Anna University Chennai 2007 M.Sc Computer Science Mathematics - Question Paper

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Name of the Candidate :
7 seven 0 3
M.Sc. DEGREE EXAMINATION, 2007
( MATHEMATICS )
( 1st YEAR )
( PAPER - IV )
140. ANALYTICAL MECHANICS AND TENSOR ANALYSIS
( Revised Regulations )
May ] [ Time : three Hours
Maximum : 100 Marks
part Œ A (8 × five = 40)
ans any 8 ques.. every ques. carries 5 marks.
1. Derive the equations of motion of a particle in rectangular Cartesian co-ordinates.
2. Show that the 1st component of angular momentum of a rigid body rotating with 1 point fixed is Aw one ΠFw two ΠEw three , in the usual notation.
3.3. define the spinning top.
4. Derive the Hamilton ΠJacobi formula for a harmonic oscillator.
5. discuss the gyroscopic couple.
6. Write a note on Foucults pendulum.
7. What do you mean by a contraction of a mixed tensor ?
8. Show that every tensor can be written as the sum of a symmetric tensor and a skew-symmetric tensor.
9. describe the Christoffel symbols.
10. obtain the value of
.
part Œ B (3 × 20 = 60)
ans any 3 ques.. every ques. carries TWENTY marks.
11. (a) explain the motion of a particle relative to a rotating frame.
(b) A circular disk of radius a and mass m is supported on a needle point at its centre,
it is set spinning with angular velocity about a line making an angle a with the normal to the disk. obtain the angular velocity of the disk at any following time.
12. (a) explain the motion of a spherical pendulum.
(b) obtain the formula of motion of a charged particle in uniform electric and magnetic fields which are perpendicular to every other.
13. (a) explain the motion of rolling disk.
(b) State and prove Hamilton™s principle.
14. (a) State and discuss the quotient legal regulations of tensors.
(b) compute the Christoffel symbols in spherical coordinate system.
15. (a) State and prove Ricci™s theorem.
(b) Prove that
R ijkl, m + R ijlm, k + R ijmk, l = 0.

Earning:   Approval pending.