mid-term quick review

Newton's Laws

1. 1D & 2D Kinematics

Fundamental Definitions

Average velocity is defined as displacement over time: \(v_{avg} = \Delta x / \Delta t\).
Instantaneous velocity is the time derivative of position: \(v = dx/dt\).
Instantaneous acceleration is the time derivative of velocity: \(a = dv/dt = d^2x/dt^2\).
To find velocity from variable acceleration, integrate: \(v(t) = \int_{t_i}^t a(t') dt' + v_i\).
To find position from variable velocity, integrate: \(x(t) = \int_{t_i}^t v(t') dt' + x_i\).

Constant Acceleration Formulas

Velocity as a function of time: \(v(t) = v_0 + at\).
Position as a function of time: \(x(t) = x_0 + v_0t + \frac{1}{2}at^2\).
Time-independent kinematic equation: \(x(t) - x_0 = (v_f^2 - v_0^2)/(2a)\).

Projectile & Circular Motion

Total range of a projectile on level ground: \(s = (v_0^2\sin2\phi_0)/g\).
Total time of flight for a projectile on level ground: \(t = (2v_0\sin\phi_0)/g\).
Uniform circular motion features constant angular velocity: \(\omega = v/r\).
Centripetal acceleration for uniform circular motion always points radially inward: \(\vec{a} = -(v^2/r)\hat{r}\).
Non-uniform circular motion includes both tangential and centripetal acceleration components: \(\vec{a} = (d\omega/dt)r\hat{\phi} - \omega^2r\hat{r}\).

2. Error Propagation & Significant Figures

Significant Figures

Addition and Subtraction: The final result must be rounded to the same decimal place as the number with the least precise decimal place.
Multiplication and Division: The final result must have the same number of significant figures as the term with the fewest significant figures.

Uncertainty Propagation

Absolute uncertainty for addition (\(y = x_1 + x_2\)): \(\Delta y = \sqrt{(\Delta x_1)^2 + (\Delta x_2)^2}\).
Relative uncertainty for multiplication/division (\(y = x_1 x_2\) or \(y = x_1 / x_2\)): \(\Delta y / |y| = \sqrt{(\Delta x_1/x_1)^2 + (\Delta x_2/x_2)^2}\).

3. Newtonian Dynamics & Forces

Core Laws & Reference Frames

Newton's Second Law relates force and acceleration: \(\vec{F} = m\vec{a}\).
Galilean velocity transformation between frames: \(\vec{v}' = \vec{v} - \vec{v}_0\).
Fictitious force required in an accelerating (non-inertial) reference frame: \(F_{\text{fictitious}} = -m\vec{a}\).

Drag & Friction

Low-speed (linear) drag applies to highly viscous fluids or slow speeds: \(\vec{R} = -b\vec{v}\).
Terminal velocity under low-speed drag: \(v_t = mg/b\).
High-speed (quadratic) drag applies when turbulence is a factor: \(\vec{R} = -\frac{1}{2} C \rho A v^2 \hat{v}\).
Terminal velocity under high-speed drag scales with the square root of the object's radius: \(v_t = \sqrt{2mg / (C \rho A)}\).
Maximum static friction prevents initial motion: \(f_{s,\max} = \mu_s F_N\).
Kinetic friction opposes ongoing motion: \(f_k = \mu_k F_N\).

4. Numerical Integration Methods

Integration Method	Position Update Formula	Truncation Error
Euler	\(\vec{r}_{n+1} = \vec{r}_n + \vec{v}_n\Delta t\)	\(O(\Delta t^2)\)
Verlet	\(\vec{r}_{n+1} = 2\vec{r}_n - \vec{r}_{n-1} + \vec{a}_n \Delta t^2\)	\(O(\Delta t^4)\)
Velocity Verlet	\(\vec{r}_{n+1} = \vec{r}_n + \vec{v}_n \Delta t + \frac{1}{2}\vec{a}_n \Delta t^2\)	\(O(\Delta t^4)\)

5. Work, Energy, and Momentum

Work and Energy Theorems

Mechanical work definition for a constant force: \(W = \vec{F} \cdot \vec{d}\).
Work done by a varying force in 1D: \(W = \int_{x_i}^{x_f} F_x dx\).
Work done by a spring obeying Hooke's Law: \(W = -\frac{1}{2}k(x_f^2 - x_i^2)\).
Definition of kinetic energy: \(K = \frac{1}{2}mv^2\).
The Work-Kinetic Energy Theorem states that net work equals the change in kinetic energy: \(K_f - K_i = W\).

