OpenStax University Physics/V1/Formulas (master)



Introduction: 

'''1. Units_and_Measurement''': The base SI units are mass: kg (kilogram); length: m (meter); time: s (second).

Percent error is $$(\delta A/A)\times 100%$$ 

'''2. Vectors''': Vector $$\vec A = A_x \, \hat i+ A_y \, \hat j + A_z \, \hat k$$ involves components (Ax,Ay,Az) and unit vectors. &#9645; If $$\vec A + \vec B =\vec C$$, then Ax+Bx=Cx, etc, and vector subtraction is defined by $$\vec B =\vec C - \vec A$$. &#9645; 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 $$\theta=\tan^{-1}{(y/x)} $$. &#9645; Scalar multiplication $$\alpha\vec A = \alpha A_x \hat i     +  \alpha A_y \hat j+...\quad$$ &#9645; Any vector divided by its magnitude is a unit vector and has unit magnitude: $$|\hat V| = 1$$ where $$\hat V \equiv \vec V/V$$ &#9645; Dot product $$\vec A\cdot\vec B = AB\cos\theta = A_xB_x+A_yB_y+...\quad$$ and $$\vec A\cdot\vec A = A^2$$ &#9645; Cross product $$\vec A = \vec B \times \vec C\Rightarrow$$ $$A_\alpha = B_\beta C_\gamma -C_\gamma A_\beta$$ where $$(\alpha,\beta,\gamma)$$ is any cyclic permutation of $$(x,y,z)$$, i.e., (&alpha;,&beta;,&gamma;) represents either (x,y,z) or (y,z,x) or (z,x,y). &#9645; Cross-product magnitudes obey $$A=BC\sin\theta$$  where $$\theta$$ is the angle between $$\vec B$$ and $$\vec C$$,  and $$\vec A\perp\{\vec B,\vec C\}$$ by the right hand rule. &#9645; Vector identities $$ \;c (\mathbf{A}+\mathbf{B})=c\mathbf{A}+c\mathbf{B}\quad$$ &#9645; $$ \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}\quad$$ &#9645; $$ \mathbf{A}+(\mathbf{B}+\mathbf{C})= (\mathbf{A}+\mathbf{B})+\mathbf{C}\quad$$ &#9645; $$ \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}\quad$$ &#9645; $$ \mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A}\quad$$ &#9645; $$ \left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C} +\mathbf{B}\cdot\mathbf{C} \quad$$ 

'''3. Motion_Along_a_Straight_Line''': &#9645; Average velocity $$\bar v = \Delta x /\Delta t  \rightarrow v=dx/dt$$ (instantaneous velocity) &#9645; Acceleration $$\bar a = \Delta v/\Delta t \rightarrow a=dv/dt$$. &#9645; WLOG set $$\Delta t=t\;$$ and $$\Delta x = x-x_0\;$$ if $$t_i=0$$. Then $$\Delta v = v-v_0$$, and $$\; v(t) = \int_0^t a(t')dt' + v_0 $$, $$\;x(t) = \int_0^t v(t')dt' + x_0 = x_0 + \bar v t $$ &#9645; At constant acceleration: $$\bar v = \tfrac{v_0+v}{2},\quad v=v_0+at,\quad x=x_0+ v_0t+\tfrac 1 2 at^2,\,$$ $$ v^2=v_0^2+2a\Delta x$$. &#9645; For free fall, replace $$x\rightarrow y$$ (positive up) and $$a\rightarrow -g$$, where $$g$$ = 9.81 m/s2 at Earth's surface). 

