Wright State University Lake Campus/University Physics Volume 1/Equations

WSU Lake

Math
Metric prefixes on page 5.

&#9645; Arclength $$s$$ &#9645; $$r$$ radius &#9645; $$\theta = s/r$$ (in radians) &#9645; $$C_\odot = \,2\pi r$$ Circumference of circle &#9645; $$A_\odot = \,\pi r^2$$ Area of circle &#9645; $$A_\bigcirc = 4\pi r^2$$ Area of sphere &#9645; $$V_\bigcirc = \frac{4}{3}\pi r^3$$ Volume of sphere &#9645; $$\vec\mathcal F\cdot d\vec s $$$$=\mathcal F_xdx+\mathcal F_ydy = \mathcal F_rdr+\mathcal F_\theta rd\theta$$

Symbols
&#9645; $$\vec r = x\hat i + y\hat j + z\hat k = x\hat x + y\hat y + z\hat z$$ Position &#9645; $$\Delta\vec\ell, d\vec\ell, \Delta\vec s, d\vec s $$ Line element (better than $$d\vec r$$) &#9645; $$\vec v =d\vec r/dt$$ Velocity [m/s] &#9645; $$\vec a =d\vec v/dt$$ Acceleration [m/s] &#9645; $$\vec F=m \vec a$$ Force [N], Mass [kg], Acceleration [m/s2] &#9645; $$f$$ Friction force [N] or frequency [Hz=s&minus;1] &#9645; $$\mu_{s,k}$$ (static, kinetic) coefficient of friction &#9645; $$\vec N$$ normal force &#9645; $$g$$ gravitational constant (Earth: 9.8m/s2) G≈$6.67 m^{3}·kg^{−1}·s^{−2}$ &#9645; $$W$$ Work [J=Nm=kg(m/s)2] &#9645; $$P$$ Power [W=J/s] &#9645; $$\vec p$$ Momentum &#9645; $$p$$ Pressure (N/m2) &#9645; $$KE (PE)$$ Kinetic (Potential) Energy [J=Nm=Ws] &#9645; $$(U_g, U_s)$$ Potential energy (gravity, spring) &#9645; $$F=-k_s x$$ defines spring constant [N/m] &#9645; $$T$$ Tension [N] or period [s] or temperature [K] &#9645; $$Q$$ Heat [J=Ws] &#9645; $$\theta$$ Angle (radians) &#9645; $$\omega=d\theta/dt$$ Angular velocity (speed) [s&minus;1] &#9645; $$\alpha=d\omega/dt$$ Angular acceleration [s&minus;2] &#9645; $$I=\Sigma m_ir_i^2$$ Moment of inertia &#9645; $$\tau =I\alpha$$ Torque &#9645;$$L=I\omega$$ Angular momentum

Equations
&#9645; $$KE = \frac 1 2 m v^2\,$$ &#9645; $$U_g = mgy+\mathcal C$$ &#9645; $$U_s = \frac 1 2 k_s x^2$$ &#9645; $$\sum KE_f + \sum PE_f = \sum KE_i + \sum PE_i -Q$$ &#9645; $$W = F \ell\cos\theta=\vec F\cdot\vec\ell\Rightarrow\sum \vec F\cdot\Delta\ell\Rightarrow\int \vec F\cdot\Delta\ell$$ &#9645; $$P=\frac{\vec F \cdot \vec\Delta \ell}{\Delta t}=\vec F\cdot\vec v$$ is power &#9645; $$\vec p = m\vec v$$ Momentum &#9645; $$\sum \vec p_f = \sum \vec p_f + \int \vec F_{ext} dt$$ &#9645; $$m_1v_1+m_2v_2=(m_1+m_2)v_f$$ Inelastic collision &#9645; $$\omega T = 2\pi \Leftrightarrow fT=1$$ frequency-period &#9645; $$a=v^2/r=\omega v = \omega^2 r$$ centripetal acceleration &#9645; $$\sum \vec F_j=0=\sum\vec \tau_j$$ $$\tau=rF_\perp$$ statics &#9645; $$f_s\le\mu_sN ,\; f_k=\mu_kN$$ friction &#9645; $$\vec F_{12}=G\tfrac{m_1m_2}{r^2}\hat r_{12}$$ Newtonian gravity &#9645; $$\vec F_{12}=G\tfrac{m_1m_2}{r^2}\hat r_{12}$$Newtonian gravity &#9645; $$g=G\tfrac{M}{r^2}$$ planet's surface gravity

