Physics Notes - Units of Measurement

# Physics ## Lesson 1.1 Units of Measurement ### Measurement * In the past, Egyptians and Babylonians use their body parts to estimate the length of an object. * A process of assigning a quantity to describe a property of an object by comparing it with a standard. * A standard should be universal...

# Physics ## Lesson 1.1 Units of Measurement ### Measurement * In the past, Egyptians and Babylonians use their body parts to estimate the length of an object. * A process of assigning a quantity to describe a property of an object by comparing it with a standard. * A standard should be universal and does not change with time. ### International System or SI * It is a standard system of measurement for the fundamental quantities. Abbreviated from Système International. Also called as "metric system". ### SI Base Units | Quantity | Symbol | SI unit | Symbol | |---|---|---|---| | Time | t | Second | s | | Length | 1, x, r | Meter | m | | Mass | m | Kilogram | kg | | Electric Current | I, i | Ampere | A | | Thermodynamic Temperature | T | Kelvin | K | | Amount of Substance | n | Mole | mol | | Luminous Intensity | Iv | Candela | cd | #### Second > The time required for 9,192, 671, 170 cycles of microwave radiation in cesium-137 atoms. #### Meter > The distance traveled by light in a vacuum in 1/299, 792, 458 sec. #### Kilogram > Define by taking the fixed numerical value of Planck constant h to be 6.62607015 \* 10^-34 J\*s. #### Ampere > The numerical value of the elementary charge e (1.602176634 \* 10^-19 C). #### Kelvin > Fixed numerical valve Boltzmann constant k to be 1.380 649 \* 10^-23 J\*K^-1, equivalent to kg\*m^2\*s^-2\*K^-1. #### Mole (mol) > One mole contains exactly 6.022 140 76 \* 10^23 elementary entities. Fixed value of Avogadro constant. #### Candela (cd) > The luminous efficacy of monochromatic radiation with frequency of 540 \* 10^12 Hz, ked, to be 683 lm\*W^-1, when expressed in the unit Im\*W^-1 equal to cd\*sr\*W^-1 or cd\*sr\*kg^-1\*m^-2\*s^3. ## Conversion of Units * Units can be treated as algebraic quantities that can cancel each other. **Example:** Suppose you want to convert 5.0 inches to centimeters given that 1 inch is equivalent to 2.54 centimeters. 5.0 in = (5.0 in \* 2.54 cm) / in = 12.7 cm * The unit inch is placed in the denominator so that it cancels the unit from the original value. * The remaining unit "centimeter" is the desired result. ## Lesson 1.2 Scientific Notation and Scientific Figures ### Scientific Notation * Very large and very small numbers can be expressed using scientific notation sometimes called powers of 10 notation. All numbers can be expressed in the form of equation 1.2.1. ### Addition and Subtraction of Scientific Notation. * To add and subtract, make sure that the expressions have similar terms. **Ex:** 8.5 \* 10^3 kg + 3.6 \* 10^3 kg (common factor 10^3) (1) (8.5 \* 10^3 kg) \* (3.6 \* 10^3) = (8.5 + 3.6) x 10^3 kg (2) (8.5 + 3.6) x 10^3 kg = 9.1 x 10^3 kg **If exponents are not the same** **Ex:** 5.5 \* 10^3 kg + 3.6 \* 10^5 kg. (1) 10^5 (larger exponent) can be written as 10^2 \* 10^3. Therefore, 3.6 \* 10^5 = 3.6 \* 10^2 \* 10^3 kg = (3.6 \* 10^2) \* 10^3 kg = 360 \* 10^3 kg (2) Factor out the common factor (5.5 \* 10^3 kg) + (360 \* 10^3 kg) = (5.5 + 360) x 10^3 kg (3) Add N (5.5 + 360) x 10^3 kg = 365.5 x 10^3 kg ### Multiplication and Division with Scientific Notation * In multiplying, calculate the dimensions of: **Ex:** 1.5 \* 10^2 m \* 2.1 \* 10^4 m (1) 1.5 * 2.1 = 3.15 (2) Add the exponents, 10^2 * 10^4 = 10^6 (3) 3.15 * 10^6 m^2 * In Division, Ns are divided as is, while the exponents are subtracted. **Ex:** 2.1 \* 10^4 m / 1.5 \* 10^2 m (1) 2.1 / 1.5 = 1.4 (2) Subtract exponent: 10^4 - 10^2 = 10^2 (3) 1.4 \* 10^2 m ## Lesson 1.3 Uncertainties and Deviations in Measurement ### Measurement * **Error:** difference between the true value and the measured value. * **Uncertainty:** net effect of error; quantifies the doubt that exists in any reported value of the measurement. ### Accuracy and Precision * **Accuracy:** indicates how close the measured value to the true value. * **Precision:** describes the scatter or the variability of a set of measurements made. ## Lesson 1.4 Sources and Types of Error in Measurement ### Errors in Measurement * In measurement, error signifies an inevitable uncertainty that is present in all types of error. * It cannot be completely eliminated even if one is conducting an experiment carefully. ### Two main types of error: * **Systematic error** * **Random error** #### Random Error * Occur when repeated measurements produce randomly different results. * Observed after repeated measurements. #### Systematic Error * Errors that remains constant or changes in regular fashion even if measurements are