U1 Physics Notebook PDF

1 Unit 1 Physics Notebook of Sarah Sellier  2 Table of Contents Experiment Information 9 Graphs: 9 Quantities and Units...

1 Unit 1 Physics Notebook of Sarah Sellier  2 Table of Contents Experiment Information 9 Graphs: 9 Quantities and Units 10 SI Base Units 10 Prefixes 10 Submultiples 10 Multiples 11 Definitions of Quantities 12 The Mole (Avogadro’s Constant) 12 Derived Quantity 12 Dimensional Analysis 12 Importance of Dimensional Analysis 12 Significant figures and Decimal Places 14 Multiplication and Division 14 Addition and Subtraction 14 Errors and Uncertainty 14 Experimental Error 15 Precision and Uncertainty 15 Accuracy vs Precision 15 Human Reaction Time 16 Methods for Addressing Human Reaction Time: 16 Treating Errors 16 Random and Systematic Errors 16 Vectors 17 Resultant Vectors 18 Motion 19 Quantities Describing Motion 19 Equations of Motion 19 Graphs Describing Motion 20 Generic Situations 20 A Bouncing Ball 21 Projectile Motion (Parabolic Motion) 22  3 Newton’s Laws 23 The First Law 23 The Second Law 23 The Third Law 23 Uses of F = ma 23 The Lift Problem 23 A Body on an Inclined Plane 24 Principles Derived From Newton’s Laws 25 The Principle of Conservation of Linear Momentum 25 Newton’s Experimental Law of Impact 25 Impulse 25 Why Cars Crumple 25 Terminal Velocity 26 Archimedes’ Principle 26 Upthrust 26 Resistive Forces 27 Frictional force: 27 Drag Force and Air Resistance 27 Centre of Gravity of an irregular shaped Lamina 27 Forces in Equilibrium 28 The Ladder and the Wall 28 The triangle of forces 28 Moments 28 Circular Motion 29 To find Linear Velocity: 29 Centripetal Acceleration of a Body 29 Centripetal Force 30 Horizontal Circle 30 Vertical Circle 30 The Conical Pendulum 31 Moments 32 Couple 32 Work, Energy and Power 32 Energy Conversion 32 Examples of Energy Conversion 33 Energy Conservation 33 To Conserve Energy 33 Alternative Sources of energy in the Caribbean 34  4 Work 35 Potential Energy 35 Gravitation 35 Newton’s Law of Gravitation 36 Equipotential Lines 36 Satellites 37 Geostationary 37 Global Positioning (GPS) 37 Gravitational Potential 37 Gravitational Field 38 Gravitational Field strength 38 Kepler’s 3rd Law 38 Simple Harmonic Motion 40 NB: 40 Examples of Simple Harmonic Motion 40 *Personal Notes 41 Graphs and Proofs 41 Variation of ‘v’ with displacement, ‘x’ rather than time ‘t’ 42 Interchange of Potential and Kinetic Energy for a Simple Pendulum during SHM 43 For Minimum Displacement 43 For Maximum Displacement 43 A look at energy interchange for each Quarter Cycle 44 variation of Kinetic and Potential Energy with Displacement, x 44 Proving a Body Executes SHM 44 The Simple Pendulum 45 The Mass on the Vertical Spring 45 Mass on two identical springs in series 46 Mass on two identical springs in parallel 46 Resonance and Types of Oscillations 47 Forced Oscillations 48 Resonance 48 Sharpness of Resonance and Frequency Response 49 Examples: 49 Waves 49 Recall: 50 Particles and Waves 51 Electromagnetic Waves and Spectrum 52 Progressive Waves 53  5 Polarization 53 Examples of Polarization 53 Transverse and Longitudinal Waves 53 Stationary Waves 54 Harmonics 54 Stationary and Progressive