OpenStax University Physics/V1/Formulas (master)

File:OpenStax University Physics Volume 1 Formulas.pdf

These formulas print to 4 pages

<section begin=Introduction/>Introduction:

metric prefixes
da	h	k	M	G	T	P	E	Z	Y
deca	hecto	kilo	mega	giga	tera	peta	exa	zetta	yotta
1E+01	1E+02	1E+03	1E+06	1E+09	1E+12	1E+15	1E+18	1E+21	1E+24

d	c	m	µ	n	p	f	a	z	y
deci	centi	milli	micro	nano	pico	femto	atto	zepto	yocto
1E-01	1E-02	1E-03	1E-06	1E-09	1E-12	1E-15	1E-18	1E-21	1E-24

<section begin=Units_and_Measurement/>1. Units_and_Measurement: The base SI units are mass: kg (kilogram); length: m (meter); time: s (second). ^[1]

Percent error is $(δ A / A) \times 100 %$ <section end=Units_and_Measurement/>

2. Vectors: Vector

\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

involves components (A_x,A_y,A_z) and ^[2] unit vectors.^[3] ▭ If

\vec{A} + \vec{B} = \vec{C}

, then A_x+B_x=C_x, etc, and vector subtraction is defined by

\vec{B} = \vec{C} - \vec{A}

.

▭ The two-dimensional displacement from the origin is $\vec{r} = x \hat{i} + y \hat{j}$ . The magnitude is $A \equiv | \vec{A} | = \sqrt{A_{x}^{2} + A_{y}^{2}}$ . The angle (phase) is $θ = \tan^{- 1} (y / x)$ . ▭ Scalar multiplication $α \vec{A} = α A_{x} \hat{i} + α A_{y} \hat{j} + . . .$ ▭ Any vector divided by its magnitude is a unit vector and has unit magnitude: $| \hat{V} | = 1$ where $\hat{V} \equiv \vec{V} / V$ ▭ Dot product $\vec{A} \cdot \vec{B} = A B \cos θ = A_{x} B_{x} + A_{y} B_{y} + . . .$ and $\vec{A} \cdot \vec{A} = A^{2}$ ▭ Cross product $\vec{A} = \vec{B} \times \vec{C} \Rightarrow$ $A_{α} = B_{β} C_{γ} - C_{γ} A_{β}$ where $(α, β, γ)$ is any cyclic permutation of $(x, y, z)$ , i.e., (α,β,γ) represents either (x,y,z) or (y,z,x) or (z,x,y). ▭ Template:Nowrap magnitudes obey $A = B C \sin θ$ where $θ$ is the angle between $\vec{B}$ and $\vec{C}$ , and $\vec{A} ⊥ {\vec{B}, \vec{C}}$ by the right hand rule. ▭ Vector identities $c (𝐀 + 𝐁) = c 𝐀 + c 𝐁$ ▭ $𝐀 + 𝐁 = 𝐁 + 𝐀$ ▭ $𝐀 + (𝐁 + 𝐂) = (𝐀 + 𝐁) + 𝐂$ ▭ $𝐀 \cdot 𝐁 = 𝐁 \cdot 𝐀$ ▭ $𝐀 \times 𝐁 = - 𝐁 \times 𝐀$ ▭ $(𝐀 + 𝐁) \cdot 𝐂 = 𝐀 \cdot 𝐂 + 𝐁 \cdot 𝐂$ ^[4] <section end=Vectors/>

<section begin=Motion_Along_a_Straight_Line/>3. Motion_Along_a_Straight_Line: ^[5] ▭ Average velocity $\bar{v} = Δ x / Δ t \to v = d x / d t$ (instantaneous velocity) ▭ Acceleration $\bar{a} = Δ v / Δ t \to a = d v / d t$ . ▭ WLOG set $Δ t = t$ and $Δ x = x - x_{0}$ if $t_{i} = 0$ . Then $Δ v = v - v_{0}$ , and $v (t) = \int_{0}^{t} a (t^{'}) d t^{'} + v_{0}$ , $x (t) = \int_{0}^{t} v (t^{'}) d t^{'} + x_{0} = x_{0} + \bar{v} t$ ^[6] ▭ At constant acceleration: $\bar{v} = \frac{v_{0} + v}{2}, v = v_{0} + a t, x = x_{0} + v_{0} t + \frac{1}{2} a t^{2},$ $v^{2} = v_{0}^{2} + 2 a Δ x$ . ▭ For free fall, replace $x \to y$ (positive up) and $a \to - g$ , where $g$ = 9.81 m/s² at Earth's surface). <section end=Motion_Along_a_Straight_Line/>

