I. Adjust the meter rule and support:

Here are the steps for the experiment and the answers to your questions:

i. Adjust the meter rule and support:
Set up two retort stands with clamps. Suspend the meter rule horizontally using two loops of thread. Ensure the threads are vertical. For the initial setup, attach one thread loop at the $50.0 \text{ cm}$ mark and the other at the $95.0 \text{ cm}$ mark of the meter rule, as shown in the diagram. Adjust the height of the clamps so the meter rule is horizontal and can oscillate freely.

ii. Oscillate the end of the rule:
Gently twist one end of the meter rule through a small angle in the horizontal plane and release it. This will cause the rule to oscillate (twist) about its vertical axis. Ensure the oscillations are small and remain in the horizontal plane.

iii. Determine the time, $t$, for 20 oscillations:
Using a stopwatch, start timing when the rule passes a specific point in its oscillation (e.g., its maximum displacement or equilibrium position) and count "zero". Count 20 complete oscillations, stopping the stopwatch precisely when the rule passes the same point for the 20th time. Record this time $t$.

iv. Calculate the period $T$ and record the distance $L$:
The period $T$ of one oscillation is calculated by dividing the total time $t$ by the number of oscillations (20).

$$T = \frac{t}{20}$$

The distance $L$ between the loops is the difference between the two marks where the threads are attached.
For the initial setup:

$$L = 95.0 \text{ cm} - 50.0 \text{ cm} = \boxed{45.0 \text{ cm}}$$

v. Repeat the experiment with the loops at the specified marks:
Repeat steps (ii) and (iii) for the following pairs of thread positions:

• $10.0 \text{ cm}$ and $90.0 \text{ cm}$
• $15.0 \text{ cm}$ and $85.0 \text{ cm}$
• $20.0 \text{ cm}$ and $80.0 \text{ cm}$
• $25.0 \text{ cm}$ and $75.0 \text{ cm}$

vi. Determine $T$ and record $L$ in each case:
For each pair of positions from step (v), calculate the distance $L$ and the period $T$ as described in step (iv). You will create a table of values for $L$, $t$, $T$, and $\frac{1}{T}$.

Example data table structure:

Loop positions (cm)	$L$ (cm)	$t$ for 20 oscillations (s)	$T = t/20$ (s)	$\frac{1}{T}$ ($s^{-1}$)
50.0 and 95.0	45.0	(measured value)	(calculated)	(calculated)
10.0 and 90.0	80.0	(measured value)	(calculated)	(calculated)
15.0 and 85.0	70.0	(measured value)	(calculated)	(calculated)
20.0 and 80.0	60.0	(measured value)	(calculated)	(calculated)
25.0 and 75.0	50.0	(measured value)	(calculated)	(calculated)

vii. Plot a graph with $\frac{1}{T}/s^{-1}$ on the vertical axis and $L/cm$ on the horizontal axis:
Using the calculated values from your table:

• Draw a horizontal axis (x-axis) and label it $L/\text{cm}$.
• Draw a vertical axis (y-axis) and label it $\frac{1}{T}/s^{-1}$.
• Choose appropriate scales for both axes to utilize most of the graph paper.
• Plot each data point $(\text{L, } \frac{1}{T})$.
• Draw the best-fit straight line through the plotted points. The line should ideally pass through the origin $(0,0)$ or very close to it.

viii. Deduce from your graph the value of $L$ when $T = 1.5s$:
First, calculate the corresponding value for $\frac{1}{T}$:

$$\frac{1}{T} = \frac{1}{1.5 \text{ s}} = 0.67 \text{ s}^{-1}$$

On your graph, locate $0.67 \text{ s}^{-1}$ on the vertical axis. Draw a horizontal line from this point to intersect your best-fit line. From the intersection point, draw a vertical line down to the horizontal axis. Read the value of $L$ at this point. This will be your deduced value of $L$.

ix. State two precautions you took to obtain accurate results:

• Ensure small angular displacement: The rule should be twisted through a small angle to ensure simple harmonic motion and avoid wobbling.
• Avoid parallax error: When reading the meter rule marks and the stopwatch, ensure your eye is perpendicular to the scale to avoid misreading.
• Repeat readings: Take multiple readings for the time $t$ for each $L$ and calculate the average to minimize random errors.
• Start/stop timing accurately: Start and stop the stopwatch when the rule passes a clearly defined reference point in its oscillation.