PHY 115
Wave and Oscillation
♦ Compound Pendulum: any shape rigid body...
♦ Differential Equation for Simple Harmonic Motion:
(1) Force along the string = Mgcosθ
(2) Force perpendiculer to the string = Mgsinθ
The component Mgcosθ balance the tension.
T=Mgcosθ
The Force active on the oscillating particle is,
F=−Mgcosθwhen,θ→0, the sinθ=θF=−Mgθ
We know,
F=ma
The linear displacement,
y=lθ⟹dydt= dθdt⟹d2ydt2=a= d2θdt2
Comparing equation (1) and (2),
F=Mld2θdt2
Now using equation (1)
Mld2θdt2=−Mgθ⟹d2θdt2+glθ=0
comparing with differential equation of SHM,
ω2=gl⟹(2πT)2=gl⟹t=2π√lg
HW: Find out T for compound pendulum.
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