## Sunday, December 30, 2018

2014
(November)
MATHEMATICS
(Major)
Course: 504
(Mechanics and Integral Transforms)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
GROUP – A
(MECHANICS)
(Marks: 25)

1. (a) Write the value of the pitch of the wrench . 1
(b) Define screw. 2
(c) Prove that a system of forces can be reduced to a single force acting through an arbitrary chosen point and a couple whose axis passes through that point.
Or
Find the equation of null plane of a given point referred to coordinate system .
2. (a) Define virtual work.
(b) Write the name of one force which can be omitted in forming the equation of virtual work.
(c) Establish the relation between and for a common catenary.
(d) State and prove the principle of virtual work for a system of coplanar forces acting at different points of a rigid body.
(e) Derive the intrinsic equation of common catenary. 5
Or
A regular hexagon ABCDEF consists of six equal uniform rods, each of weight w, freely jointed together. The hexagon rests in a vertical plane and AB is in contact with a horizontal table. If C and F be connected by a light string, then find the tension of the string.
(b): Dynamics
(Marks: 25)
3. (a) Define radial velocity of a particle. 1
(b) Define the amplitude of a simple harmonic motion. 1
(c) Find the radial and transverse velocity components of a particle. 6
Or
A particle describes the curve with a constant velocity. Find the components of velocity among radius vector and perpendicular to it.
4. (a) Write the name of the orbit of a particle moving under a central force. 1
(b) If a particle moves upward in a resisting medium, then write the direction along which the resisting force acts.                                                                                                                                                1
(c) A particle describes the curve under a force to the pole. Find the law of the force.    5
Or
A particle falls under gravity from rest in a medium whose resistance varies as the velocity. Find the relation between and .
5. (a) Define effective force on a particle. 1
(b) Let be the coordinates of a point mass . Then write the moment of inertia of the point mass with respect to the origin. 1
(c) Prove the theorem of perpendicular axes of moment of inertia. 3
(d) Find the moment of inertia of a plane lamina of length and breadth about a line through its centre and parallel to . 5
Or
Deduce the general equation of motion of a rigid body from D’Alembert’s principle.

GROUP – B
(INTEGRAL TRANSFORMS)
(Marks: 30)
6. (a) Write the value of . 1
(b) Find . 2
(c) Evaluate . 2
(d) Evaluate (any one): 3
1. 2. 7. (a) Write the value of . 1
(b) Evaluate: 2+2=4
1. .
2. .
(c) Evaluate . 3
Or
Evaluate .
8. (a) If , then write the value of . 1
(b) Solve , using Laplace transform, with conditions . 3
(c) Solve using Laplace transform with conditions . 5
Or
Solve , using Laplace transform with conditions .
(d) Solve using Laplace transform with conditions 5
Or
Solve when  and .

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