Mechanics Formulae for Physics Proficiency Exams

 

Forces, work, energy and momentum.

 

v = dx/dt;   a = dv/dt;    v = v₀ + at;    x = x₀ + v_ot + at²/2;    v² = v_o² + 2aD;

W = mg;  F = ma;  f = mN;   W = F ^. D;   K = mv²/2;  

U_g = mgh;   U_s = kx²/2;  F_x = -dU/dx;  x_cm = S m_ix_i / Sm_i; 

p = mv;   F = dp/dt; 

 

Rotational motion.

 

q = s/r;  w = dq/dt;   a = dw/dt; ;  a_c = v²/r;  K = Iw²/2;   I = S m_ir_i²;  t = Ia;  t = r x F; 

L = m(r x v);  L = Iw;  SF = 0;  St = 0; 

 

Oscillators, gravity.

 

F = -kx:  E = ½kx²;  y = A cos(wt + f);  w = 2p/T;  T = 1/f;  w = (k/m)^1/2;    T = 2p ( l/g)^1/2; 

F = GM₁M₂/r²;   G = 6.67 x 10^-11 N m²/kg²;    U_g = -GMm/r;  K = GMm/2r;

 T² = r³ (4p²)/(GM)

 

Fluids, waves, sound.

 

r = m/v;   p = F/A;   p = rgh;  p + 0.5 rv² + rgh = const;  

y = A sin (kx - wt);  k = 2p/l;  w = 2p/T;  v = lf = w/k;   v = (t/m) ^1/2;  

b =10 log(I/I_o) dB;    f' = f (v+v_D)/(v+v_S) ; 

 

Thermo, heat, kinetic theory.

 

T_F= 1.8 T_C + 32;  DL/L = a DT;   Q = cmDt;  Q = Lm;   DE_int = Q_in - W_done; 

DQ/Dt = kA DT/L;  KE_av = 1.5 kT where k = R/N_A =  1.38 x 10^-23 J/K;

 

Constants. 

 

N_A = 6.02 x 10²³;   R = 8.31 J/mol K

 Electricity and Magnetism Formulae for Physics Proficiency Exams

 

 

Constants:

 

e = 1.6 x 10^-19 C;  k = 1/(4pe_o) = 9 x 10⁹ N m²/C²; m_o = 4 p x 10 ^-7 T m/A;

c = 3 x 10⁸ m/s;   1 eV = 1.6 x 10^-19 J; m_e = 9.11 x 10^-31 kg

 

Electric fields, resistance, circuits, capacitors:

 

F = kQ₁Q₂/r²;    E = F/q;    F_E =  E ^. dA = Q_encl/e_o;    DV = -E ^. dr;  V = kq/r

-E = i dV/dx + j dV/dy + k dV/dz;     i = V/R;    R = rL/A;   r - r_o = r_o a DT;

P = iV;  Kirchhoff: S_loopV = 0; i_in = i_out;  

q=CV;   C = e_oA/d ;U_c = ½ CV²; q = q_o e^-t/RC or q = q_f (1 - e^-t/RC); t_c = RC;

 

Magnetic fields and electromagnetism:

 

F_B = qv x B;    qvB = mv²/r;     F_B = iL x B;

dB = (m₀/4p)(i ds x r)/r³;     B_line = m₀i/2pr;  B_circle = m₀i/2r; B ^. ds = m₀i;   

B_sol = m₀ni; 

E = -dF_B/dt;  E = -L di/dt;  L_sol = m₀n²A;   t_L = L/R;   U_L = ½ Li²;   w² = 1/LC;

c = (e_om_o)^-1/2;    E_max = B_maxc;    S = E x B/m_o;   Dp = DI/c;

 

Optics:

 

n₁ sin q₁ = n₂ sin q₂;   1/p + 1/i = 1/f = 2/r;  1/f = (n-1) [r₁^-1-r₂^-1]; 

sin q = 1.22 l/d; 

c = (e_om_o)^-1/2;    E_max = B_maxc;    S = E x B/m_o;   Dp = DI/c;

