Question

A uniform stick of mass M and length L is pivoted at one end. a.)Find the period of oscillation for small angular displacement. b.) Find the period of oscillation if the stick is pivoted about point P a distance x from the center of mass.

Answer 1

T = 2π/ω = 2π√(I/gML),
where I is the moment of inertia of a uniform stick of mass M and length L is pivoted at one end.
I=(ML²)/3.
So
T = 2π/ω = 2π√(I/gML) = 2π√(((ML²)/3)/gML) = 2π√(L/3g).
The period of oscillation if the stick is pivoted about point P a distance x from the center of mass

I=(ML²)/12+Mx².

And we have
T = 2π/ω = 2π√(I/gML) = 2π√(((ML²)/12 + Mx²)/gML) = 2π√((L²/12+x²)/gL).