To solve the problem using the provided derivation in the first image for the period of the pendulum in the small angle approximation, let's summarize and apply the steps accordingly:

Given data:

• $M_1 = M_2 = 9 \, \text{kg}$
• Length of the rod $L = 7.7 \, \text{m}$
• Pivoted rod with point masses at positions as shown
• Gravitational acceleration $g = 9.8 \, \text{m/s}^2$

Moment of Inertia Calculation:

The total moment of inertia $I$ is calculated by: $I = I_{M_1} + I_{M_2}$ For $M_1$ at a distance $L$ and $M_2$ at a distance $\frac{1}{3}L$ from the pivot: $I = M_1 L^2 + M_2 \left(\frac{L}{3}\right)^2$ Substituting values: $I = 9 \times (7.7)^2 + 9 \times \left(\frac{7.7}{3}\right)^2$ Calculating: $I = 9 \times 59.29 + 9 \times 6.587$ $I = 533.61 + 59.283$ $I \approx 592.893 \, \text{kg} \cdot \text{m}^2$

Center of Mass Distance Calculation:

Given in the image as: $r_{cm} = \frac{1}{2}L \left( \frac{14}{9} \right)$ Substituting $L = 7.7 \, \text{m}$: $r_{cm} = \frac{1}{2} \times 7.7 \times \frac{14}{9}$ $r_{cm} \approx 5.988 \, \text{m}$

Torque and Angular Acceleration:

The net torque $\tau$ about the pivot: $\tau = -2Mg r_{cm} \sin \theta \approx -2 \times 9 \times 9.8 \times 5.988 \times \theta$ Calculating: $\tau \approx -1056.4416 \, \theta \, \text{N} \cdot \text{m}$ Using the relationship for angular acceleration: $\alpha = \frac{\tau}{I} = \frac{-1056.4416 \, \theta}{592.893}$ $\alpha \approx -1.782 \, \theta$

Period of the Pendulum:

For small oscillations, the period is given by: $T = 2\pi \sqrt{\frac{I}{\left(2Mg r_{cm}\right)}}$ Substituting the calculated values: $T = 2\pi \sqrt{\frac{592.893}{2 \times 9 \times 9.8 \times 5.988}}$ Calculating: $T = 2\pi \sqrt{\frac{592.893}{1056.4416}}$ $T = 2\pi \sqrt{0.561}$ $T \approx 2\pi \times 0.749$ $T \approx 4.706 \, \text{s}$

Final Result:

The period of the pendulum is approximately $T \approx 4.706 \, \text{s}$.

Would you like any further details or explanations?

Here are 5 related questions for further exploration:

1. How does the moment of inertia change if the masses were different?
2. What effect would a different length $L$ have on the period?
3. Can we explore the energy conservation approach for this system?
4. How does changing the pivot point impact the motion?
5. What approximations are made in the small angle assumption, and when do they break down?

Tip: Remember, for small-angle oscillations, the restoring torque is approximately proportional to the angular displacement, simplifying the dynamics to harmonic motion.