<section begin=Motion_in_Two_and_Three_Dimensions/>4. Motion_in_Two_and_Three_Dimensions: Instantaneous velocity: $\vec{v} (t) = v_{x} (t) \hat{i} + v_{y} (t) \hat{j} + v_{z} (t) \hat{k} = \frac{d x}{d t} \hat{i} + \frac{d y}{d t} \hat{j} + \frac{d z}{d t} \hat{k}$ ▭ $\vec{v} (t) = \lim_{Δ t \to 0} \frac{Δ \vec{r}}{Δ t} = \lim_{Δ t \to 0} \frac{\vec{r} (t + Δ t) - \vec{r} (t)}{Δ t}$ , where $\vec{r} (t) = x (t) \hat{i} + y (t) \hat{j} + z (t) \hat{k}$ ▭ Acceleration $\vec{a} = a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}$ , where $a_{x} (t) = d v_{x} / d t = d^{2} x / d t^{2}$ . ^[7] ▭ Uniform circular motion: position $\vec{r} (t)$ , velocity $\vec{v} (t) = d \vec{r} (t) / d t$ , and acceleration $\vec{a} (t) = d \vec{v} (t) / d t$ : $\vec{r} = A \cos ω t \hat{i} + A \sin ω t \hat{j},$ $\vec{v} = - A ω \sin ω t \hat{i} + A ω \cos ω t \hat{j},$ $\vec{a} = - A ω^{2} \cos ω t \hat{i} - A ω^{2} \sin ω t \hat{j} .$ Note that if $A = r$ then $| \vec{a} | = a_{C} = ω^{2} r = v^{2} / r$ where $v \equiv | \vec{v} | = ω r$ . ^[8] ▭ Relative motion: ^[9] ${\vec{v}}_{P S} = {\vec{v}}_{P S^{'}} + {\vec{v}}_{S^{'} S}$ , ^[10] <section end=Motion_in_Two_and_Three_Dimensions/>

<section begin=Newton's_Laws_of_Motion/>5. Newton's_Laws_of_Motion: ^[11] $m \vec{a} = d \vec{p} / d t = \sum {\vec{F}}_{j}$ , where $\vec{p} = m \vec{v}$ is momentum, ^[12] $\sum {\vec{F}}_{j}$ is the sum of all forces This sum needs only include external forces ^[13] ${\vec{F}}_{A B} = - {\vec{F}}_{B A}$ .^[14]

▭ Weight $= \vec{w} = m \vec{g}$ . ▭ normal force^[15] $| \vec{N} | = N = m g \cos θ$ ^[16] ▭ ^[17] $F = - k x$ where $k$ is the spring constant. <section end=Newton's_Laws_of_Motion/>

<section begin=Applications_of_Newton's_Laws/>6. Applications_of_Newton's_Laws: $f_{s} \leq μ_{s} N and f_{k} = μ_{k} N$ : $f =$ friction, $μ_{s, k} =$ coefficient of (static,kinetic) friction, $N =$ normal force. ▭ Centripetal force $F_{c} = m v^{2} / r = m r ω^{2}$ for uniform circular motion. Angular velocity $ω$ is measured in radians per second. ^[18]▭ Drag equation $F_{D} = \frac{1}{2} C ρ A v^{2}$ where $C =$ Drag coefficient, $ρ =$ mass density, $A =$ area, $v =$ speed. Holds approximately for large Reynold's number^[19] <section end=Applications_of_Newton's_Laws/>

<section begin=Work_and_Kinetic_Energy/>7. Work_and_Kinetic_Energy: Infinitesimal work^[20] $d W = \vec{F} \cdot d \vec{r} = | \vec{F} | | d \vec{r} | \cos θ$ leads to the path integral $W_{A B} = \int_{A}^{B} \vec{F} \cdot d \vec{r}$ ▭ Work done from A→B by friction $- f_{k} | ℓ_{A B} |,$ gravity $- m g (y_{B} - y_{A}),$ and spring $- \frac{1}{2} k (x_{B}^{2} - x_{A}^{2})$ ▭ Work-energy theorem: ^[21] $W_{n e t} = K_{B} - K_{A}$ where kinetic energy $= K = \frac{1}{2} m v^{2} = \frac{p^{2}}{2 m}$ . ▭ Power $= P = d W / d t = \vec{F} \cdot \vec{v}$ . <section end=Work_and_Kinetic_Energy/>