Fluid_Mechanics
&#9645; Mass density $$\rho=m/V\;$$

&#9645; Pressure $$P=F/A\;$$

Buoyant force $$B$$ equals weight of the displaced fluid. If $$W$$ is the weight of a cylindrical object, the displaced volume is $$A\Delta h$$

&#9645; $$B=\rho_{flu}(A\Delta h )g$$ and &#9645; $$W = \rho_{obj}(A\Delta h )g = M_{obj}g$$

&#9645; Pressure vs depth/height (constant density)$$\,p=p_o+\rho gh \Leftarrow dp/dy = -\rho g$$

&#9645; Volume flow rate $$Q=dV/dt\;$$

&#9645; Continuity equation $$\rho_1A_1v_1=\rho_2A_2v_2$$$$\Rightarrow A_1v_1=A_2v_2\text{ if }\rho=const.\;$$

&#9645; Bernoulli's principle $$p_1+\tfrac 1 2 \rho v_1^2 + \rho gy_1=p_2+\tfrac 1 2 \rho v_2^2 + \rho gy_2$$

Oscillations
&#9645; Simple harmonic motion $$x(t)=A\cos(\omega t+\phi),\, $$ $$v(t)=-A\omega\sin(\omega t+\phi),\,$$ $$ a(t)=-A\omega^2\cos(\omega t+\phi)$$

&#9645; $$\,x_{max}=A,\;v_{max}=A\omega,\;a_{max}=A\omega^2 $$, where $$ \omega=\sqrt{k/m};\,$$ and $$k$$ is spring constant

&#9645; Energy $$E_{Tot}=\tfrac 1 2 kx^2 + \tfrac 1 2 mv^2 = \tfrac 1 2 mv_{max}^2 = \tfrac 1 2 kx_{max}^2$$


 * $$v_s =\sqrt{  \frac{T}{273}    }\cdot 331 \text{m/s}$$ speed of sound (T in Kelvins). $$v_s=\sqrt{\frac{\gamma k_B T}{m}}$$ where $$\gamma \approx 1.4$$
 * $$v =\sqrt{  \frac{F}{\mu}    }$$ Speed of a stretched string wave: $$F$$ is the tension and $$\mu$$ is the linear mass density (kilograms per meter).
 * $$f\lambda = v_p$$ relates the frequency, f, wavelength, &lambda;,and the the phase speed, vp of the wave (also written as vw)   This phase speed is the speed of individual crests, which for sound and light waves also equals the speed at which a wave packet travels.
 * $$L = \frac{n\lambda_n}{2}$$ describes the n-th normal mode vibrating wave on a string that is fixed at both ends (i.e. has a node at both ends). The mode number, n = 1, 2, 3,..., as shown in the figure.
 * Beat frequency: The frequency of beats heard if two closely space frequencies, $$f_1$$ and $$f_2$$, are played is $$\Delta f = |f_2-f_1|$$.
 * Musical acoustics: Frequency ratios of 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, 8/5 are called the (just) "octave", "fifth",    "fourth", "major-sixth", "major-third", "minor-third", and "minor-sixth",  respectively.

Constants and conversions

 * Boltzmann's constant = kB≈ 1.38 × 10-23 J K−1, and the gas constant is R = NAkB≈8.314 J  K−1  mol−1, where NA≈ 6.02 × 1023 is the Avogadro number.
 * Boltzmann's constant can also be written in eV and Kelvins: kB ≈8.6 × 10-5 eV/deg, where 1eV=1.602x10&minus;19 Joules
 * $$T_C=T_K-273.15$$ converts from Celsius to Kelvins, and $$T_F=\frac 9 5 T_C+32$$ converts from Celsius to Fahrenheit.
 * 1 amu = 1 u ≈ 1.66 × 10-27 kg is the approximate mass of a proton or neutron.