Waves 55 Light 55 Diffraction Through a Single Slit 56 Diffraction Grating (NB: this is valid for all interfering waves) 56 Young’s Double Slit Experiment 58 Conditions to be Met for this Experiment to be Valid 58 Sound 58 General Properties 59 Concerning Musical Instruments 60 Experiments to Determine the Speed of Sound 61 Kundt’s Tube 61 Resonance Tube 62 Hearing 62 Frequency Response and Intensity 64 In Summary Equations to Note: 65 Lenses 65 Sight (The Human Eye) 65 Definitions 65 Eye Defects 65 A Brief Recapitulation of Thermal Physics 68 Thermometers 68 The Perfect Thermometer 70 The Thermodynamic Scale 71 Absolute Zero 71 Internal Energy 71 Heat Capacity and heat of fusion/vaporization 72 Energy Transfer 73 Modes 73 Conduction 73 Important Definitions: 74 Conductivity of Matter: 74 Lagging 74 Convection 76  6 Radiation 76 The Blackbody 76 Stefan’s Law 76 Net energy Radiated Per Second 77 The Greenhouse effect 77 Applications of thermal Energy Transfer 77 The Vacuum Flask 77 The Solar Water Heater 78 Applications of Conduction 78 Applications of Convection 78 Applications of Radiation 78 Ideal Gases 78 Basic Assumptions for the Kinetic Theory of Gases 79 Equations Describing Gases 80 Relationship Between Pressure of a Gas and the Mean Squared Speed 80 Mean Square Speed 80 Gas Equations 80 Derivation of Formula for Average Translational Kinetic Energy of a Molecule for an Ideal Gas 81 NBs: 81 Thermodynamics 82 Internal Energy 82 Internal Energy in 3 States of Matter 82 Internal Energy at Change of Phase 82 Laws of Thermodynamics 83 First Law 83 Laws and Stuffz 83 First Law 83 Gases 85 Work Done by A Gas (In Expanding) 85 Graphs 85 Molar Heat Capacities of Gases 87 Elasticity 87 Young’s Modulus 89 An Experiment to Determine Young’s Modulus 89 Graphs 91 Finding Strain Energy from Graphs 92 7  8 Module 1  9 Experiment Information Graphs: 𝑦 = 𝑚𝑥 Straight line starting at origin 𝑦 = 𝑚𝑥 + 𝑐 Straight line with a y intercept 𝑛 Exponential curve. This can be converted to linear form using 𝑦 = 𝐴𝑥 (logging everything) 𝑙𝑜𝑔𝑦 = 𝑙𝑜𝑔𝐴 + 𝑛 𝑙𝑜𝑔 𝑥 logarithms, such that it is put into the form y=mx+c. It is now in the form Y = mX + C where Y = logy, m = n, x = logx and c = logA. NB: 𝑛 = 𝑙𝑜𝑔𝑦/𝑙𝑜𝑔𝑥 Y = asinx or y = acosx The maximum value of either of these graphs are given as the multiplier of the trig ratio.  10 Quantities and Units SI Base Units SI Quantity Symbol SI Base Unit Symbol Mass m kilogram k Time t second s Length l metre m Temperature T Kelvin K Electric Current I Ampere A Amount of Substance n mole mol Luminous Intensity Iv candela cd. Prefixes Submultiples Prefix Value Meaning Centi c 10^-2 Milli m 10^-3 Micro μ 10^-6 Nano n 10^-9 Pico p 10^-12  11 Multiples Prefix Value Meaning Kilo k 10^3 Mega M 10^6 Giga G 10^9 Tera T 10^12 T G M k 0 c m μ n p 12 9 6 3 0 -2 -3 -6 -9 -12 Aka remove (difference) amount of zeroes Aka add (difference) amount of zeroes When converting units to a power (e.g. units squared or cubed) multiply the exponents given in this table by the value of that power. Eg: Convert 4.5 x 10^1 cm to m. To convert cm to m ordinarily you would multiply by 10^-2. To convert cm^2 to m^2 you would multiply by not 10^-2 but 10^2(-2) = 10^-4. In compound units where units are in increments of powers of 10 you may find it useful to work out the overall power of 10 which, when multiplied by the original value, will give the answer. (Rather than converting incrementally). Eg 2: Convert 7.68 x 10^2 kg/m^3 to g/cm^3 Kg to g: x 10^3 m^3 to cm^3: x 10^2x3 mass/volume = mass x volume^-1 = 10^3 x 10^-6 = 10^-3 Therefore if you multiply to the original value by 10^-3 it will be converted.  