A 1-kg mass (the blue mass) is connected to a 9-kg mass (the green mass) by a massless rod 67 cm long, as shown in the figure. A hole is then drilled in the rod 40.2 cm away from the 1-kg mass, and the rod and masses are free to rotate about this pivot point, P. Calculate the period of oscillation for the masses if they are displaced slightly from the stable equilibrium position

To calculate the period of oscillation for the masses when they are displaced slightly from the stable equilibrium position, we need to use the principles of simple harmonic motion.

Step 1: Find the effective length of the pendulum.
The effective length of the pendulum is the distance between the pivot point (P) and the center of mass of the system for small angular displacements. In this case, the center of mass is at the midpoint between the 1-kg and 9-kg masses.

The total length of the rod is 67 cm. Since the hole is drilled 40.2 cm away from the 1-kg mass, the effective length can be calculated as:
Effective Length = 67 cm - 40.2 cm = 26.8 cm.

Step 2: Find the equivalent spring constant (k) of the system.
For a simple pendulum, the equivalent spring constant is given by:
k = m * g / L,
where m is the reduced mass of the system, g is the acceleration due to gravity, and L is the effective length.

The reduced mass (m) is given by:
m = (m1 * m2) / (m1 + m2),
where m1 and m2 are the masses of the 1-kg and 9-kg masses, respectively.

m1 = 1 kg
m2 = 9 kg
m = (1 kg * 9 kg) / (1 kg + 9 kg) = 0.9 kg.

Now, substitute the values of m, g, and L into the formula to find the equivalent spring constant (k).

k = (0.9 kg * 9.8 m/s^2) / (26.8 cm) = 0.325 N/m.

Step 3: Find the period of oscillation.
The period of oscillation (T) for a simple harmonic motion can be calculated using the formula:
T = 2π * √(m / k),
where m is the reduced mass and k is the equivalent spring constant.

Substitute the values of m and k into the formula to find the period of oscillation.

T = 2π * √(0.9 kg / 0.325 N/m) ≈ 3.64 s.

Therefore, the period of oscillation for the masses when they are displaced slightly from the stable equilibrium position is approximately 3.64 seconds.