Physics: Intro and Measurement

• These study notes cover Module 1 and Module 2

Introduction to Physics

• Physics studies the universe's basic principles, including motion, forces, and energy
• Physics is the most basic of the natural sciences
• The word "physics" originates from the Greek word "physikos" (φυσικός)
• Physics is divided into classical and modern physics
• The specific course focuses on classical physics

Measurement

• Measurement compares an unknown quantity to a standard
• Measurement assigns a quantitative value to data or phenomena
• Measurement includes magnitude, units, and uncertainty using standard instruments
• There are two systems of measurement: English units/imperial and SI units
• English units are historical, commonly used in the USA
• SI units, or Système International, are modern, used worldwide, based on increments of 10
• English system units:
  • Length: inch (in), foot (ft), yard (yd), mile (mi)
  • Mass: ounce (oz), pound (lb), ton (ton)
  • Capacity: teaspoon (tsp), cup (c), pint (pt), quart (qt), gallon (gal)
• Metric system uses core units with prefixes for larger/smaller measurements
  • Core units: meter (m), gram (g), second (s), liter (L)
  • Common prefixes:
    • tera- (T) - 10^12
    • giga- (G) - 10^9
    • mega- (M) - 10^6
    • kilo- (k) - 10^3
    • hecto- (h) - 10^2
    • deka- (da) - 10^1
    • deci- (d) - 10^-1
    • centi- (c) - 10^-2
    • milli- (m) - 10^-3
    • micro- (μ) - 10^-6
    • nano- (n) - 10^-9
    • pico- (p) - 10^-12

Conversion

• Conversion uses dimensional analysis
• Conversion factor converts units
• Conversion factors depend on units used
• English conversion examples:
  • 12 in = 1 ft
  • 3 ft = 1 yd
  • 5280 ft = 1 mi
  • 16 oz = 1 lb
  • 2000 lbs = 1 ton
  • 48 tsp = 1 c
  • 2 c = 1 pt
  • 2 pt = 1 qt
  • 4 qt = 1 gal
• SI units are derived by multiplying the unit with the relevant numerical meaning
• 1000 mg = 1 g because milli- is 10^-3
• Worked example:
  • Femur bone density is 1.85 g/cm³
  • Density of the femur in kg/m³ is calculated using multiple conversion factors:
    • 1. 85 g/cm³ * (1 kg / 1000 g) * (100 cm / 1 m) * (100 cm / 1 m) * (100 cm / 1 m)
    • = 1.85 * 10^6 kg/m³ * (1/10^3)
    • = 1850 kg/m³
  • Using three cm/m units cancels the cm³
  • Femur density in mg/mm³ is derived:
    • 1. 85 g/cm³ * (1 mg / 0.001 g) * (1 cm / 10 mm)³
    • = 1.85 * (1 mg / 0.001) * (1 / 1000 mm)
    • = 1.85 mg/mm³

Significant Figures

• Significant figures (SF) represent meaningful measurement digits
• Rules for significant figures:
  • Rule 1: Counting, constants, and conversion factors have unlimited SF
    • Example: 12 boys, g = 9.8 m/s², 1 ft = 12 in
  • Rule 2: All nonzero digits are significant.
    • Example: 22 m (2 SF), 5423 (4 SF), 13 in (2 SF)
  • Rule 3: Zeros at the end of a number are not significant.
    • Example: 100 (1 SF), 1000 g (1 SF), 2400 m (2 SF)
  • Rule 4: Zeros between nonzero numbers are significant.
    • Example: 101 (3 SF), 101201 L (6 SF), 403 m (3 SF)
  • Rule 5: Zeros to the right of a decimal point are significant.
    • Example: 10.10 (4 SF), 105.40 g (5 SF), 101.240 m (6 SF)
  • Rule 6: Zeros at the end of a number and to the right of the decimal point are significant.
    • Example: 100.00 (5 SF), 1.0 g (2 SF), 500.00 L (5 SF)
  • Rule 7: Leading zeros are not significant.
    • Example: 00023 (2 SF), 0.000000001 mL (1 SF), 0.00000000000023 mg (2 SF)
  • Rule 8: Sums/differences use the least number of decimal places in original numbers
    • Example: 23.24 - 20.1 = 3.1, 23.245 + 20.20 = 43.45, 23.24 + 20.1 = 43.3
  • Rule 9: Products/quotients use the least number of SFs in the original numbers.
    • Example: 20.2 * 3 = 60, 20.42 * 3.23 = 66.0

Extension

• When multiplying with conversion factors, express the answer with the least number of SFs
  • Example:
    • 5 ft * (12 in / 1 ft) = 60 in
    • 4 ft * (12 in / 1 ft) = 50 in (instead of 48 in)
• Significant figures are important as they determine measurement accuracy and precision