12 Definitions of Quantities The Mole (Avogadro’s Constant) The mole is the amount of substance that contains the same number of particles as there are in 12g of Carbon 12 (carbon 12 has an atomic mass number of 12. Therefore this is equivalent to saying if you have the RAM/RFM in grams of a substance you have Avogadro’s constant of atoms/molecules). 1 mole contains 6.02 x 1023 particles (Avogadro’s constant). Derived Quantity A derived quantity is made by multiplying/dividing base quantities. All equations in physics are homogenous. This means that the units on one side of the equation are equal to the units on the other side of the equation. Dimensional Analysis For the purpose of dimensional analysis these symbols are used to denote the base quantities. Base Physical Quantity Symbol Mass [M] Time [T] Length [L] Temperature [θ] Electric Current [A] Importance of Dimensional Analysis 1. To deduce the dimensions of a derived quantity 2. To ensure the homogeneity of an equation 3. To predict the forms of equations.  13 Example: Derived Quantity Dimension Volume [L]³ speed [LT⁻¹] force [MLT⁻²] NB: Although any equation that is not homogenous is incorrect, a homogenous equation may also be incorrect, as it may fail to account for constants. Example: Equation of motion: S = ut + ½ at2 M = (ms-1)(s)+(ms-2)(s2) M = m+m 𝑎 Problem: (𝑝 + 2 ) 𝑣 Remember you cannot add/subtract any unit that is not the same. This implies that any constant being added/subtracted from a given unit must be equal to that unit. 𝑎 Pressure (P) 2 𝑣 [ 3] a 𝐿 −2 = 𝐹𝑜𝑟𝑐𝑒 𝐴𝑟𝑒𝑎 −6 = −1 −2 a 𝐿 [ ] [𝑀𝐿 𝑇 ] a = 5 −2 [𝑀𝐿 𝑇 ]  14 Significant figures and Decimal Places Multiplication and Division When numbers are multiplied and divided, the final answer has a number of significant figures that equals the smallest number of significant figures in any of the original factors. Eg: 14.72 x 8.36 = 123.0592 = 123 Addition and Subtraction When numbers are added or subtracted the number of decimal places in the result should equal the smallest number of decimal places in any term in the sum. Eg: 123.45 + 5.4 = 128.85 = 128.8 Errors and Uncertainty Suppose the value of some quantity, x is measured in an experiment. If x is subjected to an error, ∆𝑥, then ∆𝑥is said to be the maximum possible error or absolute or actual error and the measurement will be recorded as 𝑥 ± ∆𝑥. The maximum actual error or the maximum relative error can be represented as a fraction or as a percentage. △𝑥 The maximum fractional error is 𝑥 △𝑥 The maximum percentage error (%x) = 𝑥 × 100  15 Experimental Error Human Error When the experimenter makes a mistake; the experimenter’s error. Do NOT quote human error as a source of experimental error. Eg. Incorrectly setting up an