Download SGBAU BSc 2019 Summer 4th Sem Mathematics Classical Mechanics Question Paper

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B.Sc. Part-II (Semester-IV) Examination

MATHEMATICS

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(Classical Mechanics)

Paper—VIII

Time : Three Hours] [Maximum Marks : 60

Note :— (1) Question No. 1 is compulsory and attempt it once only.

(2) Solve ONE question from each unit.

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1. Choose the correct alternative :
   1. Each planet describes __ having the sun at one of its foci. 1
      1. An ellipse
      2. A circle
      3. A hyperbola
      4. None of these

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   2. If a bead is sliding along the wire then the constraint is __ . 1
      1. Holonomic
      2. Non-holonomic
      3. Superfluous
      4. None of these

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   3. For an inverse square law, the virial theorem reduces to __ . 1
      1. 2T=-nV
      2. 2T=nV
      3. 2T=V
      4. 2T=-V

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   4. The virtual work on a mechanical system by the applied forces and reversed effective forces is . 1
      1. Zero
      2. One
      3. Negative
      4. None of these

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   5. The shortest distance between two points in a space is . 1
      1. A circle
      2. A straight line
      3. An ellipse
      4. A parabola

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   6. If H is the Hamiltonian of the system then a generalized coordinate q, is said to be cyclic if . 1
      1. ≠ 0
      2. >0
      3. =0
      4. <0

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   7. A square matrix A is said to be orthogonal if 1
      1. A=AT
      2. AT=A-
      3. A=A"
      4. None of these

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   8. The general displacement of a rigid body with one point fixed is a rotation about some axis. 1
      1. One
      2. Two
      3. Three
      4. None of these

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   9. The sum of the finite rotations depends on the ~ of the rotation. 1
      1. Degree
      2. Order
      3. Both Degree and Order
      4. None of these

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   10. A particle moving in a space has _ degrees of freedom. 1
       1. One
       2. Two
       3. Three
       4. Four

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UNIT—I

2. (a) Derive the lagrange’s equations of motion in the form :
   dt| aq aq,
   for conservative system from D Alembert’s principle. 6
   

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   (b) A bead is sliding on a uniformly rotating wire in a force-free space, then show that the acceleration of the bead is ¥=rw?’, where w is the angular velocity of rotation. 4
3. (p) Two particles of masses m and m, are connected by a light inextensible string which passes over a small smooth fixed pulley. If m > m,, then show that the common acceleration of the particles is . 5
   (q) Obtain the equations of motion of a simple pendulum by using D’Alembert’s principle. 5

UNIT—II

4. (a) For a central force field, show that Kepler's second law is a consequence of the conservation of angular momentum. 5
   

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   (b) Prove that if the potential energy is a homogeneous function of degree —1 in the radius vector F. then the motion of a conservative system takes place in a finite region of space only if the total energy is negative. 5
5. (p) Prove that in a central force field the areal velocity is conserved. 5
   (q) Show that if a particle describes a circular orbit under the influence of an attractive central force directed towards point on the circle, then the force varies as the inverse fifth power of the distance. 5

UNIT—III

6. (a) Show that the functional :
   

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   Iy(ol= [ {2y +y (9} dx
   defined in the space c [0, 1] is continuous on the function y,(x) = x in the sense of first order proximity. 5
   (b) Find the extremals of I[y(x)]= j[yz +y‘2 +2ye*dx . 5
7. (p) Find the extremals of the functional :
   yeol= [r16y? -y? +x’]ix. 5
   

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   (q) Write down the Euler-Ostrogradsky equation for the functional :
   lztx 1= [ (2) +{ 2| +12a806,y)p dx dy 5
   

UNIT—IV

8. (a) Show that Hamilton’s principle can be derived from D’Alembert’s principle. 5
   

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   (b) Define Hamiltonian H. Derive the Hamilton’s equations for the Hamiltonian H of the system. 1+4
9. (p) Deduce the Hamilton’s equations of motion of a particle of mass m in Cartesian coordinates (X, y, 2). 5
   (q) Define Routhian, prove that a cyclic coordinate will not occur in the Routhian R. 1+4

UNIT—V

10. (a) Prove that if A is any 2 x 2 orthogonal matrix with determinant | A | =1, then A is a rotation matrix. 5
    

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    (b) Define infinitesimal rotation. Prove that infinitesimal rotations commute. 1+4
11. (p) Show that two complex eigenvalues of an orthogonal matrix representing a proper rotation are e*®, where ¢ is the angle of rotation. 5
    (q) Prove that the general displacement of a rigid body with one point fixed is a rotation about some axis. 5

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