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

Stress and Strain

Young’s Modulus

Young’s modulus $E$ is a measure of the stiffness of a material. It is defined as the ratio of normal stress $\sigma$ to normal strain $\varepsilon$ in the linear elastic region:

$$E = \frac{\sigma}{\varepsilon}$$

where

• $\sigma = \dfrac{F}{A}$ is the applied force $F$ divided by the cross‑sectional area $A$ of the wire.
• $\varepsilon = \dfrac{\Delta L}{L_0}$ is the extension $\Delta L$ divided by the original length $L_0$ of the wire.

Objective

To describe an experiment that determines the Young’s modulus of a metal wire using the static loading method.

Apparatus

• Metal wire (uniform cross‑section, length $L_0$ ≈ 1 m)
• Clamp or fixed support
• Hook or pulley system
• Set of calibrated masses (or a force sensor)
• Micrometer or vernier calliper (to measure wire diameter)
• Vernier height gauge or travelling microscope (to measure extension $\Delta L$)
• Stopwatch (optional, for checking equilibrium)

Experimental Procedure

1. Measure the diameter $d$ of the wire at several points with the micrometer and calculate the average. Compute the cross‑sectional area $A = \pi d^{2}/4$.
2. Secure one end of the wire to the fixed clamp. Attach the other end to the hook/pulley.
3. Record the initial length $L_0$ of the unloaded wire between the two supports.
4. Gradually add masses to the hook, allowing the system to come to rest after each addition.
5. For each load, read the extension $\Delta L$ from the height gauge or microscope. Record the corresponding total force $F = mg$, where $m$ is the total mass added and $g = 9.81\ \text{m s}^{-2}$.
6. Continue adding masses until the extension reaches about 0.5 %–1 % of $L_0$ (to stay within the elastic limit). Then remove the loads in reverse order, recording the extensions again to check for hysteresis.
7. Plot a graph of applied force $F$ (or stress $\sigma$) against extension $\Delta L$ (or strain $\varepsilon$). The slope of the linear region gives $E$.

Data Recording Table

Load (kg)	Force $F$ (N)	Extension $\Delta L$ (mm)	Stress $\sigma$ (Pa)	Strain $\varepsilon$ (dimensionless)
0	0	0.00	0	0
0.5	4.905	0.12
1.0	9.810	0.25
1.5	14.715	0.38

Calculations

For each measurement:

$$\sigma = \frac{F}{A}, \qquad \varepsilon = \frac{\Delta L}{L_0}$$

The Young’s modulus is obtained from the slope $m$ of the straight‑line fit to a plot of $\sigma$ versus $\varepsilon$:

$$E = m$$

Alternatively, using $F$ versus $\Delta L$:

$$E = \frac{F L_0}{A \Delta L}$$

Average the values of $E$ obtained from the different loads and calculate the standard deviation to assess experimental uncertainty.

Sources of Error and Uncertainty

• Inaccurate measurement of wire diameter → error in area $A$.
• Parallax error when reading extensions.
• Wire not perfectly vertical → additional bending stresses.
• Temperature changes affecting material properties.
• Loading beyond the elastic limit causing permanent deformation.

Suggested Diagram