Accuracy & Precision

• Accuracy is how close a measurement is to the real value
• Precision is how consistent measurements are
• Measurement errors can be random or systematic
  • Random error:
    • Causes measurements to vary randomly
    • Due to instrumental limitations, environmental factors, or procedure variations
    • Lacks precision but clusters around the accurate value
  • Systematic error:
    • Causes measurements to vary predictably
    • Due to observational errors, calibration errors, or environmental interference
    • May be precise, but not accurate
• Percentage Error Formula:
  • % error = (|experimental value - true value| / true value) * 100
  • Example:
    • Resistor with color code of 240 Ohms measures 245 Ohms
    • % error = (|245-240| / 240) * 100 = 2.08%

Scientific Notation

• Scientific notation indicates significant figures
• Scientific notation form: c x 10^n
  • c is any number from 1 to 10 (not including 10)
  • n is some integer
• Converting 156000 to scientific notation:
  • Imagine a decimal point at the end (156000.)
  • Move the decimal point until only one digit is before it (1.56000)
  • The point moved 5 places to the left
  • Decimal is now c. Since decimal moved left, n is positive
  • 156000 = 1.56 * 10^5
• Converting 0.000053 to scientific notation:
  • Move decimal point (000005.3) 6 places to the right
  • Decimal is c. Since the decimal moved right, n is negative
  • 0.000053 = 5.3 * 10^-6

Vectors

• Physical quantities are classified as either vector or scalar
  • Vector: Fully expressed by magnitude and direction
  • Scalar: Expressed by magnitude alone
• Magnitude is a number plus the unit of measurement
• Vector vs Scalar Quantities:
  • Scalars include distance, mass, speed, time, work, and density
  • Vectors include displacement, force, velocity, acceleration, torque, and momentum
• Vector quantities are represented by scaled vector diagrams
• Vector diagrams depict vectors with arrows drawn to scale in a specific direction

Components of Vector Diagram

• Vector is represented by an arrow with 3 parts:
  • Arrowhead: Indicates the direction of the vector
  • Length: Represents the magnitude of the vector
  • Tail: Represents the origin of the vector
• Devise a suitable scale:
• Worked Example 1:
  • A student walks 20 m from his classroom to the next building in a direction of 30° West of North
  • Representing this displacement using a vector diagram, a scale of 1 cm = 4 m can be used
  • Dimensional analysis is used to find the actual length
• Both the magnitude and direction of the vector must be clearly labeled
• Worked Example 2: (30 degrees West of North)
  • The diagram shows the magnitude is 20 m
  • The direction is 30 degrees West of North

Vector Addition

• Resultant Vector (R) is the sum of two or more vectors represented by a single vector
• Vector addition can be solved using the graphical and analytical methods
• Graphical Vector Addition Methods:
  • Head-to-Tail Method
  • Parallelogram Method
• Analytical Vector Addition Methods:
  • Pythagorean Theorem
  • Component Method

Vector Addition - Graphical Method

• Head-to-Tail Method example:
  • A person walks 9 blocks east and 5 blocks north
  • Using the head-to-tail method determines that the resultant displacement is 10.3 blocks at an angle of 29.1 degrees north of east
• Steps:
  1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor
  2. Draw an arrow to represent the second vector (5 blocks to the north) and place the tail of the second vector at the head of the first vector
  3. If there are more than two vectors, continue this process for each vector to be added
  4. Draw an arrow from the tail of the first vector to the head of the last vector, and the resulting measurement is the sum of the other vectors
  5. To get the magnitude of the resultant, measure its length with a ruler, or use the Pythagorean theorem
  6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor, or use trigonometric relationships
     • Remember :The reference direction is the last direction in the phrase "direction of direction"
• Using the graphical, head-to-tail method, the resultant vector is 10.3 units north of east

Vector Addition - Analytical Method

• Pythagorean Theorem is used to determine the magnitude:
  • Worked example:
    • Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east
    • Determining Eric's resulting displacement:
      • c² = a² + b²
      • c² = (11km)² + (11km)²
      • c² = 121km² + 121km²
      • c² = 242 km²
      • √c² = √242 km²
      • c = 15.6
      • R = 15.6km
    • Eric is 15.6 km away from his original position
• Trigonometric functions are used to determine the direction
  • SOHCAHTOA
  • Using the sine function, the resultant displacement is 15.6 km,45 degrees North of East
• Worked example 2:
  • A motorboat leaves the shore at a velocity of 4m/s due east
  • The water current :in the river moves due north at a velocity of 3m/s
  • What would be the resultant velocity of the boat?
    • Magnitude:
      • c² = a² + b²
      • c² = (4.0m/s)² + (3.0m/s)²
      • c² = 16 m²/s² + 9 m²/s²
      • c² = 25 m²/s²
      • √c² = √25 m²/s²
      • c = 5.0
      • R = 5.0 m/s
    • Direction:
      • tan(θ) = opposite/adjacent
      • tan(θ) = (3/4)
      • θ = tan⁻¹(3/4)
      • θ = 36.9
    • Resultant velocity of the boat is 5.0 m/s, 36.9 degrees from the +x axis or North of East

