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