Applied Mathematics

APPLIED MATHEMATICS. PAPER-I TIME ALLOWED: 3 HOUR MAXIMUM MARKS: 100 Note: Attempt FIVE questions in all, including QUESTION NUMBER 8 which is COMPULSORY. Select at least TWO questions from EACH SECTION. All questions carry equal marks. SECTION-I 1. (a) Using the triangle law of addition of vectors, deduce the Sine formula and the Cosine formula. (b) If 4= Sin (Kr) then show that 2y+K2 Y =0 r 2. (a) Prove that the system of spherical polar coordinates is an orthogonal system. (b) Evaluate the surface integral ?-a r.ds where s is a closed r³ surface and a is constant. 3. (a) State the laws of friction. A. body weighing 40 lb. is resting on a rough horizontal plane and can just be moved by a force of 10 lb. wt. acting horizontally. Find the coefficient of friction. (b) A weightless tripod, consisting of three legs of equal length l, smoothly jointed at the vertex, stands on a smooth horizontal plane. A weight W hangs from the apex. The tripod is prevented l joining the midpoints of the legs. Show that the tension in each string is v2 W 2 3 v3 SECTION-II 4. (a) A particle is moving along the parabola x2 4ay with constant speed v. Determine the tangential and the normal components of its acceleration when it reaches the point whose abscissa is v5 a. (b) (i) Define ;Simple Harmonic Motion. (ii) A particle moves in a straight line with acceleration kv3. If its initial velocity is u, find the velocity and the time spent when the particle has traveled a distance x. 5. (a) An aero plane is flying with uniform speed vo in an arc of a vertical circle of radius a, whose center is at a height h vertically above a point 0 of the ground. If a bomb is dropped from the aero plane when at a height Y and strikes the ground at 0, show that Y satisfies the equation: ga2 KY2+Y(a2—2hK)+K(h2-a2)=O, where K=h+ ———------2v02 (b) Show that when a particle moves under a central force, the areal velocity is constant.

6. (a) State Kepler’s laws. Prove that the speed at any point of a Central Orbit is given by vp=h, where h is the areal speed and p is the perpendicular distance from the centre of force of the tangent at that point. Hence fine the expression for v when a particle, subject to the inverse square law of force, describes (i) an elliptic, (ii) a parabolic, (iii) a hyperbolic Orbit. (b) ‘Find the moment of inertia of a uniform solid sphere of mass m and radius a. 7. (a) State Poisson’s Hypothesis. (b) Write notes on: (i) Collision of two spheres and Coefficient of Restitution. (ii) Moments and Products of Inertia. (iii) Rectilinear Motion. COMPULSORY QUESTION Write only correct choice in the answer book. Do not reproduce the questions: (1) A set of vectors a1,a2,a3…..an is said to be linearly independent if the vector equation ?a1+µa2+van=0 has: (a) One or two trivial solutions. (b) Only the trivial solution ?=0, µ=0…,v=0. (c) Three or more non-trivial solutions. (d) None of these. (2) If a and b are parallel or anti parallel, then; (a) a x b ? 0 (b) a x b = -b x a = b (c) a x b = 0 (c) .None of these. (3) If a, b, c are coplanar, then b x c will be: (a) Normal to the plane in which a, b, c will lie. (b) Not normal to the plane in which a, b, c will lie. (c) Parallel to the plane in which a, b, c will lie. (d) None of these. (4) Let B be vector field defined in a region of space R and let it and its partial derivatives be continuous at points of R. If div B = 0 at all points of R, then: (a) B is said to be ir-rotational in R.

(b) B is said to be sole-noidal in R. (c) None of these. (5) A particle of mass m moves in a circle of radius r with constant speed v, the force F acting on the particle is: mV mV2 (a) F= ——— (b) F= —— r2 r2 mv2 (c) F = ———- (d) None of these. r (6) Every set of particles has: (a) more than two centres of mass (b) One and only one center of mass (c) Two centres of mass (d) None of these (7) A stone is let ffall freely from a height of 100 ft. The time it takes on reaching the ground is: 375 (a) — Sec. (b) — Sec. (c) — Sec. (d) none of these. 242 (8) The rate of increase of the kinetic energy of a particle is equal to (a) the power applied to the particle. (b) The momentum of the particle. (c) None of these. (9) The orbit of a particle moving under a central force is necessaril a: (a) space curve (b) plane curve (c) none of these (10) The centre of mass of a hallow right circular curve of semi-vertical angle a and height h is: 572 (a) —h (b) —h (c) —h (d) None of these. 433 (11) Let V be a Vector, then if div v ?0, then V is called: (a) Solenoidal vector. (b) Irrotational vector. (c) None of these. (12) Let r be a position vector of a particle, then:

(a) div r=3 (b) div r=2 (c) div r=0 (d) None of these. (13) Cylindrical coordinate system is: (a) an orthogonal coordinate system. (b) Not an orthogonal coordinate system. (c) None of these. dt. (14) let t be a unit vector, then t.----------- is: dt (a) perpendicular (b) parallel (c) None of them. (15) let r be a position vector, then: (a) Curl (rn r) = 1 (b) Curl (rn r)= 3 (c) Curl (rn r) =0 (d) None of these (16) Centre of mass of a hallow right circular cone of semi-vertical angle a and height h is: 321 (a) —h (b) —h (c) —h (d) None of these. 433 (17) Homiltonian of the system of particles is equal to (with usual notation): (a) T+V (b) T-V (c) None of these. (18) when a particle moves under a central force, the areal velocity is (a) constant (b) variable (c) None of these. (19) let F is a variable force, then the work done is: (a) ?F.d r (b) ?F´ d.r (c) ?Fdr (d) None of these. (20) Centre of mass of a thin rod of length l is: lI (a) – (b) – 43 I

(c) – (d) None of these. **************************