Vector Addition – Component Method

• Method is used for adding two or more vectors
  • Worked Example:
    • d₁ = 36 m, 34° N of E and d₂ = 23 m, 64° W of N
  1. Determine the size and direction of the vectors that you wan to add:
     • Vector breakdown to the coordinate quadrants in the images
  2. Find the x- and y-components for the first/second vectors using SOH CAH TOA
     • The results of the vector components are in the images
  3. Add all the x- components and add all the y-components of your two vectors, but do not add an x- component to the y-component:
     • ∑x = 9.2 m and ∑y = 30.1 m
  4. Use the Pythagorean theorem to get the magnitude of the total 2D displacement:
     • d = √(dx)² + (dy)² = √(9.2 m)² + (30.1 m)² = √84.6 m² + 906.0 m² = √990.6 m² = 31.5 m
  5. Use the tangent function to get the angle:
     • tanθ = ∑y/∑x = 30.1 / 9.2, or tanθ = 30.1m / 19.2m, θ = tan⁻¹|30.1 / 19.2| = 73.0°, Quadrant 1

Motion In One Dimension

• Motion can be described quantitatively using 5 values:
  1. Distance
  2. Displacement
  3. Speed
  4. Velocity
  5. Acceleration
• Scalar and Vector values:
  • Distance is a scalar quantity
  • Displacement is a vector quantity

Distance & Displacement

• Distance:
  • Is a scalar quantity
  • Does not follow the operations of signed numbers
  • Refers to the path's total length taken by an object from its origin/initial position to the final position
• Displacement:
  • Is a vector quantity
  • Follows the operations of signed numbers
  • Refers to the straight-line distance between its initial and final position

Speed & Velocity

• Speed and Velocity:
  • Speed is a scalar quantity
  • Velocity is a vector quantity
• Speed is how fast an object moves
• Velocity is the speed in a given direction
• Average speed = total distance/total time
• Average velocity = total displacement/total time
• Practice questions and answers in the images

Acceleration

• Acceleration is the change in velocity per unit time
• a = acceleration is expressed with the formula: ∆v/t = v(f) - v(i) / t
• There are three cases when an object experiences an acceleration:
  • Change in speed
  • Change in direction
  • Change in both speed and direction

Uniformly Accelerated Motion

• Accelerating objects are changing their velocity by a constant amount each second
• This is referred to as a constant acceleration
• Accelerating objects are changing their velocity by a changing amount each second
• This is referred to as a non-constant acceleration
• An object that is changing velocity, whether by constant/varying amount is accelerating
• An object with a constant velocity is not accelerating
• If a body/object maintains a constant change in velocity in a given interval of time along a straight line, the motion is described to be uniformly accelerated
• Kinematic equations used in analyzing problems related to uniformly accelerated motion:
  • d = v(i)·t + (1 / 2)·a·t²
  • vf² = vi² + 2·a·d
  • a = v(i) + a·t
  • d = (vi+vf / 2)·t
• Worked example:
  • An airplane accelerates down a runway at 3.20 m/s² for 32.8 s until it finally lifts off the ground
  • What is the distance traveled before takeoff?
    • d = 1721.34 m

Free Fall Motion

• Nearly 400 years ago, Galileo Galilei proposed all objects fall equally towards the earth despite size, shape, and mass
• Objects move freely under gravity and undergo free fall motion
• Objects thrown upward or downward and those released from rest are all freely falling bodies
• Any free-falling object at a certain constant speed experiences a constant acceleration and is directed downwards
• All objects, light or heavy, experience the same acceleration
• Examples of velocity and acceleration vectors during free-fall:
• velocity changes - acceleration never changes
• velocity decreasing when rising - acceleration constant
• velocity increasing when falling - acceleration directed downward if rising and directed down when falling
  • The equations involved in analyzing free fall motion
    • d = v(i) + v(f) / 2
    • d = v(i)·t + (1 / 2)·g·t²
    • d = v(f)² - v(i)² / 2·g
    • vf = vi + gt
    • vf² = vi² + 2gd
• Example formulas:
• displacement
• time
• initial velocity
• final velocity
• gravity