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gina-plett / PHYS. 131 Final

April 16, 2025 21:58
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PHYS. 131 Final

A comprehensive cheat sheet covering key concepts in fluid mechanics, oscillations, waves, sound, interference, and nuclear physics, designed for quick reference and effective problem-solving.

Fluids: Statics and Dynamics

Fluid Properties & Pressure

Density (ρ):
Mass per unit volume. ρ = m/V (kg/m³)

Pressure (P):
Force per unit area. P = F/A (N/m² or Pa)

Hydrostatic Pressure:
Pressure at depth h in a fluid.
P = hρg, where g is the acceleration due to gravity (9.8 m/s²)

Gauge Pressure:
Pressure relative to atmospheric pressure.
P_gauge = P_absolute - P_atm

Absolute Pressure:
Total pressure including atmospheric pressure.
P_absolute = P_gauge + P_atm

Atmospheric Pressure:
Standard atmospheric pressure.
1 atm = 101325 Pa

Pascal’s Principle:
Pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and the walls of the container.

Hydraulic Systems:
F₁/A₁ = F₂/A₂ (Force amplification)

Buoyant Force (F_b):
Upward force exerted by a fluid on an immersed object.
F_b = V_displaced * ρ_fluid * g

Archimedes’ Principle:
The buoyant force on an object is equal to the weight of the fluid displaced by the object.

Apparent Weight:
Weight of an object in a fluid.
W_apparent = W_object - F_b

Manometer:
Measures pressure differences via column height difference.

Barometer:
Uses fluid column to measure atmospheric pressure.

Blood Pressure:
Measured in mmHg, systolic/diastolic readings.

Flow Rate (Q):
Volume of fluid passing a point per unit time.
Q = V/t = Av

Continuity Equation:
For incompressible fluids: A₁v₁ = A₂v₂

Streamlines:
Lines representing the path of fluid particles in a steady flow.

Viscosity (η):
A measure of a fluid’s resistance to flow.

Fluid Dynamics

Bernoulli’s Equation:
Relates pressure, velocity, and height in a flowing fluid.
P + ½ρv² + ρgh = constant

Torricelli’s Law:
Speed of fluid exiting an opening at depth h.
v = √(2gh)

Viscosity Formula:
F = η * (VA/L), where F is the viscous force, A is the area, V is the velocity, and L is the distance between layers.

Poiseuille’s Law:
Q = [(P₂ - P₁)πr⁴] / (8ηL), where Q is the flow rate, r is the radius of the pipe, and L is the length of the pipe.

1/2ρv^2
(1/2mv)/v = KE/V

ρgh
mgh/V = PEg/V

Oscillations and SHM

Simple Harmonic Motion (SHM)

Definition: Periodic motion where the restoring force is proportional to the displacement.
F = -kx

Hooke’s Law:
F = -kx (Restoring force of a spring)

Displacement (x):
x(t) = A cos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is phase constant.

Angular Frequency (ω):
ω = √(k/m) = 2πf = 2π/T

Velocity (v):
v(t) = -Aω sin(ωt + φ)
V_max = Aω

Acceleration (a):
a(t) = -Aω² cos(ωt + φ) = -ω²x
A_max = Aω²

Period (T):
Time for one complete oscillation. T = 2π√(m/k)

Frequency (f):
Number of oscillations per unit time. f = 1/T

Total Energy (E):
E = KE + PE = ½mv² + ½kx² = ½kA²

Potential Energy (PE):
PE = ½kx²

Kinetic Energy (KE):
KE = ½mv² = ½mω²(A² - x²)

SHM Graphs:
Displacement leads to velocity and acceleration graphs. Amax = -xmax

Vertical Spring Systems:
Add mg to equilibrium; use same SHM formulas.

Horizontal Spring Systems:
Normal SHM setup

L=L+L0
Change in L = x (displacement)= L+L0, so L = x - L0, h = L - sqrt(x2-L2)

U=elastic potential energy is same as PE
Gravitational force = mg

Pendulums & Damping

Simple Pendulum:
T = 2π√(L/g), where L is length and g is gravitational acceleration.

Restoring force:
F= mgθ = -mg/L*s

Damped Oscillations:
Amplitude decreases over time due to energy loss.

Damping Equation:
ma = -bv - kx, where b is the damping coefficient.

Underdamped:
Oscillates with gradually decreasing amplitude.

Overdamped:
Returns to equilibrium slowly without oscillating.

Critically Damped:
Returns to equilibrium as quickly as possible without oscillating.

Time Constant (τ):
τ = m/b (Time for amplitude to decrease by a factor of e)

Energy Loss:
Damped oscillator loses energy twice as fast as amplitude.

