A 1-kg mass (the blue mass) is connected to a 9-kg mass (the green mass) by a massless rod 70 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 can use the concept of simple harmonic motion. In this case, the system consists of two masses connected by a rod, acting as a simple pendulum.
The period of a simple pendulum is given by the formula:
T = 2π√(L/g)
Where:
T = Period of oscillation
π (pi) = Mathematical constant, approximately equal to 3.14159
L = Length of the pendulum
g = Acceleration due to gravity
In this question, the length of the pendulum can be found by calculating the distance between the pivot point P and the center of mass of the system. In this case, the pivot point is located 40.2 cm away from the 1-kg mass.
To find the center of mass of the system, we need to consider the masses and their respective distances from the pivot point. The 1-kg mass is located 40.2 cm away, and the 9-kg mass is located 29.8 cm away from the pivot point.
To find the center of mass, we can use the weighted average formula:
(x1m1 + x2m2) / (m1 + m2)
Where:
x1 = Distance of 1-kg mass from the pivot point (40.2 cm)
x2 = Distance of 9-kg mass from the pivot point (29.8 cm)
m1 = Mass of 1-kg mass (1 kg)
m2 = Mass of 9-kg mass (9 kg)
Using the formula, the center of mass of the system is calculated as follows:
(40.2 cm * 1 kg + 29.8 cm * 9 kg) / (1 kg + 9 kg)
(40.2 kg*cm + 268.2 kg*cm) / 10 kg
308.4 kg*cm / 10 kg
30.84 cm
Now, we can substitute the length of the pendulum (L = 30.84 cm) and the acceleration due to gravity (g = 9.8 m/s^2) into the formula:
T = 2π√(L/g)
T = 2π√(30.84 cm / 100 cm/1m / 9.8 m/s^2)
T = 2π√(30.84 / 980)
T = 2π√(0.0314)
T = 2π * 0.177
T = 1.114 seconds
Therefore, the period of oscillation for the masses is approximately 1.114 seconds.