<section begin=Potential_Energy_and_Conservation_of_Energy/>8. Potential_Energy_and_Conservation_of_Energy: Potential Energy: $Δ U_{A B} = U_{B} - U_{A} = - W_{A B}$ ; PE at $\vec{r}$ WRT ${\vec{r}}_{0}$ is $Δ U = U (\vec{r}) - U ({\vec{r}}_{0})$ $U = m g y + 𝒞$ (gravitational PE Earth's surface. $U = \frac{1}{2} k x^{2} + 𝒞$ (ideal spring) ▭ Conservative force: $\oint {\vec{F}}_{cons} \cdot d \vec{r} = 0$ . In 2D, $\vec{F} (x, y)$ is conservative if and only if $\vec{F} = - (\partial U / \partial x) \hat{i} - (\partial U / \partial y) \hat{j} ⟺ \partial F_{x} / \partial y = \partial F_{y} / \partial x$ ▭ Mechanical energy is conserved if no non-conservative forces are present: $0 = W_{n c, A B} = Δ (K + U)_{A B} = Δ E_{A B}$ <section end=Potential_Energy_and_Conservation_of_Energy/>

<section begin=Linear_Momentum_and_Collisions/>9. Linear_Momentum_and_Collisions: $\vec{F} (t) = d \vec{p} / d t, where \vec{p} = m \vec{v}$ is momentum. ▭ Impulse-momentum theorem $\vec{J} = F_{a v e} Δ t = \int_{t_{i}}^{t_{f}} \vec{F} d t = Δ \vec{p}$ . ▭ For 2 particles in 2D $If {\vec{F}}_{e x t} = 0 then \sum_{j = 1}^{N} {\vec{p}}_{j} = 0 \Rightarrow p_{f, α} = p_{1, i, α} + p_{2, i, α}$ where (α,β)=(x,y) ▭ Center of mass: ${\vec{r}}_{C M} = \frac{1}{M} \sum_{j = 1}^{N} m_{j} {\vec{r}}_{j} \to \frac{1}{M} \int \vec{r} d m,$ ${\vec{v}}_{C M} = \frac{d}{d t} {\vec{r}}_{C M}$ , and ${\vec{p}}_{C M} = \sum_{j = 1}^{N} m_{j} {\vec{v}}_{j} = M {\vec{v}}_{C M} .$ ▭ $\vec{F} = \frac{d}{d t} {\vec{p}}_{C M} = m {\vec{a}}_{C M} = \sum_{j = 1}^{N} m_{j} {\vec{a}}_{j}$ ^[22] <section end=Linear_Momentum_and_Collisions/>

<section begin=Fixed-Axis_Rotation/>10. Fixed-Axis_Rotation:

$θ = s / r$ is angle in radians, $ω = d θ / d t$ is angular velocity; ▭ $v_{t} = ω r = d s / d t$ is tangential speed. Angular acceleration is $α = d ω / d t = d^{2} θ / d t^{2}$ . $a_{t} = α r = d^{2} s / d t^{2}$ is the tangential acceleration. ▭ Constant angular acceleration $\bar{ω} = \frac{1}{2} (ω_{0} + ω_{f})$ is average angular velocity. ▭ $θ_{f} = θ_{0} + \bar{ω} t = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2} .$ ▭ $ω_{f} = ω_{0} + α t .$ $ω_{f}^{2} = ω_{0}^{2} + 2 α Δ θ .$ ▭ Total acceleration is centripetal plus tangential: $\vec{a} = {\vec{a}}_{c} + {\vec{a}}_{t} .$ ▭ Rotational kinetic energy is $K = \frac{1}{2} I ω^{2},$ where $I = \sum_{j} m_{j} r_{j}^{2} \to \int r^{2} d m$ is the Moment of inertia. ▭ parallel axis theorem $I_{p a r a l l e l - a x i s} = I_{c e n t e r o f m a s s} + m d^{2}$ ▭ Restricting ourselves to fixed axis rotation, $r$ is the distance from a fixed axis; the sum of torques, $\vec{τ} = \vec{r} \times \vec{F}$ requires only one component, summed as $τ_{n e t} = \sum τ_{j} = \sum r_{⊥_{j}} F_{j} = I α$ . ▭ Work done by a torque is $d W = (\sum τ_{j}) d θ$ . The Work-energy theorem is $K_{B} - K_{A} = W_{A B} = \int_{θ_{A}}^{θ_{B}} (\sum_{j} τ_{j}) d θ$ . ▭ Rotational power $= P = τ ω$ . <section end=Fixed-Axis_Rotation/>

