A physical pendulum is a disk of radius R, swinging about an axis perpendicular to its plane and at distance λ from its centre. (a) Show that ω = √[g $\lambda$ /(R²/2+ $\lambda^2$ )]. (b) For what $\lambda$ is this frequency maximum?
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Question

A physical pendulum is a disk of radius R, swinging about an axis perpendicular to its plane and at distance λ from its centre. (a) Show that ω = √[g $\lambda$ /(R²/2+ $\lambda^2$ )]. (b) For what $\lambda$ is this frequency maximum?

Select all that apply

G1 Physics · Accepted Answer

Moment of inertia about the pivot is I=M(R²/2+λ²). Small-angle torque is -Mgλθ, so ω=√[gλ/(R²/2+λ²)]. Maximizing ω² is equivalent to maximizing λ/(R²/2+λ²); differentiation gives λ=R/√2. At this value the equivalent pendulum length is √2R. Hence all four statements are correct.

A physical pendulum is a disk of radius R, swinging about an axis perpendicular to its plane and at distance λ from its centre. (a) Show that ω = √[g λ \lambda λ /(R²/2+ λ 2 \lambda^2 λ 2 )]. (b) For what λ \lambda λ is this frequency maximum?