Excigma
SocialProjectsNotes

Formulae


Non-uniform acceleration

Rotational Quantities

θ=s/r\theta = s/r
ω=ΔθΔt=v/r\omega = \frac{\Delta\theta}{\Delta t} = v/r
α=ΔωΔt=a/r\alpha = \frac{\Delta\omega}{\Delta t} = a/r

Circular motion

Fc=mv2r=mrω2F_c = \frac{mv^2}{r} = mr\omega^2
ac=v2r=rω2a_c = \frac{v^2}{r} = r\omega^2
v=ωrv = \omega r
ω=2πf\omega = 2 \pi f

Circular motion - Banked corners

(Fn)2−(Fg)2\sqrt{(F_n)^2 - (F_g)^2}

Simple harmonic motion

a=−x0ω2a = -x_0 \omega^2

Displacement

x=x0cos(ωt)x = x_0 cos(\omega t)

Velocity

x=−x0ω∗sin(ωt)x = -x_0 \omega * sin(\omega t)
a=±ω(x0)2−x2a = \pm \omega \sqrt{(x_0)^2 - x^2}

Acceleration

x=−x0ω2cos(ωt)x = -x_0 \omega^2 cos(\omega t)

Simple harmonic motion + Natural frequency

Pendulum

T=2πlgT = 2 \pi \sqrt{\frac{l}{g}}
f0=12πglf_0 = \frac{1}{2\pi} \sqrt{\frac{g}{l}}

Spring

T=2πmkT = 2 \pi \sqrt{\frac{m}{k}}
f0=12πkmf_0 = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

Energy in Simple harmonic motion

Etotal=Ep+EkE_{total} = E_p + E_k

Spring:

Etotal=12k(x0)2E_{total} = \frac{1}{2} k (x_0)^2

For moving system:

Etotal=12m(v0)2E_{total} = \frac{1}{2} m (v_0)^2
Etotal=12mω2(x0)2E_{total} = \frac{1}{2} m \omega^2 (x_0)^2

At any point:

Ek=12mω2((x0)2−x2)E_k = \frac{1}{2} m \omega^2((x_0)^2 - x^2)

For Pendulums

Etotal=mgh0E_{total} = mgh_0

Thermal Physics

Kelvin

T(k)=θ(deg⁡C)+273.15T(k) = \theta (\deg C) + 273.15

Ideal gasses

N=nNaN = nN_a
Number of particles=Number of moles∗Avogadro’s number\text{Number of particles} = \text{Number of moles} * \text{Avogadro's number}
n=mMn = \frac{m}{M}
Number of moles=MassMolar mass\text{Number of moles} = \frac{\text{Mass}}{\text{Molar mass}}
ΔW=PΔV\Delta W = P \Delta V

Gas laws

Boyle's law

P∝1VP \propto \frac{1}{V}
PV=ConstantPV = \text{Constant}

Charles' law

  • TT : Temperature in Kelvin
    T∝VT \propto V
P∝TP \propto T
PV=nRTPV = nRT

Mean squared speed

Ek=12m<c2>=32nNRTE_k = \frac{1}{2} m <c^2> = \frac{3}{2} \frac{n}{N}RT

  • KK : Boltzmann constant

    Ek=32KTE_k = \frac{3}{2} KT

  • NmNm : Total mass

    PV=NKT=13Nm<C2>PV = NKT = \frac{1}{3} Nm <C^2>

Internal Energy

U=ΣEk+ΣEpU = \Sigma E_k + \Sigma E_p

Specific heat capacity

  • Δh\Delta h : Heat added from/to the environment
  • ΔQ\Delta Q : Heat added from/to the system
    Δh+ΔQ=mcΔT\Delta h + \Delta Q = mc \Delta T

Specific latent heat

Q=mLQ = mL

Sensing devices and Ultrasound

Acoustic impedance

Z=ρv\Zeta = \rho v

Ratio of reflected to incident Ultrasound

  • RR : Ratio
  • IrI_r : Intensity reflected
    R=IrI0=(z2−z1)2(z2+z1)2R = \frac{I_r}{I_0} = \frac{(z_2 - z_1)^2}{(z_2 + z_1)^2}

Attenuation of Ultrasound

TAttenuated=I0e−μxT_{Attenuated} = I_0 e^{-\mu x}

Strain Gauge

R=μLAR = \frac{\mu L}{A}

Potential divider

Vout=Vin∗RTR+RTV_{out} = \frac{V_{in} * R_T}{R + R_T}

Electronics

Op-amps

  • A0A_0 : Open-loop gain of op-amp
    A0=VoutVinA_0 = \frac{V_{out}}{V_{in}}
Vout=A0(V+−V−)V_{out} = A_0(V^+ - V^-)