'''4. Motion_in_Two_and_Three_Dimensions''': Instantaneous velocity: $$\vec v(t)=v_x(t)\hat i + v_y(t)\hat j + v_z(t)\hat k = \frac{dx}{dt}\hat i + \frac{dy}{dt}\hat j + \frac{dz}{dt}\hat k$$ &#9645; $$\vec v(t)= \lim_{\Delta t\rightarrow 0}{\tfrac{\Delta \vec r}{\Delta t}} =\lim_{\Delta t\rightarrow 0}{\tfrac{\vec r(t+\Delta t)-\vec r(t)}{\Delta t}}$$, where $$\vec r(t) = x(t)\hat i + y(t)\hat j + z(t)\hat k$$ &#9645; Acceleration $$\vec a = a_x\hat i + a_y\hat j + a_z\hat k$$, where $$a_x(t)=dv_x/dt=d^2x/dt^2$$. &#9645; Uniform circular motion: position $$\,\vec r(t)$$, velocity $$\,\vec v(t)=d\vec r(t)/dt$$, and acceleration $$\,\vec a(t)=d\vec v(t)/dt$$: $$\vec r = A\cos\omega t\hat i + A\sin\omega t\hat j \,,\,$$ $$ \vec v = -A\omega\sin\omega t\hat i + A\omega\cos\omega t\hat j \,,\,$$ $$ \vec a = -A\omega^2\cos\omega t\hat i - A\omega^2\sin\omega t\hat j\,. $$ Note that if $$A=r$$ then $$|\vec a|=a_C=\omega^2r = v^2/r$$ where $$v\equiv|\vec v|=\omega r$$. &#9645; Relative motion: $$\,\vec v_{PS}=\vec v_{PS'}+\vec v_{S'S}$$, 

'''5. Newton's_Laws_of_Motion''': $$\;m\vec a = d\vec p/dt =\sum \vec F_j$$, where $$\vec p =m\vec v$$ is momentum, $$\sum \vec F_j$$ is the sum of all forces This sum  needs only include external forces $$\vec F_{AB}=-\vec F_{BA}$$. &#9645; Weight$$=\vec w=m\vec g$$. &#9645; normal force $$|\vec N|=N=mg\cos\theta$$ &#9645; $$F=-kx$$ where $$k$$ is the spring constant. 

'''6. Applications_of_Newton's_Laws''': $$f_s\le\mu_sN \text{ and } f_k=\mu_kN$$: $$\,f=$$ friction, $$\mu_{s,k}=$$ coefficient of (static,kinetic) friction, $$N=$$ normal force. &#9645; Centripetal force$$F_c=m v^2/r=mr\omega^2$$ for uniform circular motion. Angular velocity $$\omega$$ is measured in radians per second. &#9645; Drag equation $$F_D=\tfrac 1 2 C \rho A v^2$$ where $$C=$$ Drag coefficient, $$\rho=$$ mass density, $$A=$$ area, $$v=$$ speed. Holds approximately for large Reynold's number 

'''7. Work_and_Kinetic_Energy''': Infinitesimal work $$ dW=\vec F\cdot d\vec r = |\vec F|\,|d\vec r|\cos\theta$$ leads to the path integral $$W_{AB}=\int_A^B\vec F\cdot d\vec r$$ &#9645; Work done from A&rarr;B by friction $$-f_k|\ell_{AB}|,\;$$gravity $$-mg(y_B-y_A),\;$$ and spring $$-\tfrac 1 2 k\left(x_B^2-x_A^2\right)$$ &#9645; Work-energy theorem: $$W_{net}=K_B-K_A$$ where kinetic energy $$=K=\tfrac 1 2 mv^2 = \frac{p^2}{2m}$$. &#9645; Power$$=P=dW/dt=\vec F\cdot \vec v$$. 

<section begin=Potential_Energy_and_Conservation_of_Energy/>'''8. Potential_Energy_and_Conservation_of_Energy''': Potential Energy: $$\Delta U_{AB} = U_B-U_A=-W_{AB}$$; PE at $$\vec r$$ WRT  $$\vec r_0$$ is $$\Delta U = U(\vec r)-U(\vec r_0)$$ $$U=mgy+\mathcal{C}$$ (gravitational PE Earth's surface. $$U=\tfrac 1 2 k x^2+\mathcal{C}$$ (ideal spring) &#9645; Conservative force:  $$\oint\vec F_{\text{cons}}\cdot d\vec r = 0$$. In 2D, $$\vec F(x,y)$$ is conservative if and only if $$ \vec F = -(\partial U/\partial x) \,\hat i  -(\partial U/\partial y) \,\hat j \iff \partial F_x/\partial y = \partial F_y/\partial x$$ &#9645; Mechanical energy is conserved if no non-conservative forces are present: $$0=W_{nc,AB}=\Delta (K+U)_{AB} =\Delta E_{AB}$$ <section end=Potential_Energy_and_Conservation_of_Energy/>

