# CLassical Assignment 2

```Classical Mechanics (PH411)
Assignment No. 2
Date: Sep 5th, 2019
1. Find Lagrange’s equation of motion of the bob of a simple pendulum.
2. Obtain the equations of motion for the motion of a particle of mass m in a potential
V(x,y,z) in a spherical polar co-ordinates.
3. Masses m and 2m are connected by a light inextensible string which passes over a
pulley of mass 2m and radius a. Write the Lagrangian and find the acceleration of the
system.
4. A simple pendulum has a bob of mass m with a mass m1 at the moving support
(pendulum with moving support) which moves on a horizontal line in the vertical plane in
which pendulum oscillates. Find the Lagrangian and Lagrange’s equation of motion.
5. Two equal masses m connected by a massless rigid rod of length l forming a dumbbell is
rotated in the x-y plane. Find the Lagrangian and obtain Lagrange’s equation of motion.
6. A bead of mass m slides freely on a frictionless circular wire of radius a that rotates in a
horizontal plane about a point on the circular wire with a constant angular velocity .
Find the equation of motion of the bead oscillates as a pendulum of length l = g/2 (see
Fig. 1).
7. A particle of mass m is constrained to move on the inner surface of a cone of half angle
Fig. 1
Fig. 2
 with its apex on a table. Obtain its equation of motion in cylindrical co-ordinates (, ,
z). Hence, show that the angle  is a cyclic co-ordinate (Fig. 2).
8. A rigid body capable of oscillating in a vertical plane about a fixed horizontal axis is
called a compound pendulum. (i) Set up its Lagrangian; (ii) Obtain its equation of motion;
and (iii) Find the period of pendulum (Fig. 3).
9. A particle of mass m is constrained to move on the surface of a cylinder of radius a. It is
subjected to an attractive force directed towards the origin and is proportional to the
distance of the particle from the origin. Write its Lagrangian in cylindrical co-ordinates
Fig. 3
Fig. 4
and (i) obtain its equations of motion, (ii) Show that the angular momentum about z-axis
is a constant of motion, (iii) show that the motion of the particle in the z-direction is
simple harmonic (Fig. 4).
10. Write Lagrangian of the double pendulum.
11. Show that the shortest distance between two points in a straight line.
12. Using Lagrange’s method of undetermined multiplier, find the equation of motion and
force of constraint in the case of a simple pendulum.
13. Discuss the motion of a disc of mass m and radius b rolling down an inclined plane
without slipping. Also, find the force of constraint using the Lagrange method of
undetermined multipliers (Fig. 5).
14. A particle of mass m is placed at the top of a smooth hemisphere of radius a. Find the
reaction of the hemisphere on the particle. If the particle is disturbed, at what height
does it leave the hemisphere (Fig. 6)?
Fig. 5
Fig. 6
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