## Sunday, December 30, 2018

2015
(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)
(a): Statics
(Marks: 25)

1. (a) Write the entity with which screw is associated. 1
(b) Write two characteristics of central axis. 2
(c) Find the equation of the central axis of any given system of forces. 7
Or
Find the necessary conditions for the equilibrium of a rigid body under the action of a system of forces acting at its different points.
2. (a) Define virtual work. 1
(b) Write two forces which may be omitted informing the equation of virtual work. 2
(c) Derive the Cartesian equation of the common catenary. 5
Or
A body rests in equilibrium upon another fixed body, the portions of the two bodies in contact being spheres of radii and , and the shortest line joining the centres of sphere being vertical. If the first body be slightly displaced, find whether equilibrium is stable or unstable, the bodies being rough enough to prevent sliding.
(d) Establish the relation between and for a common catenary. 2
(e) Find the work done by tension or thrust of a light road. 5

(b): Dynamics
(Marks: 25)
3. (a) Define angular velocity of a particle. 1
(b) Define frequency of simple harmonic motion. 1
(c) Find the tangential and normal components of velocity of a particle. 6
Or
The velocity components of a particle along and perpendicular to the radius vector are and . Find the acceleration along and perpendicular to the radius vector.
4. (a) Define central force. 1
(b) Write the direction of the resisting force to the motion of a particle. 1
(c) Find the law of force to the pole if the path of the particle is cardioids . 5
Or
A particle is projected upwards under gravity in a resisting medium whose resistance varies as the square of the velocity. Find the relation between velocity and distance.
5. (a) Define impressed force on a particle. 1
(b) Write the vector equation of motion of rigid body when the centre of inertia is regarded as a fixed point.       2
(c) Expression is called momental ellipsoid of a body. Justify it. 2
(d) Prove the theorem of parallel axes of moment of inertia. 5
Or
Find the moment of inertia of a uniform rod of length 2a and mass M about an axis through an extremity and perpendicular to it.

GROUP – B
(Integral Transforms)
(Marks: 30)
6. (a) Write the value of the following: 1+1
1. .
2. .

(b) Find . 1
(c) Find . 2
(d) Prove that 3
Or
Find .
7. (a) Find . 2
(b) Find . 2
(c) Find 1
(d) Find 3
Or
Find .
8. (a) Write the value of . 1
(b) Solve any two of the following by using Laplace transform: 4x2=8
1. where .
1. 2. (c) Solve: 5 Or
Solve: ***