Gain of inverting feedback loop

  • RfR_f : Resistance of inverting feedback loop
    A=VoutVin=−RfRA = \frac{V_{out}}{V_{in}} = \frac{-R_f}{R}

Gain of non-inverting feedback loop

  • RfR_f : Resistance of non-inverting feedback loop
    A=VoutVin=1+−RfR1A = \frac{V_{out}}{V_{in}} = 1 + \frac{-R_f}{R_1}

Signal attenuation

dB=10∗log(PoutPin)dB = 10 * log(\frac{P_{out}}{P_{in}})
dB=Attenuation per unit length∗LengthdB = \text{Attenuation per unit length} * \text{Length}

Signal to noise Ratio

SNR(dB)=10∗log(PSignalPNoise)SNR (dB) = 10 * log(\frac{P_{Signal}}{P_{Noise}})

Gravitational fields

Orbit

  • GG : Gravitational constant: 6.67x10−116.67x10^{-11}
    Fc(Required)=mv2r=GMmr2=Fg(Provided)F_c \text{(Required)} = \frac{mv^2}{r} = \frac{GMm}{r^2} = F_g \text{(Provided)}

Newton's law of Gravitation

Fg=GMmr2F_g = \frac{GMm}{r^2}

Gravitational field strength

g=GMr2g = \frac{GM}{r^2}

Gravitational Potential Energy

Ep=−GMmrE_p = \frac{-GMm}{r}

Gravitational Potential

ϕ=−GMr\phi = \frac{-GM}{r}

Escape Velocity

ΔEp(Gain)=Ep(∞)−Ep(Surface)\Delta E_p \text{(Gain)} = E_p (\infin) - E_p \text{(Surface)}

since

ΔEp(Gain)=ΔEk(Lost)\Delta E_p \text{(Gain)} = \Delta E_k \text{(Lost)}
GMmre=12mv2\frac{GM\xcancel{m}}{r_e} = \frac{1}{2} \xcancel{m}v^2
2GMre=vescape\sqrt{\frac{2GM}{r_e}} = v_{escape}

Gravitational Potential

Φ=−GMr\Phi = \frac{-GM}{r}

Keplar's third law

T2∝r3T^2 \propto r^3

Electric fields

F=EqF = Eq

Coulomb's law

  • ϵ0=Permittivity of free space=8.85∗10−12\epsilon_0 = \text{Permittivity of free space} = 8.85 * 10^{-12}
    F=14πϵ0∗Q1Q2r2=KQ1Q2r2F = \frac{1}{4 \pi \epsilon_0} * \frac{Q_1 Q_2}{r^2} = \frac{K Q_1 Q_2}{r^2}

Electric field strength

E=q14πϵ0r2E = \frac{q_1}{4 \pi \epsilon_0 r^2}

Electric potential between two points

ΔVAB=Kq1Δr=Kq1(1rB−1rA)\Delta V_{AB} = \frac{Kq_1}{\Delta r} = K q_1 (\frac{1}{r_B} - \frac{1}{r_A})

Potential Gradient

E=−∇VE = - \nabla V

Capacitance

C=QtotalVtotalC = \frac{Q_{total}}{V_{total}}
C∝A; C∝1dC \propto A \text{; } C \propto \frac{1}{d}

Capacitors in Series

1Ctotal=1C1+1C2+...+1Cn\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}

Capacitors in Parallel

Ctotal=C1+C2+...+CnC_{total} = C_1 + C_2 + ... + C_n

Energy in Capacitors

W=12QVW = \frac{1}{2}QV
W=12CV2W = \frac{1}{2}CV^2

Charged conducting isolated space

V=Q4πϵR=KQRV = \frac{Q}{4 \pi \epsilon R} = \frac{KQ}{R}
Since C=QV, C=4πϵ0R\text{Since } C = \frac{Q}{V} \text{, } C = 4 \pi \epsilon_0 R

Magnetic fields

Magnetic Flux density

B=FILB = \frac{F}{IL}

Force on a current-carrying conductor inside a magnetic field

  • θ\theta : Angle between BB and II
    F=BIL∗sin(θ)F = BIL * sin(\theta)

Force on a charged particle moving through a magnetic field

  • θ\theta : Angle between BB and vv
    F=Bqv∗sin(θ)F = Bqv * sin(\theta)

Hall Effect

  • nn : Density of charge carriers
  • tt : Thickness of wafer
    VH=IBnqtV_H = \frac{IB}{nqt}

Excigma's Webpage

Made by Excigma during 2021 to assist notekeeping and study.