Wright State University Lake Campus/2019-1/Phy2400/Final formula sheet

00-Mathematics for this course
 Circle's circumference (area): $$C_\odot = \,2\pi r \,(A_\odot = \,\pi r^2)$$.

Sphere's area (volume): $$A_\bigcirc = 4\pi r^2\,(V_\bigcirc = \frac{4}{3}\pi r^3)$$

Unit vectors: $$\vec A = A_x \, \hat i+ A_y \, \hat j$$. Other notations: $$(\hat x, \hat y)$$, and $$(\widehat {e_1}, \widehat {e_2})$$

$$\vec r = x\hat x + y\hat y$$. Magnitude: $$A \equiv |\vec A|= \sqrt{A_x^2 + A_y^2} \quad$$.

In first quadrant: $$\tan\theta=y/x$$ leads to $$\theta = \arctan{(y/x)} \quad$$

$$\vec A +\vec B =\vec C \Leftrightarrow$$ $$A_x + B_x =C_x$$ and $$A_y + B_y =C_y$$ 

01-Introduction
 
 * 1 kilometer = .621 miles and 1 MPH = 1 mi/hr ≈ .447 m/s
 * Typically air density is 1.2kg/m3, with pressure 105Pa. The density of water is 1000kg/m3.
 * Earth's mean radius ≈ 6371km, mass ≈ $6 kg$, and gravitational acceleration = g ≈ 9.8m/s2
 * Universal gravitational constant = G ≈ $6.67 m^{3}·kg^{−1}·s^{−2}$
 * 1 amu = 1 u ≈ 1.66 × 10-27 kg is the approximate mass of a proton or neutron.
 * 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.

02-One dimensional kinematics
 Difference: $$\delta\mathcal X$$: $$\Delta\mathcal X = \mathcal X_f - \mathcal X_i$$ Average (mean): $$\bar{\mathcal X}=\langle \mathcal X\rangle = \mathcal X_\text{ave}=\Sigma \mathcal X_i\mathcal/N \ \text{or} \ \Sigma \mathcal P_i\mathcal X_i$$. Average velocity: $$\bar v = \Delta x/\Delta t$$ 

03-Two-Dimensional Kinematics
 $$v^2=v_0^2+ 2a_x\Delta x + 2a_y\Delta y$$  ... 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:

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

04-Dynamics: Force and Newton's Laws
 $$ m\vec a = \sum\vec F_j \,$$ and $$\vec F_{ij}=-\vec F_{ji}$$.

For important forces: Normal $$(N\,or\,n)$$ is perpendicular to surface; Friction $$(f)$$ is parallel to surface; Tension $$(T)$$ is along a rope or string; Weight $$(w=mg)$$ where $$g\approx9.8m/s^2$$ at Earth's surface.

The x and y components of the three forces of tension on the small grey circle where the three "massless" ropes meet are:
 * $$T_{1x}=-T_1\cos\theta_1 $$,          $$ T_{1y}=T_1\sin\theta_1$$
 * $$T_{2x}=0 $$,                                   $$T_{2y}=-mg$$
 * $$T_{3x}=T_3\cos\theta_3 $$,             $$T_{3y}=T_3\sin\theta_3$$



05-Friction, Drag, and Elasticity
 
 * $$f_k=\mu_k N$$ is an approximation for the force friction when an object is sliding on a surface, where μk ("mew-sub-k") is the kinetic coefficient of friction, and N is the normal force.
 * $$f_s\le \mu_sN$$ approximates the maximum possible static friction that canoccurs before the object begins to slide.
 * Air drag: $$D$$ depends on velocity, size, and shape.

06-Uniform Circular Motion and Gravitation

 * $$2\pi\;rad=360\;deg=1\;rev$$ 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.


 * $$F = G \frac{mM}{r^2}= mg^*$$ is the force of gravity between two objects, where the universal constant of gravity is G ≈ 6.674 × 10-11 m3·kg−1·s−2.



