BSc Sem VI 2022 2023 Apr 2023 PHYSICS CLASSICAL MECHANICS Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate full marks

1. Q1 Attempt any two
   • (i) State and prove Kepler’s laws of planetary motion. 10
   • (ii) Discuss quantitatively the motion of a particle in an inverse square field. 10 Show that the eccentricity of the particle is given by + Give conditions on E and for different shapes of orbits
   • (iii) What is a Foucault pendulum? Obtain equation of motion for it. Hence 10 show that the pendulum precesses slowly clockwise in the northern
   • (iv) Consider a starred (rotating) reference frame rotating with angular 10 velocity w relative to the unstarred (fixed) frame with their origins O and O* coinciding. Prove that for an arbitrary vector A,
2. Q2 Attempt any two
   • (i) Starting with D’Alembert’s Principle, obtain Lagrange’s equations in 10 terms of generalized coordinates
   • (ii) What is meant by generalized co-ordinates? Derive an expression for 10 generalized velocity and generalized kinetic energy
   • (iii) A body of mass m, can move on a smooth flat horizontal table top. Itis 10 connected to a string of length £ which passes through a hole in the centre of the table. The other end of the string is connected to a mass m2 which is suspended vertically. Identify appropriate generalized coordinate for the system and obtain the equations of motion using
   • (iv) double pendulum consists of two weightless rods connected to each_ 10 other and a point of support. The masses m, and are not equal but the length of the rods are equal. Pendulums are free to swing only in one vertical plane. Write down the Lagrangian for the system
3. Q3 Attempt any two
   • (i) For a moving fluid show that 10 (Symbols have their usual meanings) Derive Bernouilis theorem. Hence for a steady flow of the fluid show 10 + (Symbols have their usual meanings)
   • (iii) | With reference to rotations of rigid body explain setting of the Euler’s 10 angles. Draw suitable diagrams. Find expression for the Lagrangian of
   • (iv) Derive an expression for the moment of inertia tensor for a rigid body 10 made up of N number of particles
4. Q4 Attempt any two
   • (1) Discuss numerical solutions of Duffing’s equation for 10 Compare the nature of odd and even harmonics
   • (ii) Consider an anharmonic oscillator with potential energy 10 + where K is the spring constant and @ is anharmonic coefficient. Discuss the potential energy curve for positive and negative values of K and a. Comment on confinement of motion
   • (iii) Discuss fixed points of a logistic map, stability of fixed points and 10 periodic attractors. Discuss logistic map for 3< <4 and explain the onset of chaos qualitatively
   • (iv) Discuss numerical solutions of Duffing’s equation for 10
5. Q5 Attempt any four
   • (i) If a body of mass 100 kg is moving with a velocity of 10m/s; estimate 05 the maximum Coriolis force experienced by the body
   • (ii) The eccentricity of a planet’s orbit about sun is 0.4. Find the ratio of the 05 lengths of the semi major to the semi minor axes of the orbit of the
   • (iii) | Write down the Lagrangian for a simple pendulum and hence find its 05 equation of motion
   • (iv) Define constraints. With good examples, explain holonamic and non- 05
   • (v) Consider a fluid flow given by Show that the fluid is 05 incompressible and non-irrotational
   • (vi) | What is a rigid body? Discuss the different types of rigid bodies with 05 reference to the symmetry present in the body Two very close initial values of x on logistic map are 0.40000 and 05
   • 0.40002 respectively. With after 20 iterations the values are
   • 0.14561 and 0.00170 respectively. Calculate Lyapunov exponent
   • (viii) Discuss the nature of phase space diagram for 1)undamped oscillator 05 and 2)damped oscillator

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