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Momentum and Conservation momentum m p = mv v Law of conservation of momentum * isolated system, sum of external forces acting on system is zero S F = 0 * collision or explosion * S(initial momentum) = S(final momentum) * S(initial momentum)k = S(final momentum)k subscript k represents x, y or z component of momentum before (initial) during After (final) CENTRE OF MASS Centre of Mass (CM) - point where the total mass of a collection of objects can be regarded as being located. Diver jumping into a swimming pool: The motion of the CM can only change with the application of an external force, and no new force is applied, the CM of the diver moves in a parabolic path. VECTORS Ax = A cosq Y Ay = A sinq Ay A A = Ax2 + Ay2 tanq = Ay / Ax q Ax X WORK W (SI unit: joules J) - measure of the amount of energy transferred into or out of a system by the action of a single applied (external) force acting through a distance: F = constant W = F d cosq F(r) = variable WP1-->P2 = F(r) . dr = F(r) cosq dr (integration limits r1 to r2) The total work done by a number of forces is Wtotal = SWi When work is done on a system to overcome inertia, it goes into kinetic energy K. The total work done Wtotal on the system produces a change in its kinetic energy DK. F Wtotal = DK K = ½ m v² q r Newton’s First Law of Motion * Defines an inertial frame of reference observer’s acceleration is zero * Force F - agent of change (SI unit: netwon, N) SF=0 a=0 aobserver = 0 v=0 v = constant Newton’s Second Law of Motion * Force F - agent of change (SI unit: netwon, N) a = SF/m external environment interaction system boundary SYSTEM internal environment, mass m aobserver = 0 SYSTEM Disturbance SF mass m response a Newton’s Third Law of Motion Forces always act in pairs, interacting between two systems such that the two forces have the same magnitude but are opposite in direction. aobserver = 0 Interaction between systems A and B FBA = - FAB SYSTEM A SYSTEM B FBA = FAB B acting on A: FAB A acting on B: FBA Impluse J (SI Unit N.s or kg.m.s-1) Impulse Change in Momentum t2 J F dt mv2 mv1 t1 An impulse exerted by a racquet changes a ball's mometum. The greater the contact time between the racquet and ball, the longer the force of the racquet acts upon the ball and hence the greater the ball's change in momentum and hence greater speed of ball flying away from the racquet. Contact J > 0, balls momentum (speed) increases NO contact J = 0 0, balls momentum (speed) is constant SUPERPOISION PRINCIPLE When two or more disturbances of the same kind overlap, the resultant amplitude at any point in the region is the algebraic sum of the amplitudes of each contributing wave. The Principle of Superposition leads to the phenomena known as interference. For example, assume that there are two monochromatic and coherent light sources (waves of a single frequency which are always "in-step" with each other). The waves from each source reaching arbitray points within a region will have traveled different path lengths and therefore will have different phases. At some points the waves will be in phase (in step difference in pathlengths Dd= ml m = 0, 1, 2, ...) and reinforce each other giving maximum disturbance at that point constructive interference. At other points, the two waves will be out of phase (out of step - difference in path lengths = (m+1/2)l m = 0, 1, 2, ...) and cancel each other - destructive interference. This region is characterized by bright and dark areas called interference fringes. Superposition and Interference for light Waves out of phase destructive interference dark fringe Dd = (m+½)l Two monochromatic & coherent light sources + Dd = m l Waves in phase constructive interference bright fringe ARCHIMEDES PRINCIPLE AND BUOYANCY An object immersed in a fluid will be "lighter", that is, buoyed up by an amount equal to the weight of the fluid it displaces. Object: mass m, weight FG,, volume V, density r r = m / V m = V r FG = m g FB Buoyant force Fluid density rF Weight FG Volume of water displaced Vd = Volume of object submerged Vs Newton's Second Law: FB + FG = m a a = FB / m - g FB = weight fluid displaced = Vd rF g = Vs rF g a = (Vs rfluid g/ V r } - g = g { (Vs / V) . (rF / r) - 1} Object partially submerged and floating ---> a = 0 Vs = ( r / rF )V greater the density of the object compared to fluid then the greater the volume of the object submerged. Object fully submerged: a = 0 r = rF object floats under water a > 0 r < rF object rises to surface a < 0 r > rF object sinks to the bottom OBJECT MOVING WITH A CONSTANT ACCELERATION (one dimensional motion) a = constant displacement, s +X time t = 0 initial velocity, u v=u+at s = u t + ½ a t2 v2 = u 2 + 2 a s average velocity <v> = s / t = (u + v) /2 time t final velocity, v OSCILLATORY MOTION Mass / spring systems, vibrations, waves, sound, light, colour, ... Period, T time for a complete cycle or oscillation (SI unit: second, s) Frequency, f number of cycles (oscillations) in one second (SI unit: hertz Hz) Angular frequency “angle swept out” in a time interval (SI unit: radian/second rad.s-1) amplitude max deviation of disturbance from equilibrium position (for simple harmonic motion, amplitude is independent of period or frequency) T=1/f f=1/T w=2pf=2p/T Simple Harmonic Motion SHM Hooke’s Law and mass / spring systems Hooke’s Law F = - k x spring constant, k F restroing force acting on mass by spring when the extension of the spring from its natural length is x Potential energy of system U = ½ k x2 SHM ma=-kx mass, m Amplitude of oscillation ,A a = - w2 x a = - (k / m) x w2 = k / m T = 2 p (m / k) x = A cos( w t + f ) v = A w sin( w t + f ) a = - A w2cos( w t + f ) = - w2 x STANDING WAVES ON A STRING (WIRE) t time, T period, A antinode, N node t = 0, T , 2T , 3T , ... t = T /2, 3T /2, 5T /2, ... A t = T /4, 3T /4, 5T /4, ... m mass of string L length of string T string tension v speed of wave along string l resonance wavelength of standing wave on string f resonance frequency of vibration of string linear density of string N mode number (integer) = m /L l = 2L / N N A N A N N = 1, 2, 3, .... v = f l = (T / ) f = (N /2L). (T / m )