07-Work and Energy
 
 * $$KE = \frac 1 2 m v^2\,$$ is kinetic energy, where m is mass and v is speed..
 * $$U_g = mgy$$ is gravitational potential energy,where y is height, and $$g = 9.80\;\frac{m}{s^2}$$ is the gravitational acceleration at Earth's surface.
 * $$U_s = \frac 1 2 k_s x^2$$ is the potential energy stored in a spring with spring constant $$k_s$$.
 * $$\sum KE_f + \sum PE_f = \sum KE_i + \sum PE_i -Q$$ relates the final energy to the initial energy. If energy is lost to heat or other nonconservative force, then Q>0.
 * $$W = F \ell\cos\theta=\vec F\cdot\vec\ell$$ (measured in Joules) is the work done by a force $$F$$ as it moves an object a distance $$\ell$$. The angle between the force and the displacement is &theta;.
 * $$\sum \vec F\cdot\Delta\ell$$ describes the work if the force is not uniform. The steps, $$\Delta\vec\ell$$, taken by the particle are assumed small enough that the force is approximately uniform over the small step.  If force and displacement are parallel, then the work becomes the area under a curve of F(x) versus x.
 * $$P=\frac{\vec F \cdot \vec\Delta \ell}{\Delta t}=\vec F\cdot\vec v$$ is the power (measured in Watts) is the rate at which work is done. (v is velocity.)

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08-Linear Momentum and Collisions
<section begin=08-Linear_Momentum_and_Collisions/>
 * $$ \vec p = m\vec v$$ is momentum, where m is mass and $$ \vec v$$ is velocity. The net momemtum is conserved if all forces between a system of particles are internal (i.e., come equal and opposite pairs):
 * $$\sum \vec p_f = \sum \vec p_i$$.
 * $$\bar F\Delta t = \Delta p$$ is the impulse, or change in momentum associated with a brief force acting over a time interval $$\Delta t$$. (Strictly speaking, $$\bar F$$ is a time-averaged force defined by integrating over the time interval.)

09-Statics and Torque
<section begin=09-Statics_and_Torque/>
 * $$\tau = rF\sin\theta,\!$$ is the torque caused by a force, F, exerted at a distance ,r, from the axis. The angle between r and F is &theta;.

The for torque is the  (N·m). It would be inadvisable to call this a Joule, even though a Joule is also a (N·m). The symbol for torque is typically τ, the Greek letter tau. When it is called moment, it is commonly denoted M. The lever arm is defined as either r, or r⊥. Labeling r as the lever arm allows moment arm to be reserved for r⊥. <section end=09-Statics_and_Torque/>

10-Rotational Motion and Angular Momentum
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11-Fluid statics
<section begin=11-Fluid_statics/> Pressure versus Depth: A fluid's pressure is F/A where F is force and A is a (flat) area. The pressure at depth, $$h$$ below the surface is the weight (per area) of the fluid above that point. As shown in the figure, this implies:


 * $$ P = P_0 + \rho g h $$

where $$P_0$$ is the pressure at the top surface, $$h$$ is the depth, and $$\rho$$ is the mass density of the fluid. In many cases, only the difference between two pressures appears in the final answer to a question, and in such cases it is permissible to set the pressure at the top surface of the fluid equal to zero. In many applications, it is possible to artificially set $$P_0$$ equal to zero, for example at atmospheric pressure. The resulting pressure is called the gauge pressure, for $$ P_{gauge} = \rho gh $$ below the surface of a body of water.

Buoyancy and Archimedes' principle Pascal's principle does not hold if two fluids are separated by a seal that prohibits fluid flow (as in the case of the piston of an internal combustion engine). Suppose the upper and lower fluids shown in the figure are not sealed, so that a fluid of mass density $$\rho_{flu}$$ comes to equilibrium above and below an object. Let the object have a mass density of $$\rho_{obj}$$ and a volume of $$A\Delta h$$, as shown in the figure. The net (bottom minus top) force on the object due to the fluid is called the buoyant force:


 * $$\rm{buoyant}\;\rm{force}=(A\Delta h )(\rho_{flu})g\,$$,

and is directed upward. The volume in this formula, AΔh, is called the volume of the displaced fluid, since placing the volume into a fluid at that location requires the removal of that amount of fluid. Archimedes principle states:


 * A body wholly or partially submerged in a fluid is buoyed up by a force equal to the weight of the displaced fluid.

Note that if $$\rho_{obj}=\rho_{flu}$$, the buoyant force exactly cancels the force of gravity. A fluid element within a stationary fluid will remain stationary. But if the two densities are not equal, a third force (in addition to weight and the buoyant force) is required to hold the object at that depth. If an object is floating or partially submerged, the volume of the displaced fluid equals the volume of that portion of the object which is below the waterline. <section end=11-Fluid_statics/>

12-Fluid dynamics
<section begin=12-Fluid_dynamics/> <section end=12-Fluid_dynamics/>
 * $$\frac{\Delta V}{ \Delta t } =\dot V=Av = Q$$ the volume flow for incompressible fluid flow if viscosity and turbulence are both neglected. The average velocity is $$v$$ and $$A$$ is the cross sectional area of the pipe.  As shown in the figure, $$v_1A_1=v_2A_2$$ because $$Av$$ is constant along the developed flow.  To see this, note that the volume of pipe is $$\Delta V=A\Delta x$$ along a distance $$\Delta x$$. And, $$v = \Delta x / \Delta t$$ is the volume of fluid that passes a given point in the pipe during a time $$\Delta t$$.
 * $$P_1 + \rho g y_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \frac{1}{2}\rho v_2^2$$ is Bernoulli's equation, where $$P$$ is pressure, $$\rho$$ is density, and $$y$$ is height. This holds for inviscid flow.

