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, we need to use the concept of the torsional pendulum. The period of oscillation of a torsional pendulum is given by the equation:
T = 2π * √(I / k)
where T is the period of oscillation, π is a mathematical constant (approximately 3.14159), I is the moment of inertia of the system, and k is the torsional constant.
1. **Moment of Inertia (I):** The moment of inertia of the system is calculated by summing the moments of inertia of the individual masses. The moment of inertia (I) for a point mass is given by the equation:
I = m * r^2
where m is the mass of the object and r is the distance of the object from the axis of rotation (in this case, the pivot point P).
For the 1-kg mass: I1 = m1 * r1^2
For the 9-kg mass: I2 = m2 * r2^2
In our case, the 1-kg mass is located 40.2 cm away from the pivot point (P), and the 9-kg mass is located 67 cm away from the pivot point (P).
I1 = (1 kg) * (0.402 m)^2
I2 = (9 kg) * (0.67 m)^2
2. **Torsional Constant (k):** The torsional constant (k) represents the stiffness of the system, and it is dependent on the physical properties of the connecting rod. Unfortunately, without additional information, we cannot determine the torsional constant. In this case, we will assume that the torsional constant is given.
3. **Calculate the Period (T):** Now we have all the required values to calculate the period of oscillation (T) using the first equation mentioned earlier:
T = 2π * √(I / k)
Plug in the calculated values of I1, I2, and k, and calculate T.
Note: The unit for r in the calculation of I should be in meters for consistency with the units of other quantities. So, convert 40.2 cm and 67 cm to meters before using them in the equation.
I hope this explanation helps you understand how to calculate the period of oscillation for the given masses in this torsional pendulum system.