Introduction Meter: distance light travels in vacuum during time interval of 1/299,792,458s Second: time for 9,192,631,770 wavelengths of radiation to be emitted by a cesium atom Kilogram: mass of a sphere carved from a single crystal of silicon-28 Prefix Pico Nano Micro Milli Centi Deci Kilo Mega Giga Tera
Symbol p n µ m c d k M G T
Sci. Not. 10-12 10-9 10-6 10-3 10-2 10-1 103 106 109 1012
Parallax: apparent shift in the position of an object when it is viewed from different angles Graphing: (dependent variable; x) as a function of (independent variable; y) Quadratic: y=ax2+bx+c Inverse: y=a/x contact force objects touch long range force gravity, magnets agents source of the force Period & Frequency Period (T): time needed to repeat one complete cycle of motion Frequency (f): number of cycles that take place in one second; 1 cycle/s = 1 Hz Period of mass oscillating on a spring is T=2π√(m/k) m: mass of the oscillating object k: spring constant (if not known, can be found using k=F/x) F: force applied to spring x: distance the spring extends Period of pendulum is T=2π√(l/g) l: length of pendulum g: acceleration due to gravity (9.80 m/s2) Force of compressed or stretched spring F = kx F: force k: spring constant x: amount spring has been stretched or compressed Scalar and Vector quantities Scalar Quantity: quantity that tells you only magnitude (mass, distance and temperature) Vector Quantity: quantity that has magnitude and direction (displacement and velocity) Distance and Displacement Distance is the total length an object has moved and has no direction associated with it. Displacement is the straight-line distance between two points, and is a vector which points from an object's initial position towards its final position. (If you end where you started, displacement is 0) Speed and Velocity Average Speed: amount of distance a moving object covers divided by the amount of time to cover that distance. v=∆d/∆t
Acceleration Average acceleration: rate of change of velocity between t0 and t1 a=v1-v0/t1-t0 Equations of motion Velocity with constant acceleration v1=v0+at Final position with constant acceleration d=do+1/2(v0+v1)t; d=do+vot+1/2at2 Final velocity with constant acceleration v12=v02+2a(d1-d0) Projectile motion Ignoring air resistance, horizontal velocity of a projectile is constant vx=∆d/t=constant Initial vertical velocity of a projectile launched horizontally is zero voy=0 Final velocity found using vfy=-gt Gravitational Field Strength A gravitational field exists for all objects that have mass. At the Earth's surface Earth’s GFS is g=9.80m/s2. Weight is the force of gravity acting on a body. Weight is the product of the mass of an object and the acceleration due to gravity that is acting on an object w=mg. Mass is a constant that does not change with location. Weight is not a constant and does change depending on the location of the object Newton’s Law of Universal Gravitation States that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. F=G(m1m2/r2) F: gravitational force in Newtons G: universal gravitational constant 6.67x10-11 N·m2/kg2 m1, m2: the 2 masses r: the distance between the centres of the 2 masses Newton’s Laws of Motion 1. Newton’s First Law of Motion (“Law of Inertia”) A body at rest stays at rest, and a body in motion stays in motion, unless it is acted on by an external force Inertia: tendency of an object to resist change Equilibrium: when the net force on an object is at 0; occurs in state of rest or constant velocity 2. Newton’s Second Law of Motion (“Law of Force and Acceleration”) A force acting on a mass causes that mass to accelerate in the direction of the force. The acceleration is directly proportional to the force and inversely proportional to the mass. 3. Newton’s Third Law of Motion (“Law of Action and Reaction”) For every action force, there is an equal and opposite reaction force. Action-reaction pairs act on different bodies, and they do not cancel each other. (Force a book exerts on a table and the force a table exerts on a book). [Whenever one body exerts a force on a second body, the second body exerts an oppositely directed force of equal magnitude on the first body] Friction Kinetic Friction: The friction between two surfaces when one surface is moving relative to the other fK: force of kinetic friction Static Friction: The friction between two surfaces when the surfaces are not moving relative to each other fs: force of static friction Force of friction = fx = µFN µ: coefficient of friction FN: normal force (the reaction force to an object's weight) µ = fx/FN fx = µ(FN)
