Consider a simple pendulum of mass m whose length changes with time as . Obtain the Lagrange equation of motion, the generalised momentum and the energy function for the system.
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A central potential is given as V(r):
(i) Obtain the criterion for a stable circular orbit.
(ii) If the central potential is given as . For what values of n are the circular orbits stable.
Consider a simple pendulum of mass m whose length changes with time as . Obtain the Lagrange equation of motion, the generalised momentum and the energy function for the system.
A pendulum of mass m is pivoted to a support of mass M sliding without friction along x-axis as shown in the figure. Obtain the Lagrange equation of motion for the given system.
Two frictionless blocks of equal masses m are connected by a massless rigid rod of length L. The system is constrained to move in the vertical plane under the action of an applied force as shown in the figure. Obtain the equation of motion of the system and calculate the condition for static equilibrium.
Two objects of equal mass are connected by a three spring with spring constants as shown in figure. Obtain the normal mode frequencies.
The orbit of a particle in a central force field is given by , obtain the force law.
Two equal of mass m attached by a thread of length b are oriented vertically as shown in the figure. The acceleration decrease according to height as , where g0 and a are constants. Calculate the vertical acceleration of the system as measured from the centre of the thread.
