Tensile Method to Measure the Elastic Modulus of Metal (2) Experimental Principle
As shown in equation (1), Young’s modulus can be calculated by measuring force (F), cross-sectional area (S), original length (L), and elongation (ΔL). The force F is determined by the added weights, while the cross-sectional area S is 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 requires precise measurement. To achieve this, an optical lever is used to magnify the tiny elongation for accurate measurement.
The optical lever system is designed to measure the small elongation ΔL. One end of the wire is fixed, and the other end is clamped in a small cylinder that slides freely on a platform. A weight is attached to the cylinder to apply tension. The optical lever has a mirror that reflects light from a scale, allowing the movement of the cylinder to be observed through a telescope. When the wire stretches, the cylinder moves down, causing the mirror to rotate slightly. This rotation changes the position of the reflected image on the scale, which can be measured with high precision.
The relationship between the elongation ΔL and the measured displacement n on the scale is derived geometrically. For small angles, the formula simplifies to:
$$
\Delta L = \frac{n \cdot a}{2D}
$$
where D is the distance from the optical lever to the scale, and a is the distance between the two feet of the optical lever. By measuring these quantities, ΔL can be calculated, enabling the determination of Young’s modulus using the formula:
$$
E = \frac{F \cdot L}{S \cdot \Delta L}
$$
Experimental Procedure:
1. Level the small platform and ensure it is horizontal in both directions. Remove the level once it is properly adjusted.
2. Attach a 2 kg weight to the lower end of the cylinder to straighten the wire. Adjust the platform height so that the upper surface aligns with the top of the cylinder, and measure the original length L of the wire.
3. Place the optical lever on the platform, ensuring the front foot rests in the groove and the rear foot sits on the cylinder. Adjust the mirror to be vertical.
4. Position the reading telescope about 1.10–1.30 meters away from the mirror. Adjust the telescope height and focus until the crosshair aligns with the scale image. Ensure there is no parallax. Record the initial reading n₀.
5. Add seven 1 kg weights sequentially, recording the corresponding readings n₠to n₇. Then remove the weights one by one, recording the reverse readings. Calculate the average for each load. Use the difference method to determine the elongation caused by 4 kg of force.
6. Measure the distance D from the front foot of the optical lever to the scale. Use a caliper to measure the length of the optical lever’s feet (a). Measure the wire diameter d five times with a micrometer and take the average.
7. Record all values, including uncertainties. Calculate Young’s modulus E using the derived formula and compute its uncertainty. Present the final result as E ± UE.
This detailed process ensures accurate measurement of the elastic modulus, highlighting the importance of precision in both experimental setup and data analysis.
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