A 1-kg mass (the blue mass) is connected to a 8-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.
The lever arm from the pivot axis to the 1 kg mass is 40.2 cm, and to the 8 kg mass is 26.8 cm. The moment of inertia about the pivot axis is
I = 1*(0.402)^2 + 8*(0.268)^2
= 0.736 kg/m^2
You will also need to know the distance L from the center of mass to the pivot axis. In this case, it is 17.23 cm = 0.1723 m. (The center of mass is 9.57 cm from the 8 kg mass).
The equation you need to use for the period can be found at
http://hyperphysics.phy-astr.gsu.edu/hbase/pendp.html
how did you figure out distance from the center of mass?
Require that the moment about the center of mass be zero, and solve for the location.
if the mass is 9 kg instead of 8, would the L be 21.9
To calculate the period of oscillation for the masses, we can use the formula for the period of a simple pendulum. However, in this case, the system is not a simple pendulum but a compound pendulum due to the addition of the 8-kg mass.
To determine the period of oscillation, we need to calculate the effective length of the compound pendulum. The effective length is the distance from the pivot point to the center of mass of the compound pendulum system.
Here's how we can calculate the effective length:
1. First, calculate the center of mass position of the compound pendulum. Since the 1-kg mass is 40.2 cm away from the pivot point, and the 8-kg mass is 67 cm away, we can use the formula:
Center of mass position = (m1 * r1 + m2 * r2) / (m1 + m2)
where m1 and m2 are the masses and r1 and r2 are the distances from the masses to the pivot point.
Center of mass position = (1 kg * 40.2 cm + 8 kg * 67 cm) / (1 kg + 8 kg)
2. Once we have the center of mass position, we can calculate the effective length of the compound pendulum by subtracting it from the total length of the rod:
Effective length = Total length - Center of mass position
Effective length = 67 cm - (center of mass position calculated in step 1)
3. Finally, we can use the formula for the period of a simple pendulum to calculate the period of oscillation:
Period = 2π * √(length / g)
where length is the effective length of the compound pendulum and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Convert the effective length from cm to meters before plugging it into the formula.
Period = 2π * √(effective length in meters / 9.8 m/s^2)
Now, plug in the values into the formula and calculate the period.