<section begin=Linear_Momentum_and_Collisions/>'''9. Linear_Momentum_and_Collisions''': $$\vec F(t) = d\vec p/dt \text{, where } \vec p=m\vec v$$ is momentum. &#9645; Impulse-momentum theorem $$\vec J = F_{ave}\Delta t = \int_{t_i}^{t_f} \vec F dt = \Delta\vec p$$. &#9645; For 2 particles in 2D $$\text{If } \vec F_{ext}=0 \text{ then } \sum_{j=1}^N\vec p_j =0 \Rightarrow p_{f,\alpha}=p_{1,i,\alpha}+p_{2,i,\alpha}$$ where (&alpha;,&beta;)=(x,y) &#9645; Center of mass: $$\vec r_{CM}= \tfrac 1 M \sum_{j=1}^N m_j\vec r_j \rightarrow \tfrac 1 M \int\vec r dm, $$ $$\vec v_{CM}= \tfrac{\,d}{dt}\vec r_{CM}$$, and $$\vec p_{CM}=\sum_{j=1}^N m_j\vec v_j =M\vec v_{CM}.$$ &#9645; $$\vec F = \tfrac{\,d}{dt}\vec p_{CM}=m\vec a_{CM}=\sum_{j=1}^Nm_j\vec a_j$$ <section end=Linear_Momentum_and_Collisions/>

<section begin=Fixed-Axis_Rotation/>'''10. Fixed-Axis_Rotation''': $$\theta=s/r\,$$ is angle in radians,$$\,\omega=d\theta/dt\,$$ is angular velocity; &#9645; $$\,v_t=\omega r=ds/dt\,$$ is tangential speed. Angular acceleration is $$\alpha=d\omega/dt=d^2\theta/dt^2\,$$. $$a_t=\alpha r= d^2s/dt^2\,$$ is the tangential acceleration. &#9645; Constant angular acceleration $$\bar\omega=\tfrac 1 2 (\omega_0+\omega_f)\,$$ is average angular velocity. &#9645; $$\theta_f=\theta_0+\bar\omega t = \theta_0+\omega_0t+\tfrac 1 2 \alpha t^2\,.$$ &#9645; $$\omega_f=\omega_0+\alpha t.\,$$ $$\omega_f^2=\omega_0^2 + 2\alpha\Delta\theta\,.$$ &#9645; Total acceleration is centripetal plus tangential: $$\vec a = \vec a_c + \vec a_t.\,$$ &#9645; Rotational kinetic energy is $$K=\tfrac 1 2 I\omega^2,\,$$ where $$I =\sum_j m_j r_j^2\rightarrow\int r^2 dm$$ is the Moment of inertia. &#9645; parallel axis theorem $$I_{parallel-axis}=I_{center\,of\,mass}+md^2$$ &#9645; Restricting ourselves to fixed axis rotation, $$r$$ is the distance from a fixed axis; the sum of torques, $$\vec\tau=\vec r\times\vec F$$ requires only one component, summed as $$\tau_{net}=\sum\tau_j=\sum r_{\perp_j}F_j=I\alpha$$. &#9645; Work done by a torque is $$dW=\left(\sum\tau_j\right)d\theta$$. The Work-energy theorem is $$K_B-K_A=W_{AB}=\int_{\theta_A}^{\theta_B}\left(\sum_j\tau_j\right)d\theta$$. &#9645; Rotational power $$=P=\tau\omega$$. <section end=Fixed-Axis_Rotation/>

<section begin=Angular_Momentum/>'''11. Angular_Momentum''': Center of mass (rolling without slip) $$d_{CM}= r\theta,\;$$ $$v_{CM}= r\omega,\;$$$$a_{MC}=R\alpha= \tfrac{mg\sin\theta/}{m+\left(I_{cm}/r^2\right)} $$ &#9645; Total angular momentum and net torque: $$d\vec L/dt = \sum \vec\tau $$ $$= \vec l_1+\vec l_2 + ...;$$ $$\vec l = \vec r\times\vec p\,$$ for a single particle. $$L_{total}=I\omega.$$ &#9645; Precession of a top $$\omega_P=mrg/(I\omega). $$<section end=Angular_Momentum/>

<section begin=Static_Equilibrium_and_Elasticity/>'''12. Static_Equilibrium_and_Elasticity''': Equilibrium $$\sum \vec F_j=0=\sum\vec \tau_j.\,$$ Stress = elastic modulus &middot; strain (analogous to Force = k &middot; &Delta; x ) &#9645; (Young's, Bulk , Shear) modulus: $$\left( \tfrac{F_\perp}{A}=Y\cdot\tfrac{\Delta L}{L_0}\,,\; \Delta p          =B\cdot\tfrac{-\Delta V}{V_0}\,,\; \tfrac{F_\parallel}{A}=S\cdot\tfrac{\Delta x}{L_0}\right) $$ <section end=Static_Equilibrium_and_Elasticity/>

