Using Searle's apparatus while determining the Young's modulus of a wire, a girl is taking measurements of different parameters : Length of the wire, L, is 30 cm, which is measured with a millimeter scale , diameter of the wire, d, is 1 mm, which is measured with a screw gauge. (Least count of the screw gause is 0.01 mm, least count of the spherometer attached to the apparatus frame is 0.005 mm, spherometer attached to the apparatus frame is 0.005 mm , spherometer measured the extension e of the wire , W is the weight of the loads.)
The parameter that should be measured more carefully is

Options

A. load on the hanger
B. length of the wire
C. diameter of the wire
D. extension of the wire

Answer

To determine which parameter should be measured more carefully while using Searle's apparatus to find the Young's modulus of a wire, we will analyze the parameters given in the question:

1. **Length of the wire (L)**: 30 cm, measured with a millimeter scale.
2. **Diameter of the wire (d)**: 1 mm, measured with a screw gauge (least count = 0.01 mm).
3. **Extension of the wire (e)**: measured with a spherometer (least count = 0.005 mm).
4. **Weight of the loads (W)**: the load applied to the wire.

### Step-by-Step Solution:

**Step 1: Understand the significance of each parameter.**
- The Young's modulus (Y) is defined as the ratio of stress to strain.
- Stress is calculated using the formula: \( \text{Stress} = \frac{F}{A} \), where \( F \) is the force (weight of the load) and \( A \) is the cross-sectional area of the wire.
- Strain is calculated using the formula: \( \text{Strain} = \frac{e}{L} \), where \( e \) is the extension and \( L \) is the original length of the wire.

**Step 2: Evaluate the precision of measurements.**
- The length of the wire (L) is crucial because it directly affects the calculation of strain.
- The diameter (d) is important for calculating the cross-sectional area (A = \( \frac{\pi d^2}{4} \)), which affects the stress calculation.
- The extension (e) is also significant as it contributes to the strain calculation.
- The weight (W) is important but typically has less variability compared to the measurements of length, diameter, and extension.

**Step 3: Analyze the least counts.**
- The least count of the screw gauge (for diameter) is 0.01 mm.
- The least count of the spherometer (for extension) is 0.005 mm.
- The millimeter scale used for measuring length may have a least count of 1 mm.

**Step 4: Determine which parameter requires the most careful measurement.**
- Since the least count of the millimeter scale is larger than the least counts of both the screw gauge and the spherometer, the measurement of length (L) is less precise.
- Given that strain is a ratio of extension to length, any error in measuring length will significantly affect the calculated Young's modulus.

### Conclusion:
The parameter that should be measured more carefully is the **Length of the wire (L)**.