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Pune University - Sem - I : Classical Mechanics, April 2010

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Total No. of Questions : 7] P880 [Total No. of Pages : 2 [3722]-101 M. Sc. PHYSICS PHY UTN - 501 : Classical Mechanics (New Course) (Sem. - I) Time : 3 Hours] [Max. Marks : 80 Instructions to the candidates : 1) Question No. 1 is compulsory and solve any four questions from the remaining. 2) Draw neat diagrams wherever necessary. 3) Figures to the right indicate full marks. 4) Use of logarithmic table and electronic pocket calculator is allowed. Q1) Attempt any four of the following : a) Obtain the equation of motion of a Atwood s machine by using Lagrangian method. [4] b) Describe the Hamiltonian and Hamilton s equation of motion for a charged particle moving in an electromagnetic field. [4] c) Two particles of masses m1 and m2 are located on a frictionless double inclined plane and connected by an inextensible string passing over a smooth peg. Use D Alembert s principle to show equation of motion is r1 (m1 + m2) = (m1sin 1 m2sin 2)g [4] d) A particle is constrained to move along the inner surface of a fixed hemispherical bowl. Calculate the number of degrees of freedom. [4] e) A bead moves on a circular wire. Specify the type of constraint and constraint force. [4] f) Prove that generating function F = q i p i g enerates the identity transformation. [4] Q2) a) What is Focault s pendulum? A vertical rod PQ is rotating with constant & angular velocity . An inextensible light spring of length l has one end attached at R of the rod while the other end S has mass m . Find the tension in the string. [8] P.T.O. b) Show that Q = lu and F2. Q3) a) Prove that under canonical transformation (q, p) to (Q, P) [F, G ] q , p b) Q4) a) sin p , P = q cotp are canonical. Find corresponding F1 q [8] [ = F, G ] [8] Q, P Show that Q = 2q et cosp and P = 2q e t sinp are canonical. [8] Evaluate the Poisson s brackets. i) [Lx, x] ii) [Lx, px] [8] b) Explain variational principle. Apply variational principle to find the equation of one dimensional oscillator. [8] Q5) a) Compare Newtonian, Lagrangian and Hamiltonian formulation and discuss the advantages and disadvantages of each. [8] b) State and prove Viral theorem. [8] Q6) a) Write down the Lagrange s equation of motion for a particle of mass m [8] falling freely under gravity near the surface of earth. b) Obtain the Hamiltonian and equation of motion for a projectile near the surface of the earth. [8] Q7) a) A bullet is fixed horizontally in the north direction with a velocity of 500 m/sec at 30 N latitude. Calculate the horizontal component of Coriolis acceleration and the consequent deflection of the bullet as it hit a target 250 meters away. Also determine the vertical displacement of the bullet due to gravity. If the mass of the bullet is 10 gm. Find the Coriolis force. [8] b) Explain artificial satellite. [4] c) Draw phase space diagram for i) Damped and Undamped harmonic oscillator. ii) A stone thrown vertically up in the field of uniform gravity. [4] rrrr [3722]-101 -2-

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Additional Info : M.Sc. PHYSICS, PHY UTN - 501 : Classical Mechanics (New Course) (Semister - I), Pune University
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