<section begin=Angular_Momentum/>11. Angular_Momentum: Center of mass (rolling without slip) $d_{C M} = r θ,$ $v_{C M} = r ω,$ $a_{M C} = R α = \frac{m g \sin θ /}{m + (I_{c m} / r^{2})}$ ▭ Total angular momentum and net torque: $d \vec{L} / d t = \sum \vec{τ}$ $= {\vec{l}}_{1} + {\vec{l}}_{2} + . . .;$ $\vec{l} = \vec{r} \times \vec{p}$ for a single particle. $L_{t o t a l} = I ω .$ ▭ Precession of a top $ω_{P} = m r g / (I ω) .$ <section end=Angular_Momentum/>

<section begin=Static_Equilibrium_and_Elasticity/>12. Static_Equilibrium_and_Elasticity: Equilibrium $\sum {\vec{F}}_{j} = 0 = \sum {\vec{τ}}_{j} .$ Stress = elastic modulus · strain (analogous to Force = k · Δ x ) ▭ (Young's , Bulk , Shear) modulus: $(\frac{F_{⊥}}{A} = Y \cdot \frac{Δ L}{L_{0}}, Δ p = B \cdot \frac{- Δ V}{V_{0}}, \frac{F_{∥}}{A} = S \cdot \frac{Δ x}{L_{0}})$ <section end=Static_Equilibrium_and_Elasticity/>

<section begin=Gravitation/>13. Gravitation: Newton's law of gravity ${\vec{F}}_{12} = G \frac{m_{1} m_{2}}{r^{2}} {\hat{r}}_{12}$ ▭ Earth's gravity $g = G \frac{M_{E}}{r^{2}}$ ▭ Gravitational PE beyond Earth $U = - G \frac{M_{E} m}{r}$ ▭ Energy conservation $\frac{1}{2} m v_{1}^{2} - G \frac{M m}{r_{1}} = \frac{1}{2} m v_{2}^{2} - G \frac{M m}{r_{2}}$ ▭ Escape velocity $v_{e s c} = \sqrt{\frac{2 G M_{E}}{r}}$ ▭ Orbital speed $v_{o r b i t} = \sqrt{\frac{G M_{E}}{r}}$ ▭ Orbital period $T = 2 π \sqrt{\frac{r^{3}}{G M_{E}}}$ ▭ Energy in circular orbit $E = K + U = - \frac{G m M_{E}}{2 r}$ ▭ Conic section $\frac{α}{r} = 1 + e \cos θ$ ▭ Kepler's third law $T^{2} = \frac{4 π^{2}}{G M} a^{3}$ ▭ Schwarzschild radius $R_{S} = \frac{2 G M}{c^{2}}$ <section end=Gravitation/>

<section begin=Fluid_Mechanics/>14. Fluid_Mechanics: Mass density $ρ = m / V$ ▭ Pressure $P = F / A$ ▭ Pressure vs depth/height (constant density) $p = p_{o} + ρ g h \Leftarrow d p / d y = - ρ g$ ▭ Absolute vs gauge pressure $p_{a b s} = p_{g} + p_{a t m}$ ▭ Pascal's principle: $F / A$ depends only on depth, not on orientation of A. ▭ Volume flow rate $Q = d V / d t$ ▭ Continuity equation $ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}$ $\Rightarrow A_{1} v_{1} = A_{2} v_{2} if ρ = c o n s t .$ <section end=Fluid_Mechanics/>

