CMI CM-I Problem Sheet 4 (2012)

16. Consider an undamped simple harmonic oscillator and trace the trajectory of a point A with initial conditions $x(t=0)\,=\,A\ ;p(t=0)\,=\,=0$ in phase space. Choose three neighbouring points in phase space and show that the area of the triangle in phase space is independent of time.

17. Consider the motion of a particle of mass M whose motion is given by
$$ M\frac{d^2z}{dt^2}\,+\,\lambda \frac{dz}{dt}\ =\ mg $$
Choose an appropriate initial condition for the motion of the particle and trace it's trajectory in phase space. Choosing three neighbouring points at time $t\,=\,0$ and find the area of the triangle in phase space and show it is not independent of time.

18.Draw the phase space plot of a few trajectories for a particle moving under the potential
$$ V(x)\ =\ -ax\ +\ bx^3 \ a,b\ >\ 0 $$
You need not solve the equation of motion. Draw the potential and use physical arguments to get the trajectories for different initial conditions. When will the trajectory be closed ?

19. Consider the motion of a particle described by $\vec{r}(t)$. Find the velocity and acceleration vectors in spherical polar coordinates $(r,\theta,\phi)$ along $(\hat{r},\hat{\theta},\hat{\phi})$.

20. A pendulum has a period of one second in vacuum. When placed in a resistive medium ( resistance is proportional to the velocity), it is observed that the amplitude on each swing becomes half that of the previous swing. What is it's new period?