 

Math:

 

The solutions to a quadratic equation:  ax² + bx + c = 0 are

Surface Area of a Sphere is: 4pr²  :   Volume is 4pr³/3

 

 

Modern Physics Formulae for Physics Proficiency Exams

 

Special Relativity: 

 

b = v/c;  g = (1 - b²)^-1/2;      T = T_og;   L = L_o/g

Binomial Expansion:  g = 1 + b²/2 + 3b⁴/8 + …

Lorentz:  x’ = g (x-bct);     y’ = y;     z’ = z;     ct’ = g (ct - bx)

velocity transform:  v_x’ = (v_x - u)/(1 - v_xu/c²)                  Doppler:    n’  =  n [ (1-b) / (1+b) ]^½

Kinematics:  E = g mc²;   pc = bgmc²;  E² = (pc)² + (mc²)²;       K = E - mc²

 

Origins of Quantum Theory: 

 

Photoelectric effect:   K_max = hn - f        :       de Broglie wavelength:  l = h / p

Blackbody radiation:  I = s T⁴;                      l_max T = 2.898 10^-3 m K

Compton scattering:  l’ - l = (hc/m_ec²) (1 - cos q)

Electron diffraction:  nl = d sin f          Heisenberg:  Dx Dp_x ~ h/2p;   DE Dt ~ h/2p

Schrödinger Equation:  -(h²/8p²m)d²y/dx² +Uy = Ey                P(x)dx = |y(x)|² dx

SHO:  E_n = (n + ½) (h/2p) w

Rutherford Scattering: N(q) = (nt/4r²) (zZ/2K)² (e²/4pe_o)² (sin (q/2))^-4

Hydrogen atom:  E_n = -13.6 eV / n²       l = (1/R) n₁²n₂²/(n₁² - n₂²)         R = 1.097 x 10⁷ m^-1

dV = r² sin q dr dq df  ;  P(r) = r² |R(r)|^{2  ;}  orbital:  m_L = -(e/2m) L  ;  m_L,z = -m_lm_B

x-rays:  E_a = 13.6 eV (Z-1)² (3/4);  

 

Statistics:  

 

p(E)_MB ~ E^1/2 /e^E/kT ;    p(E)_BE ~ E²/(e^E/kT -1);   

p(E)_FD~ E^1/2 /{exp[(E-E_F)/kT] + 1};   E_F = (h²/2m) (3N/8pV) ^2/3

 

Nuclear: 

 

R = (1.2 x 10^-15 m)A^1/3;  dN/dt = -lN;  l = 1/t = 0.693/t_1/2;  K_a =(A-4)Q/A

R = sNI_o/S;  N(t) = (R/l) (1 - e^-lt); A = lN; 

(for x + X  -> y + Y)   K_threshold = -Q (1 + m_x/m_X)

(for x + X  -> A + B + C + …)  K_threshold = -Q (m_x + m_X + m_A + m_B + m_C + …)/(2m_X)

Mossbauer Effect: recoil energy  K = E_g²/(2Mc²) , v ~ cDE/E

 

Particle:

 

leptons e m t ;  quarks [charge +2/3  u c t] [charge -1/3 d s b]

constants:  hc = 1240 eV nm;  m_e c²= 511,000 eV;  m_B = eh/8pm = 9.274 x 10^-24 J/T

 h = 6.63 10^-34 J s = 4.14 10^-15 eV s;  k_Boltz = 8.617 10^-5 eV/K;  e²/4pe₀ = 1.440 eV nm

1 u (or amu) = 931.5 MeV/c²    :   m_ec² = 511000 eV;

e = 1.6 x 10^-19 C;  k = 1/(4pe_o) = 9 x 10⁹ N m²/C² ; e_o = 8.85 x  10^-12 C²/(N m²);

N_Avogad = 6.02 x 10²³;     m₀ = 4p x 10^-7 T m/A;     hc = 1240 eV nm;