<section begin=Oscillations/>15. Oscillations: Frequency $f$ , period $T$ and angular frequency $ω :$ $f T = 1, ω T = 2 π$ ▭ Simple harmonic motion $x (t) = A \cos (ω t + ϕ),$ $v (t) = - A ω \sin (ω t + ϕ),$ $a (t) = - A ω^{2} \cos (ω t + ϕ)$ also models the x-component of uniform circular motion. ▭ For $(A, ω)$ positive: $x_{m a x} = A, v_{m a x} = A ω, a_{m a x} = A ω^{2}$ ▭ Mass-spring $ω = 2 π / T = 2 π f = \sqrt{k / m};$ ▭ Energy $E_{T o t} = \frac{1}{2} k x^{2} + \frac{1}{2} m v^{2} = \frac{1}{2} k A^{2} \Rightarrow$ $v = \pm \sqrt{\frac{k}{m} (A^{2} - x^{2})}$ ▭ Simple pendulum $ω \approx \sqrt{g / L}$ ▭ Physical pendulum $τ = - M g L \sin θ \approx - M g L θ \Rightarrow$ $ω = \sqrt{m g L / I}$ and $L$ measures from pivot to CM. ▭ Torsional pendulum $τ = - κ θ$ $\Rightarrow ω = \sqrt{I / κ}$ ▭ Damped harmonic oscillator $m \frac{d^{2} x}{d t^{2}} = - k x - b \frac{d x}{d t}$ $\Rightarrow x = A_{0} e^{\frac{b}{2 m} t} \cos (ω t + ϕ)$ where $ω = \sqrt{ω_{0}^{2} - {(\frac{b}{2 m})}^{2}}$ and $ω_{0} = \sqrt{\frac{k}{m}} .$ ▭ ^[23]Forced harmonic oscillator (MIT wiki!)] $m \frac{d^{2} x}{d t^{2}} = - k x - b \frac{d x}{d t} + F_{0} \sin ω t$ $\Rightarrow x = A e^{\frac{b}{2 m} t} \cos (ω t + ϕ)$ where $A = \frac{F_{0}}{\sqrt{m^{2} (ω - ω_{0})^{2} + b^{2} ω^{2}}}$ . <section end=Oscillations/>

<section begin=Waves/>16. Waves: ^[24] Wave speed] (phase velocity) $v = λ / T = λ f = ω / k$ where $k = 2 π / λ$ is wavenumber. ▭ Wave and pulse speed of a stretched string $= \sqrt{F_{T} / μ}$ where $F_{T}$ is tension and $μ$ is linear mass density. ▭ Speed of a compression wave in a fluid $v = \sqrt{B / ρ} .$ ▭ Periodic travelling wave $y (x, t) = A \sin (k x \mp ω t)$ travels in the positive/negative direction. The phase is $k x \mp ω t$ and the amplitude is $A$ . ▭ The resultant of two waves with identical amplitude and frequency $y_{R} (x, t) = [2 A \cos (\frac{ϕ}{2})] \sin (k x - ω t + \frac{ϕ}{2})$ where $ϕ$ is the phase shift. ▭ This wave equation $\partial^{2} y / \partial t^{2} = v_{w}^{2} \partial^{2} y / \partial x^{2}$ is linear in $y = y (x, t)$ ▭ Power in a tranverse stretched string wave $P_{a v e} = \frac{1}{2} μ A^{2} ω^{2} v$ . ▭ Intensity of a plane wave $I = P / A \Rightarrow \frac{P}{4 π r^{2}}$ in a spherical wave. ▭ Standing wave $y (x, t) = A \sin (k x) \cos (ω t + ϕ)$ For symmetric boundary conditions $λ_{n} = 2 π / k_{n} = \frac{2}{π} L$ $n = 1, 2, 3, . . .$ , or equivalently $f = n f_{1}$ where $f_{1} = \frac{v}{2 L}$ is the fundamental frequency. <section end=Waves/>

<section begin=Sound/>17. Sound: Pressure and displacement fluctuations in a sound wave $P = Δ P_{m a x} \sin (k x \mp ω t + ϕ)$ and $s = s_{m a x} \cos (k x \mp ω t + ϕ)$ ▭ Speed of sound in a fluid $v = f λ = \sqrt{β / ρ}$ , ▭ in a solid $\sqrt{Y / ρ}$ , ▭ in an idal gas $\sqrt{γ R T / M}$ , ▭ in air $331 \frac{m}{s} \sqrt{\frac{T_{K}}{273 K}} = 331 \frac{m}{s} \sqrt{1 + \frac{T_{C}}{27 3^{o} C}}$ ▭ Decreasing intensity spherical wave $I_{2} = I_{1} {(\frac{r_{1}}{r_{2}})}^{2}$ ▭ Sound intensity $I = \frac{⟨ P ⟩}{A} = \frac{{(Δ P_{m a x})}^{2}}{2 ρ v}$ ▭ ...level $10 \log_{10} I / I_{0}$ ▭ Resonance tube One end closed: $λ_{n} = \frac{4}{n} L,$ $f_{n} = n \frac{v}{4 L},$ $n = 1, 3, 5, . . .$ ▭ Both ends open: $λ_{n} = \frac{2}{n} L,$ $f_{n} = n \frac{v}{2 L},$ $n = 1, 2, 3, . . .$ ▭ Beat frequency $f_{b e a t} = | f_{2} - f_{1} |$ ▭ (nonrelativistic) Doppler effect $f_{O} = f_{s} \frac{v \pm v_{o}}{v \mp v_{s}}$ where $v$ is the speed of sound, $v_{s}$ is the velocity of the source, and $v_{o}$ is the velocity of the observer. ▭ Angle of shock wave $\sin θ = v / v_{s} = 1 / M$ where $v$ is the speed of sound, $v_{s}$ is the speed of the source, and $M$ is the Mach number.

