Question

In the determination of Young's modulus \(\left(Y=\frac{4 M L \mathrm{~g}}{\pi l d^{2}}\right)\) by using Searle's method, a wire of length \(L=2 \mathrm{~m}\) and diameter \(d=0.5 \mathrm{~mm}\) is used. For a load \(M=2.5 \mathrm{~kg}\), an extension \(l=0.25 \mathrm{~mm}\) in the length of the wire is observed. Quantities \(d\) and \(l\) are measured using a screw gauge and a micrometer, respectively. They have the same pitch of \(0.5 \mathrm{~mm}\). The number of divisions on their circular scale is 100 . The contributions to the maximum probable error of the \(Y\) measurement (A) due to the errors in the measurements of \(d\) and \(l\) are the same. (B) due to the error in the measurement of \(d\) is twice that due to the error in the measurement of \(l\). (C) due to the error in the measurement of \(l\) is twice that due to the error in the measurement of \(d\). (D) due to the error in the measurement of \(d\) is four times that due to the error in the measurement of \(l\).

Short answer

The answer is (B). The error in the measurement of diameter (d) is twice that due to the error in the measurement of length (l) because the diameter is squared in the formula for Young's modulus.

Step-by-step solution

Determine the Least Count (L.C.) of the Instruments

The least count of a measuring instrument, is the smallest value that can be measured with the instrument. For both the screw gauge and the micrometer, the least count can be calculated using the formula L.C. = Pitch / Number of divisions on the circular scale. In this case, L.C. = 0.5 mm / 100 = 0.005 mm.

Calculate the Maximum Probable Errors (M.P.E.) in Measuring d and l

The maximum probable error in a measurement is generally considered to be half of the least count of the measuring instrument. Therefore, M.P.E. for d and l can be calculated as: M.P.E. for d = M.P.E. for l = L.C. / 2 = 0.005 mm / 2 = 0.0025 mm.

Calculate the Relative Errors in d and l

The relative error is the absolute error divided by the magnitude of the measurement it refers to. For the diameter d: Relative error in d = M.P.E. in d / d. For the length l: Relative error in l = M.P.E. in l / l.

Find the Contribution of Errors in d and l to the Error in Y

Since Y is calculated using both d and l, their errors contribute to the total error in Y. The contribution to the error from d is proportional to the relative error in d, which is multiplied by 2 since the formula for Y has d squared. Similarly, the contribution to the error from l is proportional to the relative error in l.

Compare the Contributions of Errors in d and l

To determine which contributes more to the error in Y, compare the expressions from the previous step. The relative error in d would be multiplied by 2 because d appears as d squared in the formula for Y. Therefore, the error in d contributes twice as much to the error in Y as the error in l.

Key concepts

Searle's Method

Understanding Searle's method is critical for determining the Young's modulus—the measure of the stiffness of an elastic material. This technique is widely used in physics labs to measure the modulus of elasticity of a wire under tension.

The core of Searle's method lies in suspending a wire and adding weights that cause the wire to elongate. By precisely measuring the length and diameter of the wire, as well as the extension caused by different loads, students can calculate Young's modulus using the formula: \(Y = \frac{4MLg}{\pi l d^{2}}\). It's essential that measurements are taken meticulously to ensure accuracy.

To avoid parallax errors when reading scales and to gain accurate results, the wire's length, diameter, and any extensions due to loads are measured using instruments like a screw gauge or a micrometer, known for their precision. By minimizing experimental errors, Searle's method can give students a clear insight into the physical properties of materials.

Maximum Probable Error (M.P.E.)

When performing measurements, understanding the concept of Maximum Probable Error (M.P.E.) is indispensable. The M.P.E. represents a statistical estimate of the upper limit on the error within your measurements. Calculating the M.P.E. helps determine the reliability and accuracy of the experimental results.

For instance, in the context of Searle's method, the pitch of both the screw gauge and micrometer is given, allowing us to find the least count (L.C.). Since the least count of these instruments is 0.005 mm, the M.P.E. would be? \(0.005 \, mm / 2 = 0.0025 \, mm\), because it's generally accepted that the M.P.E. is half the least count. Hence, understanding this concept is key to interpreting the measurements and ensuring that the Young's modulus determined reflects the true nature of the material being tested.

Relative Error Calculation

Relative error is an important measure in experimental physics, providing perspective on how significant the errors are in relation to the actual measurements taken. This comparison is essential when evaluating the precision of an experiment, like when using Searle's method to determine Young's modulus.

To calculate the relative error, simply divide the M.P.E. by the actual measurement taken. For example, with a diameter of 0.5 mm and an M.P.E. of 0.0025 mm, the relative error in measuring the diameter is: \(\frac{0.0025}{0.5} = 0.005\). Similarly, the relative error for the extension can be calculated. These calculations give us insight into the reliability of the measurements and thus the final result for Young's modulus.

It's particularly important to note how the relative error in the diameter is squared in the Young's modulus formula, which can amplify its impact. Hence, assessing both the M.P.E. and the relative error is instrumental in addressing the precision and accuracy of the results obtained using Searle's method.