BSc Physics SEM VI 2018 19 May 2018-19 PHYSICS PAPER I CLASSICAL MECHANICS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 (a) Attempt any one:
   • (i) Derive the equation of motion of a particle of mass ‘m’ under the action of 10 central force and reduce the problem to one dimensional problem using the effective potential energy of the particle
   • (ii) A starred coordinate system rotates relative to an unstarred system fixed in 10 space. Both systems have common origin. Prove that Interpret the various terms in the equation
   • (b) Attempt any one:
   • (i) State Kepler’s laws of planetary motion. Show that each planet describes an 5 ellipse with the sun at one focus
   • (ii) A body is dropped from rest from a height of 50 m. Calculate the deviation 5 from the vertical suffered by the body, due to Coriolis force, when it reaches the surface of the earth Latitude of place = N, g = 9.8 m/s?
2. Q2 (a) Attempt any one:
   • (i) the D’Alembert’s principle and derive an expression for Lagrange’s 10 equation with one degree of freedom
   • (ii) What are ignorable coordinates? Show that if the Lagrangian function of a 10 system does not depend explicitely on time the total energy of the system is Attempt any one:
   • (i) What are generalised co-ordinates? Explain the statement ‘More the 5 constraints imposed on the system lesser the number of generalized coordinates describing the system
   • (ii) Show that momentum conjugate to a cyclic coordinate is conserved. Give 5
3. Q3 (a) Attempt any one: @) (6V) = for a fluid element. Hence derive the equation of 10 continuity for the motion of continuous matter
   • Q.P. Code: 51729
   • (ii) Asymmetric top spinning about the axis of symmetry inclined to the vertical 10 by an angle @ has its lower tip fixed. Its K.E. is given by: Show that the top will preccess without nutations if the spin
   • (b) Attempt any one:
   • (i) the total external force includes the body force and external force on 5 volume V due to the pressure exerted by surrounding fluid on it and the total external force acting on the fluid volume V of the fluid is zero, show that its linear momentum is constant i ow that the kinetic energy of a rigid body is given Show that the k gy of a rigid body is given by T 5
4. Q4 (a) Attempt any one:
   • (i) The potential energy of a one dimensional damped anharmonic oscillator is 10 given by V(x) = K (= + where K is the spring constant and is anharmonic coefficient. Discuss the potential energy curve for various combinations of K and a. Comment on confinement of motion
   • (ii) State Duffing’s equation for a driven damped anharmonic oscillator. Discuss 10 the features of the numerical solution of the Duffing’s equation for the two
   • (b) Attempt any one:
   • (i) | Find Hausdorf dimension of Cantor Set. 5
   • (ii) Show that in Logistic map for A = 0.6, the fixed point x = 0 is an attractor. 5
5. Q5 (a) Attempt any one:
   • (i) — Ifa body of mass is moving with a velocity of 200 m/s, estimate the 4 maximum Coriolis force experienced by the body
   • (ii) Halley’s comet has a period of revolution T = 76 years around the sun. 4 Determine the semi major axis of its orbit in AU =1.5 x 108 km (sun to earth distance) ) G =6.67x , Mass of the sun = 2.0 x kg
   • (b) Attempt any one:
   • (i) For a particle moving in a plane, write down the Lagrange’s equation of 4 motion in polar coordinates
   • (ii) Set up Lagrangian for a simple pendulum and obtain equation describing its 4
6. Q5 Attempt any one:
   • (i) Water is flowing out of a pipe of non-uniform cross section at a steady rate. 4 The area of cross section at the two points A and B of the tube are 0.0314m? and respectively. The pressure difference between A and B is Density of water is Find the rate of flow of water through the pipe
   • Q.P. Code: 51729
   • (i) OX, OY and OZ are the principal axes of a rigid body, at fixed point O. If 4 I, = I, and @ is in XY plane, show that the angular momentum about O is
7. Q5 (d) Attempt any one:
   • (i) Calculate the fixed point for the quadratic maps drawn for A = 0.3 and A = 3
   • (ii) With the help of Butterfly effect define chaos. 3

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