# 84-Q1: Moment of Inertia of a Bar¶

Time: $$1 \frac{1}{2}$$ hr.

## Apparatus¶

Bifilar pendulum (Wooden bar or metre rule with holes $$0.5\text{cm}$$ from ends); 2 threads ($$\sim$$ $$1.2\text{m}$$ long); 2 $$\times$$ $$100\text{g}$$ masses (flat bottom to strap to bar); 2 elastic bands; metre rule; stopclock; 2 clamps & stands; 2 G-clamps; piece of chalk; 1 sheet graph paper; (optional: spirit level). The aim of this experiment is to determine the moment of inertia, $$I$$, of a wooden bar acting as a bifilar pendulum. Proceed as follows:

1. Determine the centre of mass $$G$$ of the wooden bar, and draw the horizontal axis of the bar (from $$B$$, through $$G$$, to $$D$$).

2. Using the pieces of thread provided, suspend the wooden bar as shown above, such that $$L = d = CD \approx 100$$cm. Measure and record $$L$$ and $$d$$. (2 marks)

3. Make adjustments so that the bar is horizontal.

4. Measure distance $$x = 5$$cm from each side of $$G$$ to the variable positions $$P$$ and $$Q$$ as shown above.

5. Place the $$100\text{g}$$ masses at positions $$P$$ and $$Q$$ simultaneously. Set the wooden bar oscillating about a vertical axis through $$G$$. Record the time $$t$$ for 10 complete oscillations and calculate the correspondinb periodic time $$T$$. (2 marks)

6. Move the $$100\text{g}$$ masses along the wooden bar at increasing distance $$x$$ in intervals of $$5\text{cm}$$ from each side of $$G$$. At each stage, measure the time $$t$$ for 10 complete oscillations and determine the corresponding periodic time $$T$$. Tabulate your results. (10 marks)

7. Plot a graph of $$T^2$$ (vertical axis) against $$x^2$$ (horizontal axis). (marks: table 3, graph 12)

8. Given that:

$T^2 = \frac{16 \pi ^2 I L}{(M + m) g d^2} + \frac{16 \pi ^2 m L x^2}{(M + m) g d^2}$

Use your graph to determine $$I$$ and $$M$$; where $$L$$ and $$d$$ are expressed in SI units. $$g=9.81$$ ms$$^{-2}$$, $$m=0.2$$kg. What does $$M$$ represent? (marks: use of graph 5, $$I \text{ and } M$$ 10, $$M$$ represents? 2)

9. Mention any precautions that you took in performing this experiment. (4 marks)