written 5.8 years ago by |
→ Arc CP=ArcCP
R θ=r ϕ
∴ ϕ=R θr
→ Translatory displacement of center of cylinder = (R – r) \theta
\rightarrow Total rotational displacement of cylinder = \phi = 0 = Relative displacement.
\rightarrow KE = (KE)_{Translational} + (KE)_{rotational}
\frac{1}{2} \ m\ [ (R-r) \theta]^2 + \frac{1}{2} I_G (\phi - \theta)^2
\rightarrow PE = mg [ (R-r) – (R-r) \ cos \ \theta ] = mg \ (R-r) [ 1- cos \ \theta]
\frac{d}{dt} (KE + PE) = 0
\frac{d}{dt} [ \frac{1}{2} m [ (R-r) \theta]^2 + \frac{1}{2} I_G (\phi - \theta)^2 + mg (R-r) (1- cos \theta)] = 0
\frac{d}{dt} [ \frac{1}{2} (R-r)^2 \theta^2 + \frac{1}{2} I_G [ \frac{R\theta}{r} - \theta]^2 + mg (R-r) (1-cos \theta)] = 0
\frac{d}{dt} [ \frac{1}{2} m (R-r)^2 \theta^2 + \frac{1}{2} [\frac{1}{2} mr^2] . \frac{1}{r^2} [ R-r]^2 \theta^2 + mg (R-r) (1-cos \theta)] = 0
\frac{d}{dt} [\frac{1}{2} m (R-r)^2 \theta^2 + \frac{1}{4} m (R-r)^2 \theta^2 + mg (R-r) (1 –cos \theta)] = 0
\frac{d}{dt} [\frac{3}{4} m (R-r)^2 \theta^2 + mg (R-r) (1-cos \theta)] = 0
\frac{3}{4} m (R-r)^2 2 \theta \theta + mg (R-r) sin \theta . \theta = 0
\frac{3}{2}m (R-r)^2 \theta + mg (R-r) \theta = 0
\theta + [ \frac{mg (R-r)}{\frac{3}{2} m (R-r)^2}] \theta = 0
\theta + [ \frac{2g}{3 (R-r)}] \theta = 0
W_n = \sqrt{ \frac{2g}{2(R-r)}} rad/sec.