BSc Physics SEM VI 2016 17 2016-17 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) Consider two coordinate systems S* and S such that they have the 10 common origin. S* is rotating with angular velocity @ with respect to
   • S. Show that = + Hence obtain the Coriolis theorem Name the terms in the theorem
   • (ii) Set up the equation of motion of a particle moving in an inverse square field. 10 Show that the solution is equation of conic section: Hence derive the condition that the path of the particle will be an ellipse
   • (b) Attempt any one:
   • (i) Describe scattering of alpha particles. Hence obtain equations for the 5 differential cross-section and the impact parameter
   • (ii) State and explain Larmor’s Theorem. Discuss its possible application. 5
2. Q2 (a) Attempt any one:
   • (i) the D’Alembert’s principle in detail and derive Lagrange’s equations 10 using this principle
   • (ii) A mass ‘m’ moves in a vertical plane subject to gravitational force and to the 10 tension force of an attached string of length £. The other end of the string is held fixed in position O. The constraint on the system is that the mass always has to be at distance £ from the Fixed centre O
   • a) Derive the equation of the mass
   • b) Find the force of constraint
   • (b) Attempt any one:
   • (i) A body is moving freely in space (no force acts on it). Write down its 5 Lagrangian function and mention the cyclic co-ordinates considering a Cartesian axes frame. Hence find the constant of motion
   • (ii) Derive an expression for the kinetic energy of a system of particles in 5
3. Q3 (a) Attempt any one:
   • (i) Derive the equation of continuity for a fluid moving with a velocity ‘v’ and 10
   • (ii) Derive Euler’s equations of motion of a rigid body. Solve these equations for 10 torque free rotational motion of a symmetric body and hence show that the magnitude of the angular velocity vector is a constant
   • Q.P.Code : 02161
   • (b) Attempt any one:
   • (i) Explain the terms: streamline flow and steady flow. 5
   • (ii) Explain the rotation of the earth as a free symmetric top. 5
4. Q4 (a) Attempt any one:
   • (i) What is a logistic map? Calculate the fixed points for A = 0.7 and A = 1.4. 10 Starting with xo = 0.9, tabulate the iterations up to for these values of A and argue that these fixed points are stable attractors explaining what is meant by it. Draw necessary graphs
   • (ii) What is a fractal? Explain the construction of a Cantor set and find its 10 dimension. Also explain its emergence in the attractor of the Henon map
   • (b) Attempt any one:
   • (i) Graphically illustrate potential energy of an anharmonic oscillator constructed 5 using a hard spring and subjected to a conservative force
   • (ii) State Duffing’s equation. With the help of a necessary graph, explain the 5
5. Q5 (a) Attempt any one:
   • (i) Jupiter orbits the Sun at 5.20 AU. Find its period in years and days. 4 Mass of the Sun = 2.0 x
   • (ii) Find the horizontal component of the Coriolis force on a body of mass2 kg 4 moving in northward direction with speed 10 m/s at Mumbai (lattitude N)
   • (b) Attempt any one:
   • (i) Prove that when a coordinate qj does not occur explicitly in the Lagrnagian 4 function, the corresponding momentum pj is constant
   • (ii) Derive the following relation: 4 Where p and p, are the general and Cartesian momentum respectively
   • (c) Attempt any one:
   • (i) a fluid flow given by v=byi in a coordinate system s. Is the fluid 4
   • (ii) Three point masses of ‘m’ each are placed at the three andBof 4 the triangle. Coordinates of O,A and B are (0,0,0),(2,2,0)and (2,-2,0) ina coordinate frame (x,y,z). Show that if the system behaves as a rigid body, the axes Ox, oy and oz are the principal axes of the body
   • (d) Attempt any one:
   • (i) The symmetric tent map is defined by a function f(x) = 2x < x < 3 0.5 and f(x) = 2 when 0.5 < x < 1. Sketch this function and find the
   • (ii) Ina variation of Cantor set, a line segment is divided into five equal segments 3 and the middle one is removed. Then this process is continued on each of the remaining four segment. Find the fractal dimension of the resulting set

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