To find the period of oscillation, we need to use the formula for the period of a simple pendulum. The formula is:
T = 2Ï€ * sqrt(L/g)
Where:
T is the period of oscillation (in seconds)
L is the length of the pendulum (in meters)
g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth)
Let's solve the problem step by step:
a) Period of oscillation for Kate alone:
1. Substitute the given values into the formula:
L = 3 m
g = 9.8 m/s^2
2. Calculate the period:
T = 2Ï€ * sqrt(3/9.8)
b) Period of oscillation for the twins together:
1. The center of mass of the twins is still at the end of the rope, so the length of the pendulum remains the same.
L = 3 m
2. Calculate the period using the same formula:
T = 2Ï€ * sqrt(3/9.8)
Now, let's substitute the numbers into the equation and calculate the periods for both cases.
For Kate alone:
T = 2Ï€ * sqrt(3/9.8)
T ≈ 2π * 0.556
T ≈ 3.49 seconds
For the twins together:
T = 2Ï€ * sqrt(3/9.8)
T ≈ 2π * 0.556
T ≈ 3.49 seconds
Both Kate alone and the twins together have the same period of oscillation, which is approximately 3.49 seconds.