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/ApproachEquate the time period of a compound pendulum to that of a simple pendulum.

Time periodsCompound: 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 pitfallForgetting to use the parallel axis theorem: I_A = I_G + m h^2.

Final Answerk^2/h + h