Tensile Method to Measure the Elastic Modulus of Metal (2) Experimental Principle
As shown in equation (1), the Young’s modulus of a material can be calculated by measuring the force (F), cross-sectional area (S), original length (L), and elongation (ΔL). The force F is directly determined by the added weights. The cross-sectional area S can be calculated using the diameter (d) of the wire, measured with a micrometer. The original length L is measured with a meter ruler, but ΔL is very small and cannot be accurately measured with standard tools. To overcome this limitation, an optical lever is used to make a precise indirect measurement of ΔL.
The basic setup for measuring the small elongation ΔL is illustrated in Figure 2. One end of the wire is fixed, while the other end is clamped in the central slot of a small cylinder. This cylinder slides freely within a circular hole on a fixed platform and has a ring at the bottom for hanging weights. A light lever is placed on the platform, with its front foot resting in a groove and the rear foot on the cylinder. A scale and telescope are positioned at a distance D from the optical lever.
Initially, the mirror of the optical lever is vertical, and the image of the scale N0 is visible through the telescope. When weight is applied, the wire stretches by ΔL, causing the rear foot of the optical lever to lower by the same amount. This movement rotates the mirror by a small angle θ, reflecting the scale image to a new position Ni. The angle between the incoming and reflected rays is 2θ. Using geometric relationships, we derive:
$$
\Delta L = \frac{D}{2a} \cdot \Delta n
$$
where Δn is the change in scale reading. Since D is much larger than a, the magnification factor becomes significant, making ΔL measurable.
Once ΔL is determined, it is substituted into the original formula for Young’s modulus:
$$
E = \frac{FL}{A \Delta L}
$$
This gives the final expression for E, which is then calculated using the measured values of F, L, A, and ΔL.
Experimental Procedure:
1. Place the optical lever on a level platform and adjust it until it is perfectly horizontal.
2. Attach a 2 kg weight to the lower end of the cylinder to straighten the wire. Adjust the platform so that the top surface aligns with the cylinder, and measure the original length L of the wire.
3. Position the optical lever on the platform, ensuring the front foot is in the groove and the rear foot rests on the cylinder. Adjust the mirror to be vertical.
4. Set up the telescope at a distance of 1.10–1.30 m in front of the mirror. Align the telescope so that the crosshair matches the scale image in the mirror. Adjust the eyepiece and focus to ensure clear visibility without parallax.
5. Record the initial reading n₀. Then add weights in increments of 1 kg seven times, recording each scale reading n₠to n₇. Remove the weights in reverse order and record the corresponding readings again. Calculate average readings for each load.
6. Use a ruler to measure the distance D from the front foot of the optical lever to the telescope. Measure the length of the optical lever's feet with a caliper. Measure the wire's diameter d with a micrometer at five different points and take the average.
7. Calculate Young’s modulus E using the derived formula (6). Determine the uncertainty in E and present the result as E ± UE.
This method ensures accurate measurement of the elastic modulus by combining mechanical loading with optical amplification, allowing for high precision in small elongation measurements.
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