The angular position of a pendulum is represented by the equation θ = 0.0340 cos ωt, where θ is in radians and ω = 5.43 rad/s. Determine the period and length of the pendulum.
What equations would I use to find the period and the length.
To find the period and length of the pendulum, we will use the following equations:
1. Period (T): The period is the time taken for one complete cycle of the pendulum's motion. It is given by the formula: T = 2π/ω, where ω is the angular frequency.
2. Length (L): The length of the pendulum is the distance from the pivot point to the center of mass of the object. It can be calculated using the formula: L = g(T/2π)^2, where g is the acceleration due to gravity (approximately 9.8 m/s²).
Let's calculate the period and length using these equations.
Given:
ω = 5.43 rad/s
1. Period (T) calculation:
T = 2π/ω
T = 2π/5.43
T ≈ 1.16 s (rounded to two decimal places)
2. Length (L) calculation:
L = g(T/2π)^2
L = 9.8(1.16/2π)^2
L ≈ 0.3724 m (rounded to four decimal places)
Therefore, the period of the pendulum is approximately 1.16 seconds, and the length is approximately 0.3724 meters.