Question

Pendulum clock

Determine a period of small oscillations of a pendulum clock where weight (bob) is a cone made out of a uniform-density material. The cone is axially symmetric,
of height h; and with at base of radius R: The apex of the cone is a fixed pivot point.

(use lagrangian mechanics)

Answer 1

As Newton’s second law says that

  ΣFx=Fsinβ=mar

ΣFy=Fcosβ+(−mg)=0

These are two equations for the two unknowns F and β. The equation for ΣFy gives

F=mg/cosβ

Now ,

That’s our target expression for F in terms of β

Substituting this result into the equation for ΣFx and using sinβ/cosβ=tanβ, we find

ar=gtanβ

To relate β to the period T, we use the equation

  T=2πR/v

and in terms of the period ,   

aR=4π2R/T²

Or , R=Lsinβ