BSc Physics SEM VI ATKT 2018-19 ATKT Physics 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) up the equation of a particle in an inverse square field. Solve it and state 10 the conditions under which the path of the particle will be an ellipse, a parabola or a hyperbola
   • (ii) A starred system S* rotate with a variable angular velocity with respect 10 to an inertial system S fixed in space. Show that Hence obtain the Coriolis theorem
   • (b) Attempt any one:
   • (i) Show that when a body moves in a central force field its aerial velocity is 5
   • (ii) Calculate the time taken by the plane of oscillations of a pendulum at 5 latitude to turn through a right angle
2. Q2 (a) Attempt any one:
   • (i) Derive Lagrange’s equation of motion in several dimensions with no 10
   • (ii) Explain how the forces of constraints are determined and used in Lagrangian 10 formulation. the same taking Atwood’s machine as an example
   • (b) any one:
   • (i) Describe generalized coordinates. 5
   • (ii) a system of N particles show that kinetic energy has a form 5 Where contains quadratic terms in generalised velocity, T; contains linear term and is independent of velocities
3. Q3 (a) Attempt any one:
   • (i) What is an ideal fluid? Obtain the Euler’s equation of motion for an ideal 10 fluid. State the assumptions
   • (ii) Derive Euler’s equation of motion for a rigid body. Solve these equations 10 for torque free rotational motion of symmetric body and hence show that the magnitude of angular velocity vector is a constant
   • (b) Attempt any one: If no external force acts on the fluid element and the pressure is constant 5 throughout the medium, show that the angular momentum of the fluid element is constant
   • (ii) The Lagrangian for a symmetric top is 5 Obtain Lagrange’s equation for Euler’s angles 6 and
4. Q4 (a) Attempt any one:
   • (i) 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
   • (ii) What is logistic map? Using quadratic map obtain the equation for the slope 10 of the tangent drawn at a fixed point and hence explain the stability of the fixed points for < 1
   • (b) Attempt any one:
   • (i) Draw and explain phase space diagram for undamped, damped and driven 5
   • (ii) Find the value of d, the fractal dimension of Sierpinski Gasket. Draw the 5
5. Q5 (a) Attempt any one:
   • (i) Ifthe eccentricity of a planet’s orbit about the sun is 0.4, find the ratio of the 4 lengths of the axis to the minor axes of the orbit of the planet
   • (ii) A body of mass 1 kg is falling freely under gravity. Find the fictitious force 4 and the total force acting on the body as observed from a frame moving vertically downward with an acceleration of 2
6. Q5 (b) Attempt any one:
   • (i) Fora particle moving in a plane, obtain the Lagrange’s equation of motion 4 in polar coordinates
   • (ii) Body is moving freely in space (no force acting on body). Write down its 4 Lagrangian function and mention the cyclic coordinates considering Cartesian axes frame, hence find constant of motion
7. Q5 (c) Attempt any one:
   • (i) Velocity of a fluid is given by V = (bxt)i. Find acceleration a(x, t) of the 4 fluid element at position x and time t
   • (ii). A rigid body consists of three particles of masses 2, 1 and 4 units located at 4 (1, —1,1), (2,0,2) and (—1,1,0) respectively. Determine the elements of the moment of inertia matrix for the rigid body Attempt any one:
   • (i) Find fractal dimension of Cantor Set. 3
   • (ii). Draw the logistics maps for A = 0.8. Find the corresponding fixed points. 3

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