BSc Physics SEM V 2017 18 2017-18 Classical Mechanics Question Paper - Mumbai University | munotes

Questions asked in this paper

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• 2. Figures to the right indicate full marks

1. Q4 Symbols have usual meaning unless otherwise stated
2. Q5 Use of log-table and non-programmable calculator is allowed
   • (a) Attempt any one:
   • i) that the equation of motion of a particle of mass moving in a central force 10 If F (u) is the force in inverse square field, show that, solution of the above equation represents conic section with eccentricity +
   • ii) With the help of modified simple pendulum, explain rotational motion of earth about its 10 own axis. Hence develop the necessary theory to obtain period of the rotation
   • (b) Attempt any one:
   • i) that the path of the particle moving under the central force lies in a single plane 05 containing the center of force and the angular momentum is conserved
   • ii) Write equation of motion on rotating earth and hence define effective gravitational 05 acceleration Show that ge ~ g — w*Re sin? 0
3. Q2 (a) Attempt any one:
   • i) Adouble pendulum consists of two weightless rods connected to each other and a point of 10 support. The masses mi and m2 are not equal but the lengths of the rods are equal Pendulums are free to swing only in one vertical plane. Obtain the Lagrangian for the system
   • ii) Derive an expression for Lagrange's equations in several dimensions. 10
   • b) any one:
   • i) Abead slides without friction in shape of a cycloid with equations. 5 05 where 0 2m. Write the equation motion of the bead using Lagrange's
   • ii) Set up the Lagrangian for a simple pendulum and obtain an equation describing its motion. 05
   • Q.P. Code :02160
4. Q3 a) Attempt any one:
   • i) Derive Bernoulli's theorem and discuss how it represents the conservation law of energy for 10
   • ii) Consider a symmetric top rotating with respect to an inertial frame of reference fixed in 10 space. Considering no external torque acting on the body, discuss its motion with respect to
   • b) Attempt any one:
   • i) Explain the terms: streamline flow and tubes of flow. 05
   • ii) What are Euler's angles? Explain the order in which the transformation of the axes is carried 05
5. Q4 a) Attempt any one:
   • i) Duffing's equation for a driven damped anharmonic oscillator. Obtain its reduced form 10 by suitable rescaling. With the help of graphical representation, discuss features of the numerical solution of the equation for the following two cases: (1) andf = 0.5, (2) y=
   • ii) fractal dimension and explain by applying it to a line of unit length and a square of unit 10 area. Describe construction of Sierpinski gasket and find its fractal dimension
   • (b) Attempt any one:
   • i) do you mean by a fixed point of a map? Calculate the fixed points for (1) = 05 1.6 and (3) A =2.6
   • ii) What attractor? Explain using the attractor of the Henon map. 05
6. Q5 a) Attempt any one:
   • i) takes Neptune 165 years to orbit the Sun. Find its maximum distance from the Sunin AU. 04 (G = 6.67 x and Mass of the Sun = 2.0 x
   • ii) A body is dropped from a building of height 100 m in Mumbai (latitude Find the 04 deflection due to the coriolis force from the vertical when the body reaches the ground
   • b) Attempt any one:
   • i) . Abody is moving freely in space (no force acts on it). Write down its Lagrangian function and 04 mention the cyclic co-ordinates considering a Cartesian axes frame
   • ii) note on constraints 04
   • Q.P. Code :02160
   • c) Attempt any one:
   • i) Consider a fluid flow given by = b y Tin a coordinate system S. Is the fluid incompressible? 04
   • ii) Show that a sphere can always rotate with a constant angular velocity about an axis passing 04 through its centre in a torque free situation
   • d) Attempt any one:
   • i) Anasymmetric tent map is defined by a function f (x) =4x andf (x) = (4 03 4x)/3 when 0.25 < x< 1. Sketch this function and find the fixed points
   • ii) In avariation of Cantor set, a line segment is divided into five equal segments and the middle 03 three removed. Then this process is continued on each of the remaining two segment. Find the fractal dimension of the resulting set

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