instrument, misreading, miscalculation. Systematic Error Bias in measurement (that always affects the result of an experiment in the same direction). Result in readings that are consistently too high/low Can be caused by the environment, method of reading instrument, instrument itself. Can be easily corrected in cases such as zero errors (by rezeroing or subtracting the value given at zero from all readings) Can be eliminated by using a different experimental setup Eg. Incorrect zeroing of an instrument, clock running too fast/slow, not accounting for air resistance when measuring acceleration due to gravity by dropping an object etc. Random Error Comes from two principle sources: ○ No measurement has infinite precision ○ Human reaction time is always non-zero Precision and Uncertainty In a continuous (as opposed to a discrete) measurement some error is always present, due to the finite precision of the instrument used. As this is inherent in the instrument’s properties, that error cannot be reduced through multiple readings. Eg: A ruler is usually precise to 1mm. This means the true measurement is actually the measurement taken ±0.5mm. Accuracy vs Precision The accuracy of an instrument gives the truthfulness of its reading. Although an instrument is precise (can measure to a high degree of certainty) it may still not give the true value of the measurement.  16 Human Reaction Time The finite reaction time of a human is not considered human error, as it is a limitation of a part of the experimental process. When the variation in readings is greater than the precision of the instrument and is of a random interval, this may be the cause. Methods for Addressing Human Reaction Time: Finding an Average The random difference in measurements should “average out” and the mean would be a better estimate of the true value. Graphing Drawing a line of “best fit” helps to find the average of graphed values. If the error is truly random, values should occur equally on either side of the true line. Treating Errors Random and Systematic Errors Operation Symbol Random Errors Systematic Errors Multiplication (xy) %(xy) %x + %y %x + %y Division (x/y) %(x/y) %x - %y 𝑛 𝑛 n(%x) Powers(𝑥 ) %𝑥 Addition (x+y) ∆(𝑥 + 𝑦) ∆𝑥 + ∆𝑦 ∆𝑥 + ∆𝑦 Subtraction (x - y) ∆(𝑥 − 𝑦) ∆𝑥 − ∆𝑦 Examples: 1. An experiment to determine the voltage across a resistor is carried out. The data obtained are 𝑅 = 28. 3 ± 0. 3Ω and 𝐼 = 0. 26 ± 0. 02𝐴. The voltage across the resistor is found by: 𝑉 = 𝐼𝑅 𝑉 = 28. 3 × 0. 26 = 7. 358 𝑉 Error calculations:  17 %𝑉 = %𝐼 + %𝑅 0.02 0.3 %𝑉 = ( 0.26 × 100) + ( 28.3 × 100) %𝑉 = 7. 692% + 1. 06% = 8. 752% 8.752 ∆𝑉 = 100 × 7. 358 = 0. 644𝑉 The voltage should be reported as 7.4 ±0.6 volts. 2. Two masses are weighed separately using different instruments. 𝑚1 = 2. 38 ± 0. 01 𝑘𝑔 and 𝑚2 = 3. 74 ± 0. 02 𝑘𝑔. Total mass, M = 𝑚1 + 𝑚2 = 6. 12 𝑘𝑔 ∆𝑀 = ∆𝑚1 + ∆𝑚2 = 0. 01 + 0. 02 = 0. 03 𝑘𝑔 The total mass should be reported as 6.12 ± 0.03kg. 3. Two masses are weighed separately using different instruments. 𝑚1 = 4. 76 ± 0. 02 𝑘𝑔 and 𝑚2 = 1. 28 ± 0. 01 𝑘𝑔. Difference in mass, M = 𝑚1 − 𝑚2 = 3. 48 𝑘𝑔 ∆𝑀 = ∆𝑚1 + ∆𝑚2 = 0. 01 + 0. 02 = 0. 