<section begin=Gravitation/>'''13. Gravitation''': Newton's law of gravity $$\vec F_{12}=G\tfrac{m_1m_2}{r^2}\hat r_{12}$$ &#9645; Earth's gravity $$g=G\tfrac{M_E}{r^2}$$ &#9645; Gravitational PE beyond Earth $$U =-G\tfrac{M_Em}{r}$$ &#9645; Energy conservation $$\tfrac 1 2 mv_1^2-G\tfrac{Mm} {r_1}=\tfrac 1 2 mv_2^2-G\tfrac{Mm}{r_2}$$ &#9645; Escape velocity $$v_{esc} =\sqrt{\tfrac{2GM_E}{r}}$$ &#9645; Orbital speed $$v_{orbit}=\sqrt{\tfrac{GM_E}{r}}$$ &#9645; Orbital period $$T=2\pi\sqrt{\tfrac{r^3}{GM_E}}$$ &#9645; Energy in circular orbit $$E=K+U=-\tfrac{GmM_E}{2r}$$ &#9645; Conic section $$\tfrac{\alpha}{r}=1+e\,\!\cos\theta$$ &#9645; Kepler's third law$$T^2=\tfrac{4\pi^2}{GM}a^3$$ &#9645; Schwarzschild radius $$R_S=\tfrac{2GM}{c^2}$$ <section end=Gravitation/>

<section begin=Fluid_Mechanics/>'''14. Fluid_Mechanics''': Mass density $$\rho=m/V\;$$ &#9645; Pressure $$P=F/A\;$$ &#9645; Pressure vs depth/height (constant density)$$\,p=p_o+\rho gh \Leftarrow dp/dy = -\rho g$$ &#9645; Absolute vs gauge pressure $$\,p_{abs}=p_g+p_{atm} \;$$ &#9645; Pascal's principle: $$\,F/A\,$$ depends only on depth, not on orientation of A. &#9645; Volume flow rate $$Q=dV/dt\;$$ &#9645; Continuity equation $$\rho_1A_1v_1=\rho_2A_2v_2$$$$\Rightarrow A_1v_1=A_2v_2\text{ if }\rho=const.\;$$ <section end=Fluid_Mechanics/>

<section begin=Oscillations/>'''15. Oscillations''': Frequency $$f$$, period $$T$$ and angular frequency $$ \omega\,:\;$$ $$fT=1\,,\quad\omega T=2\pi$$ &#9645; Simple harmonic motion $$x(t)=A\cos(\omega t+\phi),\, $$ $$v(t)=-A\omega\sin(\omega t+\phi),\,$$ $$ a(t)=-A\omega^2\cos(\omega t+\phi)$$ also models the x-component of uniform circular motion. &#9645; For $$(A,\omega)$$ positive: $$\,x_{max}=A,\;v_{max}=A\omega,\;a_{max}=A\omega^2 $$ &#9645; Mass-spring $$ \omega=2\pi/T= 2\pi f=\sqrt{k/m};\,$$ &#9645; Energy $$E_{Tot}=\tfrac 1 2 kx^2 + \tfrac 1 2 mv^2 = \tfrac 1 2 kA^2 \Rightarrow $$$$v = \pm\sqrt{\tfrac k m \left(A^2-x^2\right)}$$ &#9645; Simple pendulum $$ \omega\approx\sqrt{g/L}$$ &#9645; Physical pendulum $$\tau=-MgL\sin\theta\approx -MgL\theta \Rightarrow \;$$$$\omega=\sqrt{mgL/I}$$ and $$L$$ measures from pivot to CM. &#9645; Torsional pendulum $$\tau=-\kappa\theta$$$$ \Rightarrow \omega =\sqrt{I/\kappa}$$ &#9645; Damped harmonic oscillator $$ m\tfrac{d^2x}{dt^2}=-kx-b\tfrac{dx}{dt} $$$$  \Rightarrow x=A_0e^{\frac{b}{2m}t}\cos{(\omega t + \phi)}$$ where $$ \omega=\sqrt{\omega_0^2-\left(\tfrac{b}{2m}\right)^2}$$ and $$ \omega_0=\sqrt\tfrac k m.$$ &#9645; Forced harmonic oscillator (MIT wiki!)] $$ m\tfrac{d^2x}{dt^2}=-kx-b\tfrac{dx}{dt}+F_0\sin\omega t $$$$ \Rightarrow x=Ae^{\frac{b}{2m}t}\cos{(\omega t + \phi)}$$ where $$ A= \tfrac{F_0}{\sqrt{ m^2(\omega-\omega_0)^2   +   b^2\omega^2}}$$. <section end=Oscillations/>

