2014

(November)

MATHEMATICS

(General)

Course: 501

[(A) Analysis – II, (B) Mechanics]

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

1. (a) Write the necessary conditions for a function to be analytic. 1

(b) Show that the function is not analytic. 3

(c) Derive the polar form of Cauchy-Riemann equations of an analytic function.

Or

If , then find such that is analytic.

2. (a) Define Jordan curve.

(b) Evaluate from to along the curve.

(c) State and prove Cauchy’s theorem.

Or

If is analytic inside and on the boundary C of a simply-connected region R, then prove that

3. (a) State Taylor’s theorem. 1

(b) Define pole of an analytic function. 1

(c) Define residue of an analytic function. 1

(d) Find the residue of at the point. 2

(e) Find the poles of the function

in a unit circle. 2

(f) Expand in a Laurent’s series for.

Or

Evaluate, using contour integration.

(B) Mechanics

(Marks: 45)

(a) Statics

4. (a) Define the central axis of a system of coplanar forces.

(b) Define pitch.

(c) Write the resultant of wrench of two given forces and inclined at an angle . 2

(d) Find the equation of the central axis of a system of forces acting on a rigid body. 6

Or

Prove that any system of forces acting on a rigid body can be reduced to a single force together with a couple whose axis is along the direction of the single force.

5. (a) Define virtual work. 1

(b) Define axis of catenary. 1

(c) Find the relation between and in a common catenary.

(d) Derive intrinsic equation of a common catenary.

Or

Find the work done by tension of a light rod.

(b) Dynamics

6. (a) Define frequency of a simple harmonic motion.

(b) Find the radial velocity and transverse velocity of a particle moving in a plane curve at any point.

Or

A point moves in a plane curve, so that its tangential and normal accelerations are equal. The angular velocity of the tangent is constant. Find the curve.

7. (a) Define central force. 1

(b) Find the law of force to the pole if the path of the particle is . 6

Or

A particle falls under gravity from rest in a medium whose resistance varies as the velocity. Find a relation between and.

8. (a) Define moment of inertia of a body about a line. 2

(b) Write the product of inertia of a body of mass with respect to and axes.

(c) Find the moment of inertia of a rectangular lamina about , which passes through the centre of the lamina; being parallel to one of its edges.

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