Footnotes

Template:Cot

↑ [http://wiki.ubc.ca/index.php?title=Uncertainty_and_Error&oldid=81540
↑ three orthonormal
↑
↑ ▭ $(𝐀 + 𝐁) \times 𝐂 = 𝐀 \times 𝐂 + 𝐁 \times 𝐂$ ▭ $𝐀 \cdot (𝐁 \times 𝐂) = 𝐁 \cdot (𝐂 \times 𝐀) = (𝐀 \times 𝐁) \cdot 𝐂$ ▭ $𝐀 \times (𝐁 \times 𝐂) = (𝐀 \cdot 𝐂) 𝐁 - (𝐀 \cdot 𝐁) 𝐂$ ▭ $(A \times B) \cdot (𝐂 \times 𝐃) = (𝐀 \cdot 𝐂) (𝐁 \cdot 𝐃) - (𝐁 \cdot 𝐂) (𝐀 \cdot 𝐃)$
↑ Delta as difference $Δ x = x_{f} - x_{i} \to d x \to 0$ in limit of differential calculus.
↑ , where $\bar{v} = \frac{1}{t} \int_{0}^{t} v (t^{'}) d t^{'}$ is the average velocity.
↑ ▭ Average values: ${\vec{v}}_{a v e} = \frac{Δ \vec{r}}{Δ t} = \frac{\vec{r} (t_{2}) - \vec{r} (t_{2})}{t_{2} - t_{1}}$ , and ${\vec{a}}_{a v e} = \frac{Δ \vec{v}}{Δ t} = \frac{\vec{v} (t_{2}) - \vec{v} (t_{2})}{t_{2} - t_{1}}$ ▭ Free fall time of flight $T_{o f} = \frac{2 (v_{0} \sin θ_{0})}{g},$ ▭ Trajectory $y = (\tan θ_{0}) x - [\frac{g}{2 (v_{0} \cos θ_{0})^{2}}] x^{2},$ ▭ Range $R = \frac{v_{0}^{2} \sin 2 θ_{0}}{g}$
↑ ▭ Tangential and centripetal acceleration $\vec{a} = {\vec{a}}_{c} + {\vec{a}}_{T}$ where $a_{T} = d | \vec{v} | / d t$ .
↑ ${\vec{r}}_{P S} = {\vec{r}}_{P S^{'}} + {\vec{r}}_{S^{'} S}$ ,
↑ ${\vec{v}}_{P C} = {\vec{v}}_{P A} + {\vec{v}}_{A B} + {\vec{v}}_{B C}$ , ${\vec{a}}_{P S} = {\vec{a}}_{P S^{'}} + {\vec{a}}_{S^{'} S}$
↑ Newton's 2^nd Law
↑ $m$ is mass, and
↑ because all internal forces cancel by the 3^rd law
↑ The 1^st law is that velocity is constant if the net force is zero.
↑ is a component of the contact force by the surface. If the only forces are contact and weight,
↑ where $θ$ is the angle of incline.
↑ Hooke's law
↑ ▭ Ideal angle of banked curve: $\tan θ = v^{2} / (r g)$ for curve of radius $r$ banked at angle $θ$ .
↑ $= R e = ρ v L / η$ , where $η =$ dynamic viscosity; $L =$ characteristic length. ▭ Stokes's law models a sphere of radius $r$ at small Reynold's number: $F_{s} = 6 π r η v$ .
↑ done by force:
↑ The work done on a particle is
↑ ▭ Rocket equation $m d v = - u d m \Rightarrow Δ v = u \ln (m_{f} / m_{i})$ where u is the gas speed WRT the rocket.
↑ [https://scripts.mit.edu/~srayyan/PERwiki/index.php?title=Module_3_--_Damped_and_Driven_Harmonic_Oscillations&oldid=7055
↑ [http://wiki.ubc.ca/index.php?title=Waves_and_the_Doppler_Effect&oldid=218637