repeated. ### Types of Scattering in Experiments: * **Fundamental Noise** * **Technical Noise** #### Fundamental Noise * Can you control the behavior of the launcher? * Can you control how the ball as it is launched by the launcher? #### Technical Noise * How does minute changes in the angle by which the ball launch affect your data? ### Mistakes * Similar in nature to systematic errors and are also difficult to detect. * **Least Count:** smallest division found in a measuring device. ### Error Bars * Graphical representations indicating the range of uncertainty of a specific value or data. ## Lesson 2.1 Scalar and Vector Quantities ### Physical Quantities * **Two types of physical quantities:** * **Scalar quantities** * **Vector quantities** #### Scalar quantities * A physical quantity that has magnitude (size of quantity) but no direction. * Temperature is a scalar quantity. * Described by a single number with its appropriate unit. * Other examples are mass, time, distance, speed, density, volume. #### Vector quantity * A physical quantity with both magnitude and direction. * Magnitude describes the size or the size of the physical quantity while Direction describes how the vector is oriented relatives to a reference point. ### Representing Vector Quantities * Vectors are represented by a single letter with an arrow above them. * **Speed** - scalar quantity * **Velocity** - vector quantity ### Magnitude of vectors * Magnitude of vectors is represented by the same letter used in a vector quantity * Magnitude of *v* = |*v*| * The shortest distance is defined as displacement * Displacement is a vector, it is represented by an arrow. * The length of the arrow represents the vectors magnitude. ## Lesson 2.2 Vector addition through Graphical Method ### Vectors in One Dimension * If two vectors are pointing in the same direction its parallel if two vectors have opposite directions but have the same magnitude its anti-parallel. ### Subtracting of Vectors A + (-B) = A - B ## Lesson 2.3 Components of Vectors * Sin θ = opp/hyp = Ay/A * Cos θ = adj/hyp = Ax/A * Tan θ = opp/adj = Ay/Ax * Ax = A cos θ * Ay = A sin θ **Ex:** What are the x- and y- components of a displacement vector with a magnitude of 50 m and an angle of 30°? 50 cos (30) = x component - 43.30 m 50 sin (30) = y component - 25 m **Ex:** a car has a displacement of 750 m, 45° north of west. What are the components of a displacement vector? 750 sin (45) = x component - 530.37 m 750 cos (45) = y component - 530.37 m ### Step 1: Determine whether the angles given were measured from the +x-axis > **Vector A is 10m, 20° NE** > **Vector B is 15m, 50° NE** ### Step 2: Resolve each vector into its x- and y- components * Ax = A cos θ * Ay = A sin θ ### Step 3: Add all components together * Rx = Ax + Bx * Ry = Ay + By ### Step 4: Calculate the magnitude of the vector using Pythagorean theorem. R = √R²x + Ry² R = √R²x + R²y = √(19.04m)² + (4.91m)² = 24. 18 m > The resultant vector of Denise is 24.18m, 19.04°, or 38.06 ° NE ## Lesson 2.4 Vector Addition through Analytical Method * Vectors can be added by placing them head to tail. * Graphical method, however, is prone to measurement errors **Ex. Word problem:** Denis walks everyday from her house to the school. Fist, she covers 10m, 20° north of east. Then, she walked 15m in a direction 50° north of east. What is her resultant displacement? ## Lesson 2.5 Vector Multiplication ### Scalar Product * The scalar product of a vector and another vector will result in a scalar quantity. * It is also referred to as a dot product. * A \* B = AB cos Φ * **When can the scalar product be positive?** * **When can the scalar product be negative?** * **When can the scalar product be zero?** * **What happens when vectors are parallel?** A \* B = AB cos 0 = AB * **What happens when vectors are anti parallel?** A \* B = AB cos 180 = - AB ### Scalar Product Laws * **Commutative Law** A \* B = B \* A * **Distributive Law** A \* (B + C) = A \* B + A \* C **Ex:** Find the scalar product between the two vectors if the magnitude of A is 9.0 and B is 15.0 and the angle between them is 45°. The scalar product is 95. **Ex:** Find the magnitude of vector A if the scalar product between A and B is 30 m, the magnitude of B is 10 m, and the angle between the vectors is 60°. The magnitude of A is 6 m. **Ex:** Find the angle between the two vectors A and B with magnitudes 16 and 4, respectively. The scalar product of the two vectors is 10. The angle between the two vectors is 71.17° or 70°. ### Vector Product * The following properties apply to vector product. * **Anticommutative Property** A \* B = - B \* A * **Distributive Law** A \* (B + C) = A \* B + A \* C * **Magnitude of A \* B also equals B (A sin Φ).** * **Magnitude of B) \* (Component of A perpendicular to B)**