Question

A pendulum with length I₀, oscillates with angular frequency ω=1 s⁻¹, what is the period T for the same pendulum but with length 2I₀ ?

Answer 1

The period T for a pendulum with length 2I₀, given that the angular frequency ω is 1 s⁻¹ for length I₀, is 1.414 times the original period of 2π seconds.

Step-by-step explanation:

To determine the period T for a pendulum with length 2I₀ when the angular frequency ω is given as 1 s⁻¹ for initial length I₀, we use the formula for the period of a simple pendulum:

T = 2π√(l/g)

Since ω = 2π/T, we rearrange to find T = 2π/ω. For a pendulum with length I₀ and ω = 1 s⁻¹, the period T is simply 2π seconds. For a pendulum with length 2I₀, the period T'₀ can be found by inserting this new length into the formula:

T'₀ = 2π√(2I₀/g)

Simplifying, we find:

T'₀ = 2π√(2) * √(I₀/g)

Since √(2) is approximately 1.414, the new period T'₀ will be about 1.414 times the original period T, which was 2π seconds.