13-Temperature, Kinetic Theory, and Gas Laws
<section begin=13-Temperature,_Kinetic Theory,_and_Gas_Laws/> <section end=13-Temperature,_Kinetic Theory,_and_Gas_Laws/>
 * $$T_C=T_K-273.15$$ 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).
 * NAkB=R 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.
 * $$\frac 3 2 k_B T =\frac 1 2 mv_{rms}^2$$ 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 $$\varpi=3\;$$

14-Heat and Heat Transfer
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15-Thermodynamics
<section begin=15-Thermodynamics/> A point on a PV diagram define's the system's pressure (P) and volume (V). Energy (E) and pressure (P) can be deduced from equations of state: E=E(V,P) and T=T(V,P). If the piston moves, or if heat is added or taken from the substance, energy (in the form of work and/or heat) is added or subtracted. If the path returns to its original point on the PV-diagram (e.g., 12341 along the rectantular path shown), and if the process is quasistatic, all state variables (P, V, E, T) return to their original values, and the final system is indistinguishable from its original state.
 * Pressure (P), Energy (E), Volume (V), and Temperature (T) are state variables (state functionscalled state functions). The number of particles (N) can also be viewed as a state variable.
 * Work (W), Heat (Q) are not state variables.

The net work done per cycle is area enclosed by the loop. This work equals the net heat flow into the system, $$Q_{in} - Q_{out} $$ (valid only for closed loops).

Remember: Area "under" is the work associated with a path; Area "inside" is the total work per cycle.

<section end=15-Thermodynamics/>
 * $$\Delta W= -P\Delta V$$ 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).

16-Oscillatory Motion and Waves
<section begin=16-Oscillatory_Motion_and_Waves/>
 * CALCULUS: $$(d/dx)\sin kx=k\cos kx,\quad (d/dx)\cos kx=-k \sin kx$$

= x_0\cos{\frac{2\pi t}{T}}= x_0\cos{\omega t } $$ where T is period..
 * $$ x
 * $$ x(t) = x_0\cos\left(\omega_0 t - \varphi\right)$$. For example, $$\cos\left(\omega_0 t - \varphi\right)= \sin\omega_0 t$$.


 * $$\omega_0=\sqrt{\frac{k_s}{m}}$$ (m=mass, ks=spring constant.)$$\omega_0=\sqrt{\frac{g}{L}}$$ (L=lenghth of low-amplitude simple pendulum.)


 * $$PE=\frac 1 2 k_s x^2$$ is the potential energy of a mass spring system.

Let $$ x(t) = x_0\cos\left(\omega_0 t - \varphi\right) =$$ describe position: =a_0\cos\left(\omega_0 t + ...\right)$$, where $$a_0=\omega_0 v_0=\omega_0^2 x_0$$, is maximum acceleration. <section end=16-Oscillatory_Motion_and_Waves/>
 * $$v(t) = dx/dt = -\omega_0 x_0\sin\left(\omega_0 t - \varphi\right)=v_0\cos\left(\omega_0 t + ...\right)$$, where $$v_0=\omega_0 x_0$$ is maximum velocity.
 * $$ a(t) = dv/dt =-\omega_0 v_0\cos\left(\omega_0 t - \varphi\right)
 * $$F_0=m a_0$$, relates maximum force to maximum acceleration.
 * $$E=\frac 1 2 mv_0^2=\frac 1 2 k_s x_0^2$$ is the total energy.
 * CALCULUS: x(t) obeys the linear homogeneous differential equation (ODE), $$\frac{d^2x}{dt^2} = -\omega_0^2 x(t)$$
 * $$f\lambda = v_p$$ 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.

17-Physics of Hearing
<section begin=17-Physics_of_Hearing/> <section end=17-Physics_of_Hearing/>
 * $$v_s =\sqrt{  \frac{T}{273}    }\cdot 331 \text{m/s}$$ is the the approximate speed near Earth's surface, where the temperature, T, is measured in Kelvins.
 * $$v_s =\sqrt{  \frac{F}{\mu}    }$$ is the speed of a wave in a stretched string if $$F$$ is the tension and $$\mu$$ is the linear mass density (kilograms per meter).