General Physics - Quick Reference Notes MEASUREMENT AND UNITS Physics relies on measured quantities; units express magnitude and scale. UNIT: standard quantity used to express others; SI base units define the magnitude of basic quantities. SI BASE UNITS: e x t L e n g t h = m , e x t M a s s = k g , e x t T e m p e r a t u r e = K , e x t T i m e = s , e x t A m o u n t o f s u b s t a n c e = m o l , e x t L u m i n o u s i n t e n s i t y = c d , e x t E l e c t r i c c u r r e n t = A . ext{Length}=m, ext{ Mass}=kg, ext{ Temperature}=K, ext{ Time}=s, ext{ Amount of substance}=mol, ext{ Luminous intensity}=cd, ext{ Electric current}=A. e x t L e n g t h = m , e x t M a ss = k g , e x t T e m p er a t u re = K , e x t T im e = s , e x t A m o u n t o f s u b s t an ce = m o l , e x t Lu min o u s in t e n s i t y = c d , e x t El ec t r i cc u rre n t = A . DERIVED QUANTITIES: combinations of base units (e.g., speed v = f r a c d t v= frac{d}{t} v = f r a c d t METRIC PREFIXES: M, k, ext{(mega to kilo)}, d, c, m, bc (micro), n, p with meanings 10 6 , 10 3 , 10 1 , 10 − 1 , 10 − 2 , 10 − 3 , 10 − 6 , 10 − 9 , 10 − 12 10^6, 10^3, 10^1, 10^{-1}, 10^{-2}, 10^{-3}, 10^{-6}, 10^{-9}, 10^{-12} 1 0 6 , 1 0 3 , 1 0 1 , 1 0 − 1 , 1 0 − 2 , 1 0 − 3 , 1 0 − 6 , 1 0 − 9 , 1 0 − 12 UNIT SYSTEMS: Metric (SI) and English; SI is the standard in science and steps are multiples of 10 for easy conversions. WHY CONVERT UNITS: ensure consistency across measurements and calculations. SCIENTIFIC NOTATION AND PREFIXES PURPOSE: handle very large or very small numbers in a compact form. FORM: a × 10 b a \times 10^{b} a × 1 0 b a a a b b b COEFFICIENTS AND EXPONENTS: write, read, and convert numbers using the power of ten. FROM SCIENTIFIC TO STANDARD: move decimal point to create a coefficient 1.00–9.99; adjust exponent accordingly. FROM STANDARD TO SCIENTIFIC: convert to a single-digit coefficient and an appropriate exponent. UNCERTAINTY, ERRORS, ACCURACY, PRECISION UNCERTAINTY: measured value = (true value ± uncertainty) [unit]. ERROR TYPES:RANDOM ERROR: affects precision (consistency across trials). SYSTEMATIC ERROR: affects accuracy (closeness to true value). ACCURACY: closeness of a measurement to the true value; can be determined from a single measurement. PRECISION: consistency of repeated measurements; requires multiple measurements. PERCENTAGE ERRORS: % error = ∣ measured − true true ∣ × 100 % \%\text{ error} = \left|\dfrac{\text{measured} - \text{true}}{\text{true}}\right| \times 100\% % error = true measured − true × 100% PERCENT DIFFERENCE: % difference = ∣ x < e m > 1 − x < / e m > 2 ( x < e m > 1 + x < / e m > 2 ) / 2 ∣ × 100 % \%\text{ difference} = \left|\dfrac{x<em>1 - x</em>2}{(x<em>1+x</em>2)/2}\right| \times 100\% % difference = ( x < e m > 1 + x < / e m > 2 ) /2 x < e m > 1 − x < / e m > 2 × 100% LEAST COUNT: smallest discernible division on a given instrument; used to estimate single-measure uncertainty. VARIANCE & STANDARD DEVIATION:VARIANCE: σ 2 = 1 N ∑ < e m > i = 1 N ( x < / e m > i − x ˉ ) 2 \sigma^2 = \dfrac{1}{N}\sum<em>{i=1}^N (x</em>i - \bar{x})^2 σ 2 = N 1 ∑ < e m > i = 1 N ( x < / e m > i − x ˉ ) 2 STANDARD DEVIATION: σ = σ 2 \sigma = \sqrt{\sigma^2} σ = σ 2 HOW TO EXPRESS UNCERTAINTY FOR A SET: mean ± standard deviation; interpret spread as reproducibility. SCALARS AND VECTORS SCALAR: magnitude only (e.g., mass, time, distance). VECTOR: magnitude and direction (e.g., displacement, velocity, force). REPRESENTATION: arrows; components along axes. COMPONENTS: V < e m > x = V cos θ , V < / e m > y = V sin θ V<em>x = V\cos\theta, \quad V</em>y = V\sin\theta V < e m > x = V cos θ , V < / e m > y = V sin θ V V V θ \theta θ ADDITION: sum of vectors via head-to-tail or parallelogram rule; resultant is the combined effect. MOTION: KINEMATICS (ONE-DIMENSIONAL) KEY QUANTITIES:Distance vs. Displacement: distance is path length; displacement is straight-line change in position (vector). Speed (scalar) vs. Velocity (vector). Average velocity: v ˉ = Δ x Δ t \bar{v} = \dfrac{\Delta x}{\Delta t} v ˉ = Δ t Δ x a = Δ v Δ t a = \dfrac{\Delta v}{\Delta t} a = Δ t Δ v GRAPH INTERPRETATION:Slope of Position-vs-Time gives velocity; slope of Velocity-vs-Time gives acceleration. Area under Velocity-vs-Time gives displacement. KINEMATIC EQUATIONS: a set of four equations (SUVAT) relating displacement, velocity, acceleration, and time (use when a, v, x, t are known/unknown). MOTION: PROJECTILE MOTION PROJECTILE MOTION: motion under gravity with initial velocity at an angle. AXES:x-direction: no acceleration (ignoring air resistance): a x = 0 a_x = 0 a x = 0 y-direction: downward acceleration due to gravity: a y = − g ( g ≈ 9.8 m/s 2 ) . a_y = -g\quad (g \approx 9.8\,\text{m/s}^2). a y = − g ( g ≈ 9.8 m/s 2 ) . INITIAL CONDITIONS:V 0 = 40.0 m/s , θ = 35 ∘ V_0 = 40.0\,\text{m/s}, \quad \theta = 35^{\circ} V 0 = 40.0 m/s , θ = 3 5 ∘ Components: V < e m > 0 x = V < / e m > 0 cos θ , V < e m > 0 y = V < / e m > 0 sin θ V<em>{0x} = V</em>0\cos\theta, \quad V<em>{0y} = V</em>0\sin\theta V < e m > 0 x = V < / e m > 0 cos θ , V < e m > 0 y = V < / e m > 0 sin θ Examples: V < e m > 0 x = 32.77 m/s , V < / e m > 0 y = 22.94 m/s V<em>{0x} = 32.77\,\text{m/s}, \quad V</em>{0y} = 22.94\,\text{m/s} V < e m > 0 x = 32.77 m/s , V < / e m > 0 y = 22.94 m/s OUTCOMES:Time of flight: t < e m > flight = 2 V < / e m > 0 y g t<em>{\text{flight}} = \tfrac{2V</em>{0y}}{g} t < e m > flight = g 2 V < / e m > 0 y Maximum height: h < e m > max = V < / e m > 0 y 2 2 g h<em>{\max} = \dfrac{V</em>{0y}^2}{2g} h < e m > max = 2 g V < / e m > 0 y 2 Range: R = V < e m > 0 x ⋅ t < / e m > flight = V 0 2 sin ( 2 θ ) g R = V<em>{0x} \cdot t</em>{\text{flight}} = \dfrac{V_0^2 \sin(2\theta)}{g} R = V < e m > 0 x ⋅ t < / e m > flight = g V 0 2 sin ( 2 θ ) CIRCULAR MOTION UNIFORM CIRCULAR MOTION: constant speed, velocity direction changes -> acceleration toward center. CENTRIPETAL ACCELERATION: a c = v 2 r a_c = \dfrac{v^2}{r} a c = r v 2 CIRCULAR MOTION: v = 2 π r T v = \dfrac{2\pi r}{T} v = T 2 π r ω = 2 π T \omega = \dfrac{2\pi}{T} ω = T 2 π TANGENTIAL ACCELERATION: present if speed changes (non-uniform); equals rate of change of speed. NEWTON'S LAWS NEWTON'S FIRST LAW (Law of Inertia): an object at rest stays at rest and an object in motion stays in motion unless acted upon by a net external force. NEWTON'S SECOND LAW (Law of Acceleration):Net force and acceleration are related by F net = m a \mathbf{F}_{\text{net}} = m \mathbf{a} F net = m a If force increases, acceleration increases; if mass increases, acceleration decreases. WORK AND ENERGY WORK: energy transfer by force along displacement, defined as W = F ⋅ d = F d cos θ W = \mathbf{F} \cdot \mathbf{d} = F d \cos\theta W = F ⋅ d = F d cos θ N ⋅ m \text{N} \cdot \text{m} N ⋅ m MECHANICAL ENERGY: sum of kinetic and potential energies. KINETIC ENERGY: K E = 1 2 m v 2 KE = \tfrac{1}{2} m v^2 K E = 2 1 m v 2 GRAVITATIONAL POTENTIAL ENERGY: P E grav = m g h PE_{\text{grav}} = m g h P E grav = m g h ELASTIC POTENTIAL ENERGY (spring): P E elastic = 1 2 k x 2 PE_{\text{elastic}} = \tfrac{1}{2} k x^2 P E elastic = 2 1 k x 2 F = − k x F = -k x F = − k x TOTAL MECHANICAL ENERGY: M E = K E + P E ME = KE + PE ME = K E + PE MOMENTUM MOMENTUM: p = m v p = m v p = m v Change in momentum: used in impulse problems; can be solved via Δ p = F net Δ t \Delta p = F_{\text{net}} \Delta t Δ p = F net Δ t UNIT CONVERSION PRACTICE (METHODS) REASON TO CONVERT: different units in use across problems. METHODS: factor-label method or dimensional analysis; identify knowns, desired units, factors to multiply by, and verify final units. CHECK: round final answer to appropriate precision; re-check units. SUMMARY CHECKPOINTS Distinguish scalars vs vectors; use components for vectors. Remember the core equations: v = Δ x Δ t , a = Δ v Δ t , F < e m > net = m a , W = F ⋅ d , K E = 1 2 m v 2 , P E < / e m > g r a v = m g h , M E = K E + P E , p = m v , a c = v 2 r , v = 2 π r T , g ≈ 9.8 m/s 2 v = \dfrac{\Delta x}{\Delta t}, \ a = \dfrac{\Delta v}{\Delta t}, \ F<em>{\text{net}} = m a, \ W = \mathbf{F} \cdot \mathbf{d}, \ KE = \tfrac{1}{2} m v^2, \ PE</em>{grav} = m g h, \ ME = KE + PE, \ p = m v, \ a_c = \dfrac{v^2}{r}, \ v = \dfrac{2\pi r}{T}, \ g \approx 9.8\,\text{m/s}^2 v = Δ t Δ x , a = Δ t Δ v , F < e m > net = ma , W = F ⋅ d , K E = 2 1 m v 2 , PE < / e m > g r a v = m g h , ME = K E + PE , p = m v , a c = r v 2 , v = T 2 π r , g ≈ 9.8 m/s 2 Use projectile formulas to analyze range, time of flight, and max height for given initial speed and angle.