Dynamics in One Dimension Momentum The momentum of an objects is the product of its mass and its velocity p=mv p: momentum in kgm/s, m: mass in kg v: velocity in m/s. Change in momentum: ∆p=∆(mv) Impulse The product of the average net force exerted on an object and the time interval over which the force acts. Impulse = F∆T F: average force in Newtons t: time that the force is applied. Impulse and Momentum The impulse given to an object is equal to the object's change in momentum. F∆t=∆(mv) The law of conservation of momentum Momentum for any closed system upon which there is no net external force remains the same. pi=pf Pai+Pbi=Paf+Pbf mava+mbvb=mavaf+mbvbf Elastic Forces Hooke’s Law The force needed to compress or stretch a string is directly proportional to the product of the spring constant, k, and the amount the spring has been stretched or compressed. F=k∆x F: applied (or restoring) force in Newtons k: spring constant in N/m ∆x: distortion of the spring in m Work, Energy, & Power Work Work is equal to the component of a force that is in the direction of the motion of an object & the displacement of the object W=Fcosθd Power Power is the rate of doing work or the rate at which energy is transferred; measured in watts; 1 W = 1 J/s Power = work/time P = Fcosθd/t Energy: property of an object that allows it to produce change in the environment or in itself. The work-energy theorem states that the change in energy of an object is exactly equal to the work done on it. W=∆E Kinetic Energy = KE = ½(mv2) Gravitational Potential Energy = PE = mgh where m is the mass, g the acceleration due to gravity, h is the height the object is above a reference plane. Gravitational Potential Energy (PE or Ug): PE is the stored energy in a system resulting from the gravitational interaction between masses. When we state an object’s PE we must say which plane we are using as the reference plane.PE=mgh Mechanical Energy: sum of the kinetic and potential energy Ei = Ef mghi + ½(mvi2) = mghf + ½(mvf2) Efficiency = (work output/work input) x 100%
Collisions & Explosions Inelastic Collisions • Objects deform or stick together • Kinetic energy is converted to heat • Less kinetic energy after the collision • Momentum is conserved mava+mbvb=mava’+mbvb’ Elastic Collisions • No deformation • Kinetic energy is conserved mava+mbvb=mava’+mbvb’ ½mava2+½mbvb2=½mava’2+½mbvb’2 Explosions • Object is at rest or moving at a constant velocity • It breaks into 2 or more pieces • The kinetic energy of the system increases Temperature and Thermal Energy Thermal Energy (heat)-total amount of energy of all the particles. Kinetic Molecular Theory All matter is made up of tiny particles that are in constant motion. Particles moving faster have more kinetic energy. Thermal Energy: The overall energy of motion of the particles that make up an object Conduction: heat energy is transferred to the lower temperature; when the two are at the same temperature, thermal equilibrium is reached. Convection: transfer of heat by the process of motion in a fluid. Radiation: transfer of energy by electromagnetic waves; does not require presence of matter. Kelvin: 0.0 K is absolute zero, and equal to -273 oC Specific heat Amount of energy that must be added to a material to raise its temperature 1.0 K (1.0 oC) Q=mC∆T where m is the mass, C is the specific heat and ∆T is the change in temperature. Change of state Melting Point: temperature at which a substance changes from a solid to a liquid Boiling Point: temperature at which a substance changes from a liquid to a gas All of the energy added goes to overcoming the forces that hold the molecules together. The temperature does not change. Heat of Fusion: amount of energy needed to melt one kilogram of a substance Q=mHf where m is the mass of the substance and Hf is the heat of fusion Heat of Vaporization: amount of energy needed to vaporize one kilogram of liquid Q=mHv where m is the mass of the substance and Hv is the heat of vaporization. Calorimetry Calorimeter is a carefully insulated container that does not allow heat to leave or enter Tf = (mACATAi + mBCBTBi) / mACA+mBCB
