Mechanics — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Part of the questions to be attempted together. c). (i). Any force system in the fundamental plane may be reduced either to a single force or to a couple. (ii). Find the co-ordinates of the center of the gravity of a quadrant of the area of the curve x2/3 + y2/3 = a2/3 bounded by the axes of co-ordinates. a). State and prove Pappus Theorem for the mass center of a uniform listribution along a plane curve. Hence, find the mass center of a wire bent int he form of an isosceles right-angled triangle b). A light ladder is supported on a rough floor and leans against a smooth wall. How far up the ladder can a man climb without slipping taking place? c). (i) Show that a field of force with components (X, Y) is conservative if and only if, (i). Forces P, Q, R act along the lines x = 0, y = O, x.cosety.sine =p Find the magnitude of the resultant and the equation of its line of action? a). Prove that for a particle moving in a plane and occupying the position P(r, 0)) at a time t, the radial and transverse components of velocity are (it r de) and radial and transverse components of acceleration are b). (i) Prove that the motion of the center of mass of a system of particles is equivalent to the motion of a single particle of mass equal to the total mass of the system, subject to the net external force acting on the system. (ii) State D'Alembert's Principle. Explain why D'Alembert's principle gives the impression that it reduces dynamics to statics. c). A damped harmonic oscillator is subjected to an external periodic force, and its motion is governed by the equation: where m is the mass, fu is the damping coefficient, p is the natural frequency, k is the amplitude of the external force per unit mass, and c is the angular frequency of the driving force. Derive the particular solution of the above equation for steady-state motion. Hence, show that when the driving frequency equals the natural frequency i.e., c = p, the solution reduces to a). Derive the general equation of the orbit under a central inverse-square law force, and show that the orbit is a conic section given by where / is the semi-latus rectum, and e is the eccentricity. Also, state what types of orbits are represented by the conditions e < 1, e = 1, and e > 1. b). Derive the equation of path of a particle which is projected in a medium having no resistance. Show that the path is a parabola. Find its vertex and latus rectum. Also find its maximum horizontal range. c). Define an inertial frame of reference. Show that the laws of motion are invariant under transformation from one inertial frame to another moving with uniform translational velocity. Given one inertial frame of reference, how many inertial frames of reference can exist? Justify your answer.