Question

Motion of pendulum

Answer 1

Lagrangian for the motion of pendulum is\[L=\frac{1}{2}(a^2 \dot \phi^2+a^2 \dot \theta^2 \sin^2\phi)+mga\cos\phi\]
where \[a\] is length of the string.
Reduced equation for coordinate φ is\[ma^2 \ddot \phi-ma^2\omega^2\frac{\cos\phi}{\sin^3\phi}+mga\sin\phi=0\]

At the initial moment a bob had velocity \[\sqrt 2 aΩ\] when φ was φ=π/4. Show that after this bob will move upwards if \[2\sqrt 2 aΩ^2>g\]. How would it move if \[2\sqrt 2 aΩ^2=g\]?