Compound pendulum — equivalent simple pendulum length Find the length of an equivalent simple pendulum that would have the same frequency (same time period) as the given compound pendulum. Choose the correct formula in terms of radius of gyration k about the C.G. and distance h between the pivot and C.G.

Given/Notation

• Compound pendulum: mass m, radius of gyration about C.G. = k
• Distance between pivot and C.G. = h
• Equivalent simple pendulum length = L_e

Concept/Approach
Equate the time period of a compound pendulum to that of a simple pendulum.

Time periods
Compound: T = 2\pi \sqrt{\dfrac{I_A}{m g h}}, \; \text{where } I_A = I_G + m h^2 = m(k^2 + h^2)Simple: T = 2\pi \sqrt{\dfrac{L_e}{g}}

Step-by-Step
Set equal: \; 2\pi \sqrt{\dfrac{m(k^2 + h^2)}{m g h}} = 2\pi \sqrt{\dfrac{L_e}{g}} \Rightarrow \dfrac{k^2 + h^2}{h} = L_e \Rightarrow L_e = h + \dfrac{k^2}{h}

Common pitfall
Forgetting to use the parallel axis theorem: I_A = I_G + m h^2.

Final Answer
k^2/h + h