Verify linearity. Plot σ versus ε (or F versus ΔL) after the experiment. The points used for E must lie on a straight line; mark the limit of proportionality where the curve first deviates.

6. Data‑Recording Table

7. Worked Example (single data point)

Assume:

• Diameter d = 0.50 mm → A = π(0.00050 m)²/4 = 1.96 × 10⁻⁷ m²
• L₀ = 1.00 m
• Load m = 1.00 kg → F = 9.81 N
• Measured extension ΔL = 0.25 mm = 2.5 × 10⁻⁴ m

Calculations:

1. Stress: σ = F / A = 9.81 / 1.96 × 10⁻⁷ = 5.00 × 10⁷ Pa
2. Strain: ε = ΔL / L₀ = 2.5 × 10⁻⁴ / 1.00 = 2.5 × 10⁻⁴ (dimensionless)
3. Spring constant: k = F / ΔL = 9.81 / 2.5 × 10⁻⁴ = 3.92 × 10⁴ N m⁻¹
4. Young’s modulus (from one point): E = (F L₀) / (A ΔL) = (9.81 × 1.00) / (1.96 × 10⁻⁷ × 2.5 × 10⁻⁴) = 2.00 × 10¹¹ Pa
5. Using the graph method, the slope of the straight‑line region should give a similar value; the mean of several points improves reliability.

8. Calculations and Data Analysis

1. Convert all lengths to metres (ΔL in m, L₀ in m).
2. For each load calculate:
    σ = F / A [Pa]
    ε = ΔL / L₀ [–]
    k = F / ΔL [N m⁻¹]
3. Graphical determination of E
    Plot σ (y‑axis) against ε (x‑axis).
    Fit a straight line to the points that lie in the linear region.
    The gradient of this line = E. Record the gradient and its standard error from the fit.
4. Algebraic determination of E
    For each point, compute E_i = (F L₀) / (A ΔL).
    Calculate the mean  Ē  and the standard deviation σ_E to express experimental uncertainty.
5. Uncertainty propagation (AO2)
   • Uncertainty in diameter (Δd) → ΔA = (π d Δd) / 2.
   • Uncertainty in ΔL (ΔΔL) from the gauge reading.
   • Combined relative uncertainty in σ: (Δσ/σ) = √[(ΔF/F)² + (ΔA/A)²].
   • Combined relative uncertainty in ε: (Δε/ε) = √[(ΔΔL/ΔL)² + (ΔL₀/L₀)²].
   • Since E = σ/ε, the relative uncertainty in E is the sum of the two relative uncertainties.
6. Report the final result as E = (Ē ± ΔE) Pa, where ΔE includes the propagated random error and, if required, a systematic component (e.g., temperature).

9. Sources of Error & Evaluation (AO3)

After the experiment, evaluate the data by:

• Comparing the graphical and algebraic values of E.
• Discussing whether the % hysteresis criterion was met.
• Identifying the dominant source of uncertainty and suggesting a concrete improvement for a repeat run.

10. Connection to Syllabus Topics 6.2 (Elastic & Plastic Behaviour)

• The straight‑line portion of the σ‑ε graph represents **elastic behaviour** (Hooke’s law). Mark this region clearly.
• The point where the curve first deviates from the straight line is the **limit of proportionality** – students should label it on the graph.
• If the load is increased further, the curve bends more sharply; the stress at which the material ceases to return to its original length on unloading is the **elastic limit**.
• Beyond the elastic limit, permanent (plastic) deformation occurs; the experiment deliberately avoids this region, but a brief discussion of what would be observed if it were exceeded reinforces the concept.

11. Suggested Diagram