First Law of Thermodynamics The total increase in the thermal energy of a system is equal to the work done plus the heat added. QTOTAL = W + Q Heat engine: device that can convert thermal energy into mechanical energy heat reservoir QH heat engine
work
QL
QH = W + QL
cold reservoir Heat pump heat reservoir QH heat engine
work
QL
QH = W + QL
cold reservoir Second Law of Thermodynamics Entropy: the state of disorder Natural processes go in a direction that maintains or increases entropy. Wave properties Wave: rhythmic disturbance that carries energy through matter or space Mechanical waves carry energy through matter 1. Transverse wave: a mechanical wave in which the medium moves perpendicular to the direction on the wave. Transverse waves cannot occur in gases. 2. Longitudinal wave: a mechanical wave in which the medium moves in the same direction as the wave. Longitudinal waves can occur in solids, liquids and gases. All waves can be represented using a sine wave: λ wavelength
Below the surface, ocean waves are transverse waves. At the surface they have characteristics of both transverse and longitudinal waves. The speed of a wave is calculated by either: a. dividing the distance the wave travels by the time taken. v=∆d/∆t. b. multiplying its wavelength by its frequency. v= λf The period of a wave is the time for one complete cycle and it is equal to T=1/f
Waves at boundaries When a wave hits a boundary some of the energy is transmitted through the boundary and some is reflected back. The incident wave is the wave that hits the boundary; the reflected wave is the wave that comes back. If the incident wave hits a more dense medium (one in which the wave travels faster), the reflected wave will be inverted. Superposition of waves The principle of superposition states that the displacement of a medium caused by two or more waves is equal to the algebraic sum of the two individual waves When the displacement of the waves is in the same direction, they produce constructive interference When the displacement of the waves is in the opposite direction, they cause destructive interference If waves travel through a medium at just the right speed they appear to be stationary. These are called standing waves. The part of the medium that does not move is called the node. The part that has the maximum deflection is called the antinode. Waves in two dimensions θi = angle of incidence θr = angle of reflection θi=θr (law of reflection) Diffraction and interference of waves: the spreading of waves around a barrier Doppler Effect When objects travel towards you the sounds they make seem to have a higher pitch & when objects travel away from you the sounds they make seem to have a lower pitch The frequency of the sound produced by an object moving toward an observer appears to be fo = fs[1-(vs/v)] where fs is the frequency of the source, vs is the velocity of the source, and v is the velocity of sound If the object is moving away from the observer the frequency appears to be fo = fs[1+(vs/v)] Light fundamentals Light is the range of frequencies of electromagnetic waves that stimulate the retina of the eye. They range from about 4.00x10-7m to 7.00x10-7m. Shortest is violet, longest is red. Light travels in a vacuum at 3.00x108m/s and is represented by c. Sources of light: An object that emits light is called luminous; an object that reflects light is illuminated Luminous Flux (P): rate at which visible light is emitted from a source. Unit: lumen (lm) Illuminance (E): amount of light falling on a surface; measured in lux (lx): lm/m2 E=P/4πd2 Luminous intensity: Some light sources are specified in candela (cd), or candle power. A candela is not a measure of luminous flux, but of luminous intensity. The luminous intensity of a point source is the luminous flux that falls on 1.0m2 of a sphere of one meter radius. Thus, luminus intensity is luminous flux divided by 4π; P/4π Light and matter Light: wave, particle, or electromagnetic radiation Transparent: materials that transmit light rays without distorting them Translucent: materials that transmit light rays but do not allow objects to be seen clearly Opaque: light cannot pass through
red magenta