03 𝑘𝑔 The total mass should be reported as 3.48 ± 0.03kg.  18 Vectors Scalar (magnitude) Vector (magnitude and direction) distance displacement speed Velocity mass acceleration time Weight Work done force Every vector has an x and y component. If you know the overall vector is A Newtons and the angle it is at from the axis is B degrees the y component can be found from: 𝑦 𝐴𝑠𝑖𝑛𝐵 because 𝑠𝑖𝑛𝐵 = 𝐴 While the x component can be found from: 𝑥 𝐴𝑐𝑜𝑠𝐵because 𝑐𝑜𝑠𝐵 = 𝐴 Remember that vectors have both magnitude and direction. If the vector occurs in any quadrant but the first at least one of the components is negative. Resultant Vectors To find the resultant of the vectors A and C illustrated above: Find the components of the individual vectors. 𝑥 1 = − 𝐴𝑐𝑜𝑠𝑏 (because it is in Q2) 𝑦 1 = 𝐴𝑠𝑖𝑛𝑏 𝑥 2 = 𝐶𝑐𝑜𝑠𝑑 𝑦 2 = 𝐶𝑠𝑖𝑛𝑑 Add the x components to form the resultant x 𝑥 𝑟 = 𝐶𝑐𝑜𝑠𝑑 − 𝐴𝑐𝑜𝑠𝑏 component and the y components to form the 𝑦 = 𝐶𝑠𝑖𝑛𝑑 + 𝐴𝑠𝑖𝑛𝑏 𝑟 resultant y component.  19 Consider a vector whose x and y components equal that of the resultant components Its direction could be found using tan 𝑦 𝑟 −1 𝐶𝑠𝑖𝑛𝑑 + 𝐴𝑠𝑖𝑛𝑏 𝑡𝑎𝑛θ = 𝑥 ⇒ θ = 𝑡𝑎𝑛 ( 𝐶𝑐𝑜𝑠𝑑 − 𝐴𝑐𝑜𝑠𝑏 ) 𝑟 Its magnitude could be found using 2 2 𝑍= 𝑥 +𝑦 pythagoras’ theorem. 𝑟 𝑟 2 𝑍= (𝐶𝑐𝑜𝑠𝑑 − 𝐴𝑐𝑜𝑠𝑏) + (𝐶𝑠𝑖𝑛𝑑 + 𝐴𝑠𝑖𝑛𝑏) Motion Quantities Describing Motion Distance A scalar quantity 𝑠 = 𝑙𝑒𝑛𝑔𝑡ℎ (𝑙) The distance moved by an object. Displacement A vector quantity 𝑠 (𝑜𝑟 𝑥) = 𝑙𝑒𝑛𝑔𝑡ℎ (𝑙) The distance and direction moved in reference to a fixed point. Speed A scalar quantity 𝑢 𝑜𝑟 𝑣 = 𝑙𝑒𝑛𝑔𝑡ℎ (𝑙) 𝑡𝑖𝑚𝑒 (𝑡) The rate of change of distance. Velocity A vector quantity 𝑣= 𝑙𝑒𝑛𝑔𝑡ℎ (𝑙) = 𝑠 𝑡𝑖𝑚𝑒 (𝑡) 𝑡 The rate of change of displacement. Acceleration, A vector quantity 𝑎= 𝑙𝑒𝑛𝑔𝑡ℎ (𝑙) 2 = 𝑣 𝑡 deceleration The rate of change of velocity. 𝑡𝑖𝑚𝑒 (𝑡) Equations of Motion 1. 𝑣 = 𝑢 + 𝑎𝑡 These equations describe the motion of bodies moving with 2. 𝑠 = 1 (𝑢 + 𝑣)𝑡 constant acceleration where: 2 2 2 3. 𝑣 = 𝑢 + 2𝑎𝑠 t = time 1 2 s = displacement 4. 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 u = initial velocity v = velocity at t a = the constant acceleration 20  21 Equation has no Source of equation 1 displacement From a = (v-u)/t 2. acceleration From s = (v+u)/2 t 3. time Make t the subject of the formula in 2. Substitute that equation into 1. 4. velocity Substitute 1 in 2. NB (if you’re feeling like being dumb): If any body begins from rest u = 0. Graphs Describing Motion Generic Situations Scenario A stationary A body A body A body A body body moving with moving with moving with moving with constantly constantly constantly constantly increasing increasing decreasing increasing displacement velocity velocity acceleration Displacement time or s/t graph (v) Velocity time or v/t graph (a)  22 A Bouncing Ball A rubber ball is thrown vertically upwards from the ground and falls on a horizontal smooth surface at the ground. The ball then bounces (up and down) with a decreasing velocity.  