Question

A student performs an experiment to determine the Young's modulus of a wire, exactly \(2 \mathrm{~m}\) long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be \(0.8 \mathrm{~mm}\) with an uncertainty of \(\pm 0.05 \mathrm{~mm}\) at a load of exactly \(1.0 \mathrm{~kg}\). The student also measures the diameter of the wire to be \(0.4 \mathrm{~mm}\) with an uncertainty of \(\pm 0.01 \mathrm{~mm}\). Take \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) (exact). The Young's modulus obtained from the reading is (A) \((2.0 \pm 0.3) \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\) (B) \((2.0 \pm 0.2) \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\) (C) \((2.0 \pm 0.1) \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\) (D) \((2.0 \pm 0.05) \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\)

Short answer

The Young's modulus obtained from the reading is (C) \((2.0 \pm 0.1) \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\).

Step-by-step solution

Understanding Young's Modulus and Searle's Method

Young's modulus (E) is a measure of the stiffness of a material and is given by the formula: \(E = \frac{FL}{A\Delta L}\), where F is the force applied, L is the original length of the wire, A is the cross-sectional area, and \(\Delta L\) is the extension in the length of the wire. Searle's method involves applying a known force to a wire and measuring the extension caused by the force. The original length and diameter of the wire are also measured to calculate the cross-sectional area.

Calculate the Cross-sectional Area of the Wire

The diameter of the wire is 0.4 mm, which needs to be converted to meters: \(0.4 mm = 4 \times 10^{-4}m\). The radius r is half the diameter: \(r = 2 \times 10^{-4}m\). Using the formula for the area of a circle, \(A=\pi r^2\), calculate the cross-sectional area: \(A = \pi (2 \times 10^{-4}m)^2\).

Calculate the Force Acting on the Wire

The force F applied is due to the weight of the mass hung on the wire. Using the gravitational force equation \(F = mg\), with m being the mass and g being the acceleration due to gravity. With m = 1.0 kg and g = 9.8 m/s^2, calculate the force: \(F = 1.0 kg \times 9.8 m/s^2 = 9.8 N\).

Calculate the Change in Length of the Wire

The change in length \(\Delta L\) is measured to be 0.8 mm, which should be converted to meters: \(\Delta L = 0.8 mm = 8 \times 10^{-4} m\).

Calculate Young's Modulus

Using the Young's modulus formula from Step 1 and the values from Steps 2, 3, and 4. *Note: Ensure all units are consistent (preferably meters, Newtons, and square meters).

Calculate Uncertainty in Young's Modulus

Uncertainty in Young's modulus will be influenced by the uncertainties in measurements of the diameter and extension. Use propagation of uncertainty methods accounting for partial derivatives with respect to each variable's uncertainty. Calculate uncertainty percentage for both \(\Delta A\) from the diameter's uncertainty and \(\Delta L\) from the length's uncertainty, and then use them to find the overall percentage uncertainty in E.

Present the Final Young's Modulus Value with Uncertainty

Combine the calculated Young's modulus with the overall percentage uncertainty obtained in Step 6 to express the final answer in the form of \(E \pm \text{uncertainty}\). Choose the answer that matches one of the provided options.

Key concepts

Searle's Method

Searle's method serves as an essential experimental technique for determining the Young's modulus of a material. It involves stretching a wire or a material under a known load and accurately measuring the resulting extension. By understanding the relationship between the applied force and the amount of stretch observed, one can ascertain the material's stiffness or rigidity, which is quantified by Young's modulus.

Typically, the experiment uses a wire of known dimensions, where the length and diameter are precisely measured. A load is then carefully added to the wire, inducing an extension that is meticulously recorded. The original length of the wire, the diameter (to find the cross-sectional area), and the change in length are the critical parameters in Searle's method, each requiring high precision to ensure the reliability of the results.

This experimental setup highlights the fundamental properties of materials, particularly how they deform under mechanical stress, which is invaluable in fields such as construction, materials science, and engineering.

Uncertainty in Measurements

When performing precise experiments, such as determining Young's modulus through Searle's method, accounting for uncertainty in measurements is crucial. Measurement uncertainty can arise from a variety of sources, including instrument precision, observer error, or environmental factors.

In the context of the Young's modulus experiment, the length extension and the diameter of the wire need to be measured with accuracy and precision. The stated uncertainties, ewenvironment{itemize} * ewenvironment{itemize}item for extension in the length of the wire (ewenvironment{improvement} ewenvironment{improvement}) and item for the diameter of the wire (ewenvironment{improvement} ewenvironment{improvement}), must be factored into the final calculation to express the confidence in the obtained results. These uncertainties can be seen as the data's 'error margins' that reflect the potential variability in the measurements and must be communicated alongside the final value as a standard scientific practice.

Young's Modulus Calculation

Calculating Young's modulus requires an understanding of the physical properties being measured and how they are mathematically related. According to the formula for Young's modulus ewenvironment{itemize} ewenvironment{itemize}, the determination of this modulus depends on the force applied to the wire (ewenvironment{improvement} ewenvironment{improvement}), the original length (ewenvironment{improvement} ewenvironment{improvement}), the wire's cross-sectional area (ewenvironment{improvement} ewenvironment{improvement}), and the extension caused by the force (ewenvironment{improvement} ewenvironment{improvement}).

To convert the diameter and extension from millimeters to meters, we use the appropriate conversion factors, considering that the final results should be presented in SI units. After calculating the cross-sectional area from the diameter and substituting all known values into the Young's modulus formula, the modulus can be computed, giving a precise quantification of the material's stiffness.

Propagation of Uncertainty

In experimental science, understanding how uncertainties affect the final result of a calculation is vital. Propagation of uncertainty refers to the effect that uncertainty in measured quantities has on the calculated results. Any measured variable with its associated uncertainty can affect the overall certainty of a derived quantity.

For a calculation like Young's modulus, where multiple measured quantities are involved (force, length, area, and extension), the propagation of their uncertainties must be taken into account to estimate the uncertainty of the modulus. This is often done using calculus, employing partial derivatives to approximate how variations in each measured variable influence the final result.

As the uncertainties in the diameter of the wire and the extension in length are compounded through the calculation, we arrive at a percentage uncertainty for the Young's modulus. This final uncertainty needs to be expressed alongside the modulus value, thus providing a range within which the true value of the modulus is expected to lie.