<section begin=Waves/>'''16. Waves''': Wave speed] (phase velocity) $$v=\lambda/T=\lambda f = \omega/k$$ where $$k=2\pi/\lambda$$ is wavenumber. &#9645; Wave and pulse speed of a stretched string $$=\sqrt{F_T/\mu}$$ where $$F_T$$ is tension and $$\mu$$ is linear mass density. &#9645; Speed of a compression wave in a fluid $$v=\sqrt{B/\rho}.$$ &#9645; Periodic travelling wave $$y(x,t)=A\sin(kx\mp\omega t)$$ travels in the positive/negative direction. The phase is $$kx\mp\omega t$$ and the amplitude is $$A$$. &#9645; The resultant of two waves with identical amplitude and frequency $$ y_R(x,t) = \left[2A \cos\left( \tfrac\phi 2 \right) \right]\sin\left(kx-\omega t + \tfrac\phi 2\right) $$ where $$\phi$$ is the phase shift. &#9645; This wave equation $$\partial^2 y/\partial t^2 = v_w^2\,\partial^2 y/\partial x^2$$ is linear in $$y=y(x,t)$$ &#9645; Power in a tranverse stretched string wave $$P_{ave}=\tfrac 1 2 \mu A^2\omega^2v$$. &#9645; Intensity of a plane wave $$I= P/A \Rightarrow \tfrac{P}{4\pi r^2}$$ in a spherical wave. &#9645; Standing wave $$y(x,t)=A\sin(kx)\cos(\omega t+\phi)$$ For symmetric boundary conditions $$\lambda_n=2\pi/k_n=\tfrac 2\pi L$$ $$n=1,2,3,...$$, or equivalently $$f=nf_1$$ where $$f_1=\tfrac{v}{2L} $$ is the fundamental frequency. <section end=Waves/>

<section begin=Sound/>'''17. Sound''': Pressure and displacement fluctuations in a sound wave $$P=\Delta P_{max}\sin(kx\mp\omega t +\phi)$$ and $$s=s_{max}\cos(kx\mp\omega t +\phi)$$ &#9645; Speed of sound in a fluid $$v=f\lambda=\sqrt{\beta/\rho}$$, &#9645; in a solid $$\sqrt{Y/\rho}$$, &#9645; in an idal gas $$\sqrt{\gamma RT/M}$$, &#9645; in air $$331\tfrac m s\sqrt{\tfrac{T_K}{273\,K}}=331\tfrac m s\sqrt{1+\tfrac{T_C}{273^oC}}$$ &#9645; Decreasing intensity spherical wave $$I_2=I_1\left(\tfrac{r_1}{r_2}\right)^2$$ &#9645; Sound intensity $$I=\tfrac{\langle P\rangle}{A}=\tfrac{\left(\Delta P_{max}\right)^2}{2\rho v}$$ &#9645; ...level $$10\log_{10}{I/I_0}$$ &#9645; Resonance tube One end closed: $$\lambda_n=\tfrac 4 n L,$$ $$f_n=n\tfrac {v}{4L},$$ $$n=1,3,5,...$$ &#9645; Both ends open: $$\lambda_n=\tfrac 2 n L,$$ $$f_n=n\tfrac {v}{2L},$$ $$n=1,2,3,...$$ &#9645; Beat frequency $$f_{beat}=|f_2-f_1|$$ &#9645; (nonrelativistic) Doppler effect $$f_O=f_s\tfrac{v\pm v_o}{v\mp v_s}$$ where $$v$$ is the speed of sound, $$v_s$$ is the velocity of the source, and $$v_o$$ is the velocity of the observer. &#9645; Angle of shock wave   $$\sin\theta=v/v_s=1/M$$ where $$v$$ is the speed of sound, $$v_s$$ is the speed of the source, and $$M$$ is the Mach number. <section end=Sound/>