LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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

FIFTH SEMESTER – **APRIL 2012**

# MT 5506/MT 4501 – MECHANICS – I

Date : 27-04-2012 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

__PART – A__

**ANSWER ALL QUESTIONS: (10 X 2 = 20 marks)**

** **

- State the law of parallelogram of forces.
- What is the magnitude and direction of the resultant of two forces and when and are equal in magnitude and the angle between them is 60°?
- Define torque of a force.
- State any two laws of friction.
- Define angular velocity.
- State the triangle law of velocities.
- State the principal of conservation of linear momentum.
- A body of mass 50 gm is acted upon by a constant force F=100 dynes. Find the time required to move the body through a distance of 25 cms from rest.
- Define time of flight of a projectile.
- State Newton’s experimental law on impact.

__PART – B__

**ANSWER ANY FIVE QUESTIONS: (5 X 8 = 40 marks)**

** **

- State and prove Lami’s theorem.
- A uniform plane lamina in the form of a rhombus one of whose angles is 120° is supported by two forces of magnitudes P and Q applied at the center in the directions of the diagonals so that one side is horizontal. Show that if P>Q, then P
^{2}=3Q - State and prove varignon’s theorem on moments.
- Two unlike parallel forces P and Q (P>Q) act at A and B respectively. Show that if the direction of P be reversed, the resultant is displaced through the distance .
- A and B describe circles of radii ‘a’ and ‘b’ with speeds u and v, the motion being the same way round. If the angular velocity of either w.r.t. the other is zero, prove that the line joining them subtends at the center an angle whose cosine is
- A train goes from one station to another moving during the first part of the journey with uniform acceleration ‘’ when the steam is shut off and brakes are applied, it moves with uniform retardation ‘’. if ‘a’ be the distance between the two stations, show that the time taken is .
- If V
_{1}and V_{2}be the velocities at the ends of a focal chord of a projectile’s path and ‘u’ the horizontal component of the velocity, show that . - A ball impinges directly on exactly equal and similar ball B lying on a smooth horizontal table. If e is the coefficient of restitution, prove that after impact, the velocity of B to that A is (1+e) : (1-e).

__PART – C__

__ __

**ANSWER ANY TWO QUESTIONS: (2 X 20 = 40 marks)**

* *

- a) Two weights P and Q are suspended from a fixed point O by strings OA and OB and are kept apart by a

light rod AB. If the strings OA and OB make angle a and b with the rod, show that the angle q which

the rod makes with the vertical is given by

- b) Two like parallel forces P and Q (P>Q) act at A and B respectively. If the magnitudes of the forces are

interchanged, show that the point of application of the resultant on AB will be displaces though the

distance

- a) A non uniform rod AD rests on two supports B and C at the same level where AB=BC=CD. If a weight p

is hung from A or a weight q is hung from D, the rod just tills. Show that weight of the rod is p+q and

that the centre of gravity of the rod divides AD in the ratio 2p+q : p+2q.

- b) A uniform ladder rests at the angle 45° with its upper extremity against a rough vertical wall and its

lower extremity on the ground. If m and m^{1} be the coefficients of friction between the ladder and the

ground and the wall respectively, show then the least extremity towards the wall is

where W is the weight of the ladder.

- a) Two particles of masses m
_{1 }and m_{2 }(m_{1}>m_{2}) are connected by means of a light inextensible string

passing over a light, smooth, fixed pully. Discuss the motion.

- b) A particles of mass m is projected vertically under gravity, the resistance of air being ‘mk’ times

velocity. Show that the greatest height attained by particle is where V is the

termind velocity of the particle and λV is its initial velocity.

- a) Derive the equation of the projectile in the from .
- b) Two smooth spheres of masses m
_{1 }and m_{2}moving with velocities u_{1}and u_{2}respectively in the direction

of the line of centers impinge directly. Discuss the motion. (10 + 10)

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