Physics Essentials: Vibrations, Waves, Gravity, Fields, Optics

Posted on Nov 20, 2024 in Physics

Vibratory Motion (MAS) – Radians

X(t) = a sin (wt + fi), A = x_max

w = 2π / T = 2πnu, nu = 1/T = 1/nu

v_transversal = dx / dt

a = –w² · x

x = Acoswt = acos (wt + π/2)

v = Awcos (wt + fi), v_max = Aw

v = ±w · √(A²–x²)

F = –kx, K = Mw²-√ w = k / m (kN/m)

Simple Pendulum – Theta

θ = s / l

T = 2π · √(L / g)

s = θ x L

cos θ = Lh / L

w = √(g / L)

θ(t) = θ_m sin (wt + fi) – fi then removed

Energy of an Oscillator

E_c = 1/2mv²

E_p = E_p(x), W = E_p

For a Spring

E_p = 1/2kx²

E_c = 1/2mv²

E_t = E_p + E_c = constant

In the extremes of the pendulum, E_c is zero, so E_t = 1/2kA²

Wave Motion – Radians

f(x, t) = Asen(kx – wt)

kvT = 2π

T = 2π/kv

kv = 2π / T = w

w = kv, v = wr

λ = 2π / k, k = 2π/λ

f(x, t) = Acos(kx – wt)

c = λ / T

Waves in Three Dimensions

I = P / S = P/4πr² (in W/m²)

P = E / t

I ∝ 1/r² ∝ A² ∝ NU²

A = 1 / r

Sound – Properties and Velocity

v = √(γ · RT / M) (T in K, M – molecular mass, γ – adiabatic coefficient, R = 0.082 atm·L/kg·mol)

101300 Pa = 1 atm = 760 mmHg

β (sound intensity level, dB) = 10 log (I / I_o)

Gravitation

1st Law

Planets describe ellipses with the Sun at one focus.

2nd Law

The line joining a planet and the Sun sweeps out equal areas in equal times.

3rd Law

The squares of the periods of revolution are proportional to the cubes of the semimajor axes (R³ – T² ∝).

T₁² / R₁³ = T₂² / R₂³

E_p = mgh

Elastic: E_p = 1/2kx²

E_p = –GMm / R

Gravitational Field and Force

F = GmM/R²

g = GM/r²

Gravitational potential: U = E_p / m = –GM / R (J/kg)

–GMm/R²

Circular Orbits

mv² / R = Mw² · R_t

v_orbital = √(GM / R)

v_escape = √(2GM / R) = √(2gR_T)

Energy: v_orbital = √(GM / R) → E = 1/2mv² – GMm / R = 1/2mGM/R – GMm / R = GMm / R (1/2 – 1) = –GMm / 2R

For elliptical orbits, R is replaced by 2a

a = w² · 4πR/T² = R = GM/R² → T² = 4πR³/GM

Central Forces – Momentum

Momentum (p) = mv

Angular momentum (L) = mwR²

Conservation of L: dL/dt = d / dt (r x p) = dr / dt x p + dp / dt x r = r x F = M → dL / dt = M

L = constant if no force is applied or if the force is parallel to the radius.

Electric Field

-kqq F = ‘/ d2 (K = 9.10 ⁹Nm2/C2), k = 1/4piepsilon _0,, epsilon = 1/4pik,, E = F / q (N / C or V / m ), F = Eq. ELECTROMAGNETISM-V = PE / q ‘(in V), for point charge -> V = q/4piepsilon r,, V =-Ex, for plane-capacitor V = Ed tma gauss-flux = q / epsilon (in Vm) for point charge q/4oiepsilon-E = r2, infinite-flow thread = L · landa / epsilon = E2pid x L -> E = landa/2pid epsilon, landa = q / L (linear density of q, in C / m). infinite plane-q / s = sigma-surface charge density in C/m2,, E = sigma/2epsilon, sphere-Q/4pi epsilon E = r2,, charge density (f)-f = Q / V (in C/m3.) MAGNETIC FIELD-B in Tesla (T), F = qvB,,,, when q in B-nwton-2nd law F = m · a, mv2/R- qvB => R = mv / qB,, v = rw-> w = v/R-> mv / QBR = 0 -> mw / qB = 0 -> w = qB / m (wes indep of v), application-cyclotron, spectrograph mass-Fe = qE; Fm = qvB,, Fe = Fm-> qE = qvB-> v = e / B_1, R = mv / qB ₂-> m = RqB ₂/ v, Boit-Savart law of simple-current-> dB = mu _orI/4pi · xr/r3 dL,, rectilinear current-> B = mu _orI/2pid,, FZA on electric current magnetic-F = I x L x B ((MU ₀= 1O ^-7· 4pi N / m)). INDUCTION EM-Faraday-Lentz law: V = – d flow / dt (W-weber). v max = NBSw, forces between parallel I-F / l = mu _orI ₁I ₂/ 2pid,, law ampere-1I-> closed integral of B × dl = Muoi,, +1 I-> integ closed B × mu0 dL = (I1 + I2-I3) (eg) in solenoid-B = muonium / L (N = number of turns).,, lane laplace-V = – Bav, in an increase Elektrim-Vm = Magnetic flux / Increm t,, Ve / Vs = Ne / Ns.OPTICAL-c = · 1/raiz of epsilon mu = 3.10 ⁸m / s, gamma, X, UV, visible, IR, MO, OR. q fecuencia and energy than OR.,, landa = c / nu,, E = h nu (h = 6.63 10 ^-34Js), E = 1.6 10 ^-19J = 10 ⁴eV, PROPAGATION OF LIGHT-apparent size-d / L = alpha (in rad), vel light-n = c / v (n = refractive index), the law of reflection-i = r,, law of refraction, n ₁= n ₂sin t seni,, angle limit = i _L= N1/N2,, seni / n2/n1 = v1/v2 = sin t,, parallel-sided sheet-i1 = r2 = i,, i2 = r1 = r, delta = s Chosroes sin (ir),, sin t = sin / n,, optical prism-Snell-1 sin i = n sin t, n sin r ‘= 1 sin i’,, external ang – fi = r + r ‘,, delta min = i + i’ – fi = 2i – fi (the delta min situation only occurs in symmetric-r = r ‘, i = i’), n = sin i / sin r, r = fi / 2, delta min = 2i-2r,, n . MIRRORS AND LENSES SPHERICAL-THIN-DIOPTERS in paraxial optical, and Snell’s law remains N’i-ni = ‘,, f = – n / n’-N • R,, f’ = n ‘/ n’ – N • R,, f + f ‘= R,, M _L= y’ / y = n s ‘/ n’ s,, char-image real or virtual-real-rays are cut, virt-cut extensions ,, right or upside-cda-and up, and ‘on down, upside-xa both up or down, may or smaller than the object, DIOPTERS PLANO-n’ / s’ = n / s = 0. Plane mirror-i = i ‘, n’ =-n,, s ‘= s,, M _L= ns’ / n s = 1,, y ‘= y,, spherical mirror-N’ =- N,, 1 / S ‘+ 1 / S = 2 / r,, F = F’ = r / 2, 1 / S ‘+1 / S = 1 / F (sensitive), M _L= y’ / y = n s ‘/ n’ s =-s’ / s-> y ‘/ y = – s’ / s. THIN-LENS 1 / f ‘= 1 / s’ – 1 / s = (n-1) (1/R1 – 1/r2), f =-f ‘,, P = 1 / f’ (in diopters) ,, M_L= y / s = y ‘/ s’-> y’ / y = s’ / s