Total energy of a damped oscillator:
E(t)=1/2kA(t)^2

h equation:
h = L -sqrt(L2 - x2)

Resonance:
Max amplitude when driven frequency ≈ natural frequency.

K = w^2 * m = sqrt(g/L)^2 * m = g/L * m = mg/L
Shown below as:
K = mg/L
T = 2pisqrt (m/k) = 2pisqrt (m/(mg/L))

A(t)=A0e^(-bt)

Damped equation of motion:
x(t)+bv(t)+kx(t)=0

Conservation of energy:
Convert potential energy equation 1/2kx2 to gravitational potential energy mgh.
So you get mgh = 1/2mv2 since you cannot get k constant from simple pendulum.

Traveling and Standing Waves

Traveling Waves

Transverse Wave:
Oscillation is perpendicular to the direction of wave travel.

Longitudinal Wave:
Oscillation is parallel to the direction of wave travel.

Wave Speed (v):
v = λf, where λ is wavelength and f is frequency.

Wave Function:
D(x, t) = A sin(kx - ωt + φ), where A is amplitude, k is wave number, ω is angular frequency, and φ is phase constant.

Wave Number (k):
k = 2π/λ

Angular Frequency (ω):
ω = 2πf

Phase Constant (φ):
Initial phase offset.

Phase Difference:
Δx/λ × 2π

Snapshot Graph:
D vs. x at a specific time.

History Graph:
D vs. t at a specific location.

u (linear tension)
m/L or density * area

Tension, T = mg
(mass times gravity)

v=sqrt(T/u)
u in v = sqrt(T/u) is the linear mass density (kg/m), and T is the tension

Direction of transverse wave depends on sign of wt in D(x, t).
+ive is left since this is a sin wave.

Standing Waves

Definition:
Superposition of two waves traveling in opposite directions, creating fixed nodes and antinodes.

Nodes:
Points of zero displacement.

Antinodes:
Points of maximum displacement.

Fixed Ends (Strings):
Nodes at both ends.

Open Ends (Pipes):
Antinodes at open ends.

Fundamental Frequency (f₁):
Lowest frequency of a standing wave.

Harmonics (fₙ):
Integer multiples of the fundamental frequency. fₙ = nf₁

Overtones:
Frequencies above the fundamental frequency.

Wavelength in terms of length
λ = 2L/n

Closed-closed:
add info

Open-closed:
add info

Open-open:
add info

Assume frequency an instrument needs to produce is the fundamental.

Fundamental freq: f1=v/λ = v/2L
All other freq.: v/λ = v/L = 2f1

Sound, Interference, and Nuclear Physics

Sound and Interference

Sound Intensity (I):
Power per unit area. I = P/A (W/m²)

Sound Level (β):
Measured in decibels (dB). β = 10 log₁₀(I/I₀), where I₀ = 10⁻¹² W/m²

Doppler Effect:
Change in frequency due to relative motion between source and observer.

Observer Moving:
f_o = f_s(v ± v_o)/v, where v is the speed of sound and v_o is the observer’s speed.

Source Moving:
f_o = f_s*v/(v ± v_s), where v_s is the source’s speed.

Constructive Interference:
Path difference = nλ (n = 0, 1, 2, …)

Destructive Interference:
Path difference = (n + ½)λ (n = 0, 1, 2, …)

Beats:
Periodic variations in amplitude due to interference of two slightly different frequencies.

Beat Frequency:
f_beat = |f₁ - f₂|

Speed of light in media
v=c/n

Bright fringe:
ym (displacement) = (mwavelengthLength)/d, where, d = distance between slits, L = distance between slits and screen

Constructive: delta(L)=n*wavelength

*Destructive: delta(L) = (n +1/2)wavelength

Mach number:
speed of source/speed of sound

Nuclear Physics

Atomic Number (Z):
Number of protons in the nucleus.

Mass Number (A):
Number of protons and neutrons in the nucleus.

Binding Energy (BE):
Energy required to separate a nucleus into its constituent nucleons.

BE = (Zm_p + Nm_n - m_nucleus)c²
Where m_p is proton mass, m_n is neutron mass, and m_nucleus is the nucleus mass.

Radioactive Decay:
Spontaneous emission of particles or energy from an unstable nucleus.

Decay Constant (λ):
Probability of decay per unit time.

Half-Life (T₁/₂):
Time for half of the radioactive nuclei to decay.
T₁/₂ = ln(2)/λ

Activity (A):
Rate of decay. A = λN(t)

Decay Law:
N(t) = N₀e^(-λt), where N₀ is the initial number of nuclei.

Radiation Types:
α (alpha), β (beta), γ (gamma)

Radiation Dose:
Energy absorbed per unit mass.

Dose Equivalent:
Takes into account the biological effect of different types of radiation.

N(t),A(t), lambda

Energy from decay