23 Projectile Motion (Parabolic Motion) 𝑣𝑦 = 𝑢𝑠𝑖𝑛𝑥 − 𝑔𝑡 𝑣𝑥 = 𝑢𝑐𝑜𝑠𝑥 1 2 𝑠𝑦 = 𝑢𝑡𝑠𝑖𝑛𝑥 − 2 𝑔𝑡 (from 4) 𝑠𝑥 = 𝑢𝑐𝑜𝑠𝑥𝑡 2𝑢𝑠𝑖𝑛𝑥 𝑡𝑜𝑡𝑎𝑙 𝑓𝑙𝑖𝑔ℎ𝑡 𝑡𝑖𝑚𝑒 = 𝑔 (from 𝑠𝑦) 𝑢𝑠𝑖𝑛𝑥 𝑡𝑖𝑚𝑒 𝑡𝑜 𝑚𝑎𝑥 ℎ𝑒𝑖𝑔ℎ𝑡 = 𝑔 (from 𝑣𝑦) 2 2 𝑢 𝑠𝑖𝑛 𝑥 𝑚𝑎𝑥 ℎ𝑒𝑖𝑔ℎ𝑡 = 2𝑔 (𝑠𝑦at time above) 2 𝑢 𝑠𝑖𝑛2𝑥 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑅𝑎𝑛𝑔𝑒 = 𝑔 (𝑠𝑥at total flight time) To prove that projectile motion follows the path of a parabola we need to write it in the form 2 𝑦 = 𝑎𝑥 − 𝑏𝑥  24 Newton’s Laws The First “Every body continues in its state of This expresses the concept of inertia, rest or of uniform (unaccelerated) the reluctance of a body to start moving Law motion in a straight line unless acted or to stop moving once it has started. on by some external force.” The mass of a body is a measure of its inertia. The Second “The rate of change of momentum of Mathematically this says: a body is directly proportional to the 𝑑 Law 𝐹∝ 𝑑𝑡 (𝑚𝑣) (resultant or net) external force When the mass is constant and the acting on the body and takes place in quantities on the right are expressed in the direction of the force.” the same units as the left: 𝑑𝑣 𝐹 =𝑚 𝑑𝑡 𝐹 = 𝑚𝑎 The “Newton” is the force produced −2 when an acceleration of 1𝑚𝑠 acts on a mass of 1kg. The Third “If a body A exerts a force on a body This doesn’t mean that the two forces B, then B exerts an equal and cancel out as they act on different Law bodies. oppositely directed force on A.” Uses of F = ma NB: only the resultant force in terms of mass in kg can be used in these calculations. The Lift Problem An object of mass m is suspended from a spring balance, suspended vertically from the ceiling of a lift. Note that the reading of the spring balance is determined by the tension with which the object is suspended. The tensional force, T in three scenarios can be considered: 1. The lift is moving upward with acceleration a. T must be sufficient to overcome the effect of gravity on the object (mg) and accelerate the body. In other words 𝑚𝑎 (𝑡ℎ𝑒 𝑛𝑒𝑡 𝑓𝑜𝑟𝑐𝑒) = 𝑇 − 𝑚𝑔 ⇒ 𝑇 = 𝑚(𝑎 + 𝑔). 2. The lift is moving downward with deceleration d. Despite T, gravity affecting the object (mg) would result in d. In other words 𝑑𝑚 (𝑡ℎ𝑒 𝑛𝑒𝑡 𝑓𝑜𝑟𝑐𝑒) = 𝑇 + 𝑚𝑔 ⇒ 𝑇 = 𝑚(𝑑 − 𝑔). 3. The lift is moving with constant velocity. There is no acceleration and therefore T = mg  25 A Body on an Inclined Plane The component of a body with weight mg at an incline θdegrees parallel to the plane creating the incline is given by 𝑚𝑔(𝑠𝑖𝑛(90 − 𝑥)) = 𝑚𝑔(𝑐𝑜𝑠𝑥). There are three forces acting on the body: The weight (mg) The force, P causing its acceleration if one is present The frictional force When the body is accelerating net force = P - frictional force - component of weight parallel to plane P = frictional force + mgcosx + net force When the body moves at constant velocity Net force = 0 P = frictional force + mgcosx  26 Principles Derived From Newton’s Laws The Principle of Conservation of Linear Momentum “The total linear momentum of a system of interacting (eg. colliding) bodies, on which no external forces are acting, remains constant.” Newton’s Experimental Law of Impact The relative velocity with which two bodies separate from each other after collision is related to the relative velocity of approach as follows: 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑒 × 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ This depends on the type of collision which takes place. Type of Collision Description e Completely inelastic Bodies stick together on impact 0 inelastic Some kinetic energy is lost