Template:Cob

[1] [http://wiki.ubc.ca/index.php?title=Uncertainty_and_Error&oldid=81540

[2] three orthonormal

[3] ↑

[4] ▭ $(𝐀 + 𝐁) \times 𝐂 = 𝐀 \times 𝐂 + 𝐁 \times 𝐂$ ▭ $𝐀 \cdot (𝐁 \times 𝐂) = 𝐁 \cdot (𝐂 \times 𝐀) = (𝐀 \times 𝐁) \cdot 𝐂$ ▭ $𝐀 \times (𝐁 \times 𝐂) = (𝐀 \cdot 𝐂) 𝐁 - (𝐀 \cdot 𝐁) 𝐂$ ▭ $(A \times B) \cdot (𝐂 \times 𝐃) = (𝐀 \cdot 𝐂) (𝐁 \cdot 𝐃) - (𝐁 \cdot 𝐂) (𝐀 \cdot 𝐃)$

[5] Delta as difference $Δ x = x_{f} - x_{i} \to d x \to 0$ in limit of differential calculus.

[6] , where $\bar{v} = \frac{1}{t} \int_{0}^{t} v (t^{'}) d t^{'}$ is the average velocity.

[7] ▭ Average values: ${\vec{v}}_{a v e} = \frac{Δ \vec{r}}{Δ t} = \frac{\vec{r} (t_{2}) - \vec{r} (t_{2})}{t_{2} - t_{1}}$ , and ${\vec{a}}_{a v e} = \frac{Δ \vec{v}}{Δ t} = \frac{\vec{v} (t_{2}) - \vec{v} (t_{2})}{t_{2} - t_{1}}$ ▭ Free fall time of flight $T_{o f} = \frac{2 (v_{0} \sin θ_{0})}{g},$ ▭ Trajectory $y = (\tan θ_{0}) x - [\frac{g}{2 (v_{0} \cos θ_{0})^{2}}] x^{2},$ ▭ Range $R = \frac{v_{0}^{2} \sin 2 θ_{0}}{g}$

[8] ▭ Tangential and centripetal acceleration $\vec{a} = {\vec{a}}_{c} + {\vec{a}}_{T}$ where $a_{T} = d | \vec{v} | / d t$ .

[9] ${\vec{r}}_{P S} = {\vec{r}}_{P S^{'}} + {\vec{r}}_{S^{'} S}$ ,

[10] ${\vec{v}}_{P C} = {\vec{v}}_{P A} + {\vec{v}}_{A B} + {\vec{v}}_{B C}$ , ${\vec{a}}_{P S} = {\vec{a}}_{P S^{'}} + {\vec{a}}_{S^{'} S}$

[11] Newton's 2^nd Law

[12] $m$ is mass, and

[13] use all internal forces cancel by the 3^rd law

[14] The 1^st law is that velocity is constant if the net force is zero.

[15] s a component of the contact force by the surface. If the only forces are contact and weight,

[16] where $θ$ is the angle of incline.

[17] Hooke's law

[18] ▭ Ideal angle of banked curve: $\tan θ = v^{2} / (r g)$ for curve of radius $r$ banked at angle $θ$ .

[19] $= R e = ρ v L / η$ , where $η =$ dynamic viscosity; $L =$ characteristic length. ▭ Stokes's law models a sphere of radius $r$ at small Reynold's number: $F_{s} = 6 π r η v$ .

[20] y force:

[21] The work done on a particle is

[22] ▭ Rocket equation $m d v = - u d m \Rightarrow Δ v = u \ln (m_{f} / m_{i})$ where u is the gas speed WRT the rocket.

[23] [https://scripts.mit.edu/~srayyan/PERwiki/index.php?title=Module_3_--_Damped_and_Driven_Harmonic_Oscillations&oldid=7055

[24] [http://wiki.ubc.ca/index.php?title=Waves_and_the_Doppler_Effect&oldid=218637

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OpenStax University Physics/V1/Formulas (master)

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