Spectrum: the ordered arrangement of colours from violet to red Colours by addition white light can formed by combining red, green, & blue light
yellow white
blue
Colours by Subtraction Dye: a molecule that absorbs certain wavelengths of light & transmits or reflects others; form solution Pigment: a coloured particle that absorbs certain colours & transmits or reflects others; form suspension • Primary Pigment: a pigment that absorbs only 1 primary colour from white light
cyan
green
•
o
Yellow, cyan, & magenta are primary colours of pigments
Secondary Pigment: absorbs 2 primary colours & reflects 1 o Red, green, blue Wave Motion and Geometrical Optics
Index of refraction is the rate of the speed of light in a vacuum to its speed in a material nsubstance = c/vsubstance Snell’s Law States that the ratio of the sine of the angle of incidence to the sine of the angle of reflection is a constant n = sinθi/sinθr nisinθi = nrsinθr Critical Angle and Total Internal Reflection Total internal reflection occurs when light passes from a more optically dense medium to a less optically dense medium, at an angle so great that there is no refracted ray The critical angle is the angle at which the refracted ray lies along the boundary of the 2 materials Mirage When light rays pass through at different temperatures the light is bent away from the normal. When the air temperature at the surface of the Earth is much hotter than the air temperature in the sky, light coming from the sky is bent. Regular & Irregular Reflection Regular reflection mirror; you can see an image Irregular reflection wall; you can’t see an image Lenses & Mirrors Image orientation: an image is erect when it has same orientation as object, otherwise it is inverted Virtual image: light does not converge; cannot be projected on a screen Mirror Shape
Object
Plane
any more than 2 focal lengths away from mirror b/w 1 and 2 focal lengths at the focal point b/w mirror and focal point any
Concave convex
Image Orientation upright
Virtual or Real virtual
same as object
upside-down
real
smaller than object
upside-down none upright upright
real none virtual virtual
larger than object none larger than object smaller than object
Lens/mirror equation 1/f = 1/di + 1/do Magnification m = hi/ho = di/do Lenses • 2 focal points • f is pos. for convex & neg. for concave lenses
• • •
di is pos. on the image side of the lens where images are real do is pos. on the object side of the lens concave lens: image on the same side as the object
Special Relativity Einstein’s Theory of Special Relativity describes motion of particles moving at close to the speed of light
Size
2 postulates 1. the speed of light is the same for all observers, no matter what their relative speeds 2. laws of physics are the same in any inertial (non-accelerated) frame of reference; this means the laws of physics observed by a hypothetical observer travelling with a relativistic particle must be the same as those observed by an observer who is stationary Time Dilation ∆t = ∆to / √ [1-(v2/c2)] ∆to dilated time measured by an observer who is at rest with respect to the event v relative speed between the 2 observers c speed of light Length Contraction Lengths in the direction of relative motion are shorter for the observer who is moving the amount of length L = Lo √ [1-(v2/c2)] L is the length measured by the observer who is in motion Lo is the proper length as measured by the observer who is at rest relative to the object Relativistic Momentum P = mv / √ [1-(v2/c2)] Relativistic Addition of Velocities U = (v + U ’) / [1+(v2/c2)] U is the speed of the missile relative to the observer who is in motion relative to the rocket v is the velocity of the rocket relative to the moving observer U ’ is the velocity of the missile relative to the rocket Mass Energy Equivalency E = mc2 / √ [1-(v2/c2)] (not in motion)
Eo = mc2 m = Eo/c2
Radioactivity: the remission of energy or particles from an unstable nucleus; can omit: 1. alpha particles ά helium nuclei 24He 2. beta particles β high speed electrons -10e3. gamma rays γ electromagnetic radiation; highest frequency Nuclear Fission/Fusion When nuclear fission occurs, a nucleus splits into smaller nuclei; a chain reaction can occur. Nuclear fusion occurs when light nuclei collide at high temperature & form heavier nuclei, releasing a high amount of energy
