1. **State the problem:** We analyze the compound pendulum experiment to find the acceleration due to gravity $g$ and radius of gyration $k$ from the graph of $h^2$ against $hT^2$. 2. **Recall the key equation:** From the experiment, equation (7) is $$ghT^2 = 4\pi^2 k^2 + 4\pi^2 h^2$$ 3. **Rewrite equation (7) as a linear relation:** Divide both sides by $h$ (assuming $h \neq 0$): $$gT^2 = \frac{4\pi^2 k^2}{h} + 4\pi^2 h$$ But to plot $h^2$ against $hT^2$, rearrange (7) as: $$hT^2 = \frac{4\pi^2 k^2}{g} + \frac{4\pi^2}{g} h^2$$ This is of the form: $$y = c + mx$$ where $$y = hT^2, \quad x = h^2, \quad m = \frac{4\pi^2}{g}, \quad c = \frac{4\pi^2 k^2}{g}$$ 4. **Plotting the graph:** - Plot $h^2$ (x-axis) against $hT^2$ (y-axis). - The slope $m$ and intercept $c$ can be determined from the graph. 5. **Determine $g$ and $k$ from slope and intercept:** - From slope: $$m = \frac{4\pi^2}{g} \implies g = \frac{4\pi^2}{m}$$ - From intercept: $$c = \frac{4\pi^2 k^2}{g} \implies k^2 = \frac{c g}{4\pi^2} \implies k = \sqrt{\frac{c g}{4\pi^2}}$$ 6. **Physical meaning of $k$:** - $k$ is the radius of gyration of the pendulum about the pivot point. - It represents the distribution of the pendulum's mass relative to the pivot. 7. **Estimate error in $g$:** - Error can be estimated by propagating uncertainties in slope $m$. - If $\Delta m$ is the error in slope, then $$\Delta g = \left| \frac{d g}{d m} \right| \Delta m = \frac{4\pi^2}{m^2} \Delta m$$ 8. **Two other sources of error:** - Friction at the pivot point causing damping. - Air resistance affecting oscillation period. 9. **If the whole length of the ruler is used:** - $h = (C_p - C_m)$ will vary over the entire length, increasing the range of $h$ values. - The graph of $h^2$ against $hT^2$ will cover a wider range, improving accuracy. 10. **Summary table to be filled with experimental data:** | Time for 20 Oscillations | Period $T$ (s) | $C_p$ (cm) | $h = (C_p - C_m)$ (cm) | $h^2$ (cm$^2$) | $hT^2$ (cm s$^2$) | |-------------------------|----------------|------------|-----------------------|----------------|------------------| | 1 | | | | | | | 2 | | | | | | | 3 | | | | | | | 4 | | | | | | | 5 | | | | | | This completes the analysis and answers all parts of the problem.