13-Temperature, Kinetic Theory, and Gas Laws

 * $$PV=nRT=Nk_BT$$ is the ideal gas law, where P is pressure, V is volume, n is the number of moles and N is the number of atoms or molecules. Temperature is in Kelvins.


 * $$\frac 3 2 k_B T =\frac 1 2 mv_{rms}^2$$ is the average translational kinetic energy per "atom" of a 3-dimensional ideal gas.
 * $$v_{rms}=\sqrt{\frac{3k_BT}{m}}=\sqrt{\overline{v^2}}$$ is the root-mean-square speed of atoms in an ideal gas.

14-Heat and Heat Transfer
 Here it is convenient to define heat as energy that passes between two objects of different temperature $$Q$$ The SI unit is the Joule. The rate of heat trasfer, $$\Delta Q/\Delta t$$ or $$\dot Q$$ is "power": 1 Watt = 1 W = 1J/s
 * $$Q = mc_S\Delta T$$ is the heat required to change the temperature of a substance of mass, m. The change in temperature is &Delta;T.  The specific heat, cS, depends on the substance (and to some extent, its temperature and other factors such as pressure).  Heat is the transfer of energy, usually from a hotter object to a colder one.  The units of specfic heat are energy/mass/degree, or J/(kg-degree).
 * $$Q = mL$$ is the heat required to change the phase of a a mass, m, of a substance (with no change in temperature). The latent heat, L, depends not only on the substance, but on the nature of the phase change for any given substance.  LF is called the latent heat of fusion, and refers to the melting or freezing of the substance.  LV is called the latent heat of vaporization, and refers to evaporation or condensation of a substance.
 * $$\dot Q= \frac{k_cA}{d}\Delta T$$ is rate of heat transfer for a material of area, A. The difference in temperature between two sides separated by a distance, d, is $$\Delta T$$. The thermal conductivity, kc, is a property of the substance used to insulate, or subdue, the flow of heat.

15-Thermodynamics

 * Here, Pressure (P), Energy (E), Volume (V), and Temperature (T) are the state functions.
 * The net work done per cycle is the area enclosed by the loop and equals the net heat flow into the system, $$Q_{in} - Q_{out} $$ (valid only for closed loops).
 * $$\Delta W=-F\Delta x= ( -P\cdot\mathrm{Area} ) \left( \frac{\Delta V}{\mathrm{Area}} \right) = -P\Delta V$$ is the work done on a system of pressure P by a piston of voulume V. If ΔV>0 the substance is expanding as it exerts an outward force, so that ΔW<0 and the substance is doing work on the universe; ΔW>0 whenever the universe is doing work on the system.
 * $$\Delta Q$$ is the amount of heat (energy) that flows into a system. It is positive if the system is placed in a heat bath of higher temperature. If this process is reversible, then the heat bath is at an infinitesimally higher temperature and a finite &Delta;Q takes an infinite amount of time.
 * $$\Delta E=\Delta Q-P\Delta V$$ is the change in energy (First Law of Thermodynamics).
 * $$ -\int P \ dV$$ is work done on system. $$ \oint P \ dV = Q_{in} - Q_{out}$$ is work (out) per cycle.

Original (long) formula sheet

 * 1) $$C_\odot = \,2\pi r$$ and the circle's area is $$A_\odot = \,\pi r^2$$ is its area.
 * 2) The surface area of a sphere is $$A_\bigcirc = 4\pi r^2$$ and sphere's volume is $$V_\bigcirc = \frac{4}{3}\pi r^3$$
 * 3) = .621 miles and  1 MPH = 1 mi/hr ≈ .447 m/s
 * 4) is 1.2kg/m3, with pressure 105Pa. The density of water is 1000kg/m3.
 * 5) = G ≈ $6 kg$
 * 6) = c ≈ 3×108m/s
 * 7) .... 
 * 1) .... 
 * 1) .... 



$$v^2=v_0^2+ 2a_x\Delta x + 2a_y\Delta y$$  ... in advanced notation this becomes  $$\Delta (v^2) = 2\vec a\cdot\Delta\vec\ell$$.

