LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc.** DEGREE EXAMINATION – **MATHEMATICS**

SIXTH SEMESTER – APRIL 2011

# MT 6604/MT 5500 – MECHANICS – II

Date : 07-04-2011 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

**PART – A**

**Answer ALL the questions: (10 x 2 =20)**

** **

- What is the Centre of Gravity of a compound body?
- Where does the C.G of a uniform hollow right circular cone lie?
- Define virtual work.
- What is common catenary?
- Show that frequency is the reciprocal of the periodic time in a simple harmonic motion.

- If the maximum velocity of a particle moving in a simple harmonic motion is

2ft/sec and its period is 1/5 sec, prove that the amplitude is feet.

- What is the
*p*–*r*equation of a parabola and an ellipse? - What are the radial and transverse components of acceleration?
- Define moment of inertia?
- Explain the conservation of angular momentum.

**PART –B**

**Answer any FIVE questions: (5 x 8 = 40)**

- A homogenous solid is formed of a hemisphere of radius
*r*soldered to a right circular cylinder of

the same radius. If h be the height of the cylinder, show that the center of gravity of the solid from

the common base is .

- Find the center of gravity of a uniform trapezium lamina.
- A uniform rod AB of length 2
*a*with one end A against a smooth vertical wall being supported by

a string of length 2*l*, attached to the other end of the rod B and to a point C of the wall vertically

above A. Show that if the rod rests inclined to the wall at an angle q, then cos^{2} q =.

- Derive the intrinsic equation of the common catenary.
- A second pendulum is in a lift which is ascending with uniform acceleration . Find the number of seconds it will gain per hour. Calculate the loss if

the lift were descending with an acceleration of .

- Show that the composition of two simple harmonic motions of the same period

along two perpendicular lines is an ellipse.

- Prove that the areal velocity of a particle describing a central orbit is constant.

Also show that its linear velocity varies inversely as the perpendicular distance

from the centre upon the tangent at P.

- Show that the Moment of inertia of a truncated cone about its axis, the radii of its

ends being *a* and *b*, (*a*<*b*) is .

**PART –C**

**Answer any TWO questions: (2 x 20 = 40)**

- (a) Find the centre of gravity of the area in the first quadrant bounded by the co-

ordinate axes and the curve .

(b) AB and AC are two uniform rods of length 2a and 2b respectively. If

, prove that the distance from A of the Centre of gravity of two the

rods is (10 + 10)

- (a) Show that the length of a chain whose ends are tied together and hanging over

a circular pulley of radius *a*, so as to be in contact with two thirds of the

circumference of the pulley is a .

(b) Derive the expression for velocity and acceleration of a particle moving on a

curve. (10 + 10)

- (a) A particle P describes the orbit under a central force. Find the

law of force.

(b) The law of force is and a particle is projected from an apse at a distance

Find the orbit when the velocity of projection is . (10 + 10)

- (a) State and prove Parallel axis theorem.

(b) Find the lengths of the simple equivalent pendulum, for the following:

- i) Circular wire ii) Circular disc. (10 + 10)

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