momentum p = m v m v Law of conservation of momentum * isolated system, sum of external forces acting on system is zero  F = 0 * collision or explosion *  (initial momentum) =  (final momentum) *  (initial momentum) k =  (final momentum) k subscript k represents x, y or z component of momentum before (initial) during Momentum and Conservation 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 A x = A cos  A y = A sin  A = A x 2 + A y 2 tan  = A y / A 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 cos  F( r ) = variable W P1-->P2 =  F (r). d r =  F(r) cos  dr (integration limits r 1 to r 2 ) The total work done by a number of forces is W total =  W i When work is done on a system to overcome inertia, it goes into kinetic energy K. The total work done W total on the system produces a change in its kinetic energy  K. K = ½ m v² W total =  K F  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) a observer = 0  F = 0  a = 0 v = 0 v = constant

Newton’s Second Law of Motion * Force F - agent of change (SI unit: netwon, N) a =  F / m a observer = 0 SYSTEM internal environment, mass m system boundary external environment interaction SYSTEM Disturbance  F 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. a observer = 0 SYSTEM A Interaction between systems A and B SYSTEM B A acting on B: F BA B acting on A: F AB F BA = - F AB F BA = F AB

Impluse J (SI Unit N.s or kg.m.s -1 ) Impulse  Change in Momentum 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  d= m 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+  /2) 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 + Two monochromatic & coherent light sources Waves in phase  constructive interference  bright fringe Waves out of phase  destructive interference  dark fringe  d = m  d = (m+½)

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. FGFG FBFB Object: mass m, weight F G,, volume V, density   = m / V m = V  F G = m g Volume of water displaced V d = Volume of object submerged V s Fluid density  F Weight Buoyant force

Newton's Second Law: F B + F G = m a  a = F B / m - g F B = weight fluid displaced = V d  F g = V s  F g  a = (V s  fluid g/ V  } - g = g { (Vs / V). (  F /  ) - 1} Object partially submerged and floating ---> a = 0 V s = (  /  F )V  greater the density of the object compared to fluid then the greater the volume of the object submerged. Object fully submerged: a = 0  =  F object floats under water a > 0  <  F object rises to surface a  F object sinks to the bottom

OBJECT MOVING WITH A CONSTANT ACCELERATION (one dimensional motion) a = constant +X time t = 0 time t initial velocity, u final velocity, v displacement, s v = u + a t s = u t + ½ a t 2 v 2 = u a s average velocity = s / t = (u + v) /2

OSCILLATORY MOTION 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  = 2  f = 2  / T Mass / spring systems, vibrations, waves, sound, light, colour,...

Simple Harmonic Motion SHM Hooke’s Law and mass / spring systems Hooke’s Law F = - k x 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 x 2 SHM m a = - k x a = - (k / m) x a = -  2 x  2 = k / m  T = 2   (m / k) x = A cos(  t + f ) v = A  sin(  t + f ) a = - A  2 cos(  t + f ) = -  2 x mass, m spring constant, k Amplitude of oscillation, A