Download Vector Position of Particle - Physics - Exam Paper and more Exams Physics in PDF only on Docsity! 5 Question 1 (Marks 12) (a) The vector position of a particle varies in time according to the expression r = 3.00 î – 6.00t2ĵ . Find: (i) Expressions for the velocity and acceleration as functions of time. (ii) Determine the particle's position and velocity at t = 1.00 s. (b) A particle initially located at the origin has an acceleration of a = 3.00 ĵ m.s–2 and an initial velocity of vi = 5.00 î m.s–1. Find: (i) The vector position and velocity at any time t and (ii) The coordinates and speed of the particle at t = 2.00 s. Question 2 (Marks 20) (i) Mechanical Energy is defined as the sum of kinetic and potential energy. State clearly the conditions under which mechanical energy is conserved. (ii) State clearly the conditions under which momentum is conserved. (iii) One end of a light, inextensible string, length R, is attached rigidly to a point in the ceiling. The other is attached to a mass m, whose size is very much less than R, so you may treat it as a particle. Initially, the string is extended horizontally and straight so that the mass touches the ceiling, as shown in the sketch at left (not to scale). The floor is distance R below the ceiling. Directly below the point of attachment in the ceiling lies another small mass m. Friction between the masses and the floor is negligible. floor R Rm m ceiling D floor ceiling The mass on the string is released. It travels in an arc, does not quite touch the floor, but strikes the second mass. Air resistance is negligible. The mass that swung down from the ceiling is made of sticky clay and, during and after the brief collision, the masses stick together. Together, they swing upwards in an arc and rise to a height D above the floor. Showing all working, derive an expression for D. In your derivation, state clearly any approximations you make and any principles you use. 6 Question 3 [Marks 18] The sketch shows, from two views, a simple pulley system for raising large masses. A light, inextensible cord attached to a mass m is wound, without slipping, on a cylinder of radius r. Rigidly joined to this cylinder and mounted on the same central shaft is a cylinder of radius R. The friction between the cylinders and the shaft is negligible. The two cylinders together have mass M and moment of inertia, I. Around the larger cylinder is wound, without slipping, anther light, inextensible cord, as shown. An operator pulls with tension T1 on this cord to raise the mass m. Rr T m M m T R r shaft T2 1 1 I The mass m accelerates upwards with vertical acceleration a. (i) Using Newton's laws or otherwise, write an expression for T2, the tension in the cord attached to m, terms of m, g and a. (ii) Using Newton's laws for rotation or otherwise, derive an expression for the angular acceleration, α, of the cylinders, in terms of variables given in the diagram. (iii) Showing all working, derive an expression for the acceleration a of mass m in terms of some or all of the parameters T1, m, M, I, r, R and the gravitational acceleration g.