In free fall we often set, ax=0 and  ay= -g. If angle is measured with respect to the x axis:

$$ v_x = v\cos\theta$$     $$ v_y =  v\sin\theta$$     $$ v_{x0} =  v_0\cos\theta_0$$     $$ v_{y0} =  v_0\sin\theta_0$$






 * $$T_{1x}=-T_1\cos\theta_1 $$,          $$ T_{1y}=T_1\sin\theta_1$$
 * $$T_{2x}=0 $$,                                   $$T_{2y}=-mg$$
 * $$T_{3x}=T_3\cos\theta_3 $$,             $$T_{3y}=T_3\sin\theta_3$$


 * $$2\pi\;rad=360\;deg=1\;rev$$ relates the radian, degree, and revolution.
 * $$f = \frac{\#\,\text{revs}}{\#\,\text{secs}}$$ is the number of revolutions per second, called frequency.
 * $$T = \frac{\#\,\text{secs}}{\#\,\text{revs}}$$ is the number of seconds per revolution, called period. Obviously $$fT = 1$$.
 * $$\omega = \frac{\Delta\theta}{\Delta t}$$ is called angular frequency (ω is called omega, and &theta; is measured in radians). Obviously $$\omega T = 2\pi$$
 * $$a=\frac{v^2}{r}=\omega v =\omega^2 r$$ is the acceleration of uniform circular motion, where v is speed, and r is radius.
 * $$v=\omega r =2\pi r/T$$, where T is period.

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 $$331\tfrac m s\sqrt{\tfrac{T_K}{273\,K}}=331\tfrac m s\sqrt{1+\tfrac{T_C}{273^oC}}$$


 * $$f\lambda = v_p$$ relates the frequency, f, wavelength, &lambda;,and the the phase speed, vp of the wave (also written as vw)   This phase speed is the speed of individual crests, which for sound and light waves also equals the speed at which a wave packet travels.
 * $$L = \frac{n\lambda_n}{2}$$ describes the n-th normal mode vibrating wave on a string that is fixed at both ends (i.e. has a node at both ends). The mode number, n = 1, 2, 3,..., as shown in the figure.
 * Beat frequency: The frequency of beats heard if two closely space frequencies, $$f_1$$ and $$f_2$$, are played is $$\Delta f = |f_2-f_1|$$.
 * Musical acoustics: Frequency ratios of 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, 8/5 are called the (just) "octave", "fifth",    "fourth", "major-sixth", "major-third", "minor-third", and "minor-sixth",  respectively.


 * $$T_C=T_K-273.15$$ converts from Celsius to Kelvins, and $$T_F=\frac 9 5 T_C+32$$ converts from Celsius to Fahrenheit.
 * $$PV=nRT=Nk_BT$$ is the ideal gas law, where $$P$$ is pressure, $$V$$ is volume, $$n$$ is the number of moles and $$N$$ is the number of atoms or molecules. Temperature must be measured on an absolute scale (e.g. Kelvins).
 * Boltzmann's constant = $$k_B$$≈ 1.38 × 10-23 J K−1, and the gas constant is $$R=N_Ak_B$$ ≈8.314 J  K−1  mol−1, where $$N_A$$≈ 6.02 × 1023 is the Avogadro number. Boltzmann's constant can also be written in eV and Kelvins: kB ≈8.6 × 10-5 eV/deg.
 * $$\frac 3 2 k_B T =\frac 1 2 mv_{rms}^2$$ is the average translational kinetic energy per "atom" of a 3-dimensional ideal gas.
 * $$v_{rms}=\sqrt{\frac{3k_BT}{m}}=\sqrt{\overline{v^2}}$$ is the root-mean-square speed of atoms in an ideal gas.
 * $$E = \frac{\varpi}{2}Nk_BT$$ is the total energy of an ideal gas, where $$\varpi=3\;$$ only if the gas is monatomic.