Potential Energy and Equilibrium

Change in potential energy is the negative work done by a conservative force: \(\Delta U = -W\).
Gravitational potential energy near Earth's surface: \(U(y) = mgy\).
Elastic potential energy of a spring: \(U(x) = \frac{1}{2}kx^2\).
A conservative force is the negative gradient of its potential energy: \(\vec{F} = -\nabla U\).
Stable equilibrium occurs at a potential energy minimum: \(dU/dx = 0\) and \(d^2U/dx^2 > 0\).
Unstable equilibrium occurs at a potential energy maximum: \(dU/dx = 0\) and \(d^2U/dx^2 < 0\).

Momentum and Collisions

Linear momentum definition: \(\vec{p} = m\vec{v}\).
The Impulse-Momentum Theorem relates force over time to momentum change: \(\vec{I} = \int \vec{F} dt = \Delta \vec{p}\).
In perfectly inelastic collisions, objects stick together and kinetic energy is lost.
Shared final velocity for perfectly inelastic collisions: \(\vec{v}_f = (m_1\vec{v}_{1i} + m_2\vec{v}_{2i})/(m_1 + m_2)\).
In elastic collisions, both linear momentum and kinetic energy are conserved.
Position of the Center of Mass (CM) for discrete particles: \(\vec{r}_{CM} = (\sum m_i \vec{r}_i)/M\).
Position of the CM for continuous extended objects: \(\vec{r}_{CM} = (\int \vec{r} dm)/M\).
König's Theorem states total kinetic energy is the kinetic energy of the CM plus the internal kinetic energy: \(E_K = E_{k, cm} + E_K'\).
The total momentum of any system measured within its own CM frame is always zero: \(\vec{P}' = 0\).

6. Gravity & Rocket Propulsion

Universal Gravitation

Kepler's Law of Periods relates orbital period to the semimajor axis: \(T^2 \propto a^3\).
Newton's Law of Universal Gravitation: \(F = G(m_1 m_2)/r^2\).
Outside a uniform spherical mass (\(R > R_E\)), gravity acts exactly as if all mass is concentrated at the center.
Inside a uniform spherical mass (\(R < R_E\)), the gravitational force acts like a restoring spring: \(F \propto R\).
Universal gravitational potential energy (setting zero at infinity): \(U(r) = -GMm/r\).

Orbital Mechanics & Rockets

Required velocity for a satellite in a circular orbit at radius \(r\): \(v = \sqrt{GM_E/r}\).
First Cosmic Velocity (velocity for a circular orbit at Earth's surface): \(v = \sqrt{GM_E/R_E} \approx 7.9\) km/s.
Second Cosmic Velocity (escape velocity from Earth): \(v = \sqrt{2GM_E/R_E} \approx 11.2\) km/s.
The change in velocity for a rocket depends on the mass ratio and exhaust velocity: \(v_f - v_i = v_e \ln(M_i/M_f)\).
Rocket thrust force is proportional to the mass ejection rate: \(F_{\text{thrust}} = v_e |dM/dt|\).

Oscillation and Wave

1. Rotational Kinematics & Dynamics

Kinematics & Linear Analogies

Rotational motion perfectly mirrors linear kinematics:

Variables: Displacement \(\theta\), Velocity \(\omega = d\theta/dt\), Acceleration \(\alpha = d\omega/dt\).
Constant \(\alpha\) Equations: \(\omega_f = \omega_i + \alpha t\) | \(\theta_f = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2\) | \(\omega_f^2 - \omega_i^2 = 2\alpha\Delta\theta\).
Relations to Linear: \(v = \omega r\) | Tangential \(a_t = \alpha r\) | Radial (Centripetal) \(a_r = \omega^2 r\).
Vector Form: \(\vec{v} = \vec{\omega} \times \vec{r}\).

Moment of Inertia (\(I\))

A body's resistance to rotational acceleration:

Formulas: Discrete \(I = \sum m_i r_i^2\) | Continuous \(I = \int r^2 dm\).
Parallel Axis Theorem: \(I = I_{cm} + Mh^2\) (where \(h\) is distance from COM).
Common Geometries (Memorize):
- Uniform Rod: Center \(I = \frac{1}{12}ML^2\) | End \(I = \frac{1}{3}ML^2\).
- Sphere: Solid \(I = \frac{2}{5}MR^2\) | Shell \(I = \frac{2}{3}MR^2\).
- Cylinder/Disk: Solid \(I = \frac{1}{2}MR^2\) | Shell \(I = MR^2\).

Torque (\(\tau\)), Energy, & Angular Momentum (\(L\))

Torque: \(\vec{\tau} = \vec{r} \times \vec{F}\) | Magnitude: \(\tau = Fd\) (where \(d\) is the moment arm).
Newton's 2nd Law for Rotation: \(\vec{\tau}_{net} = I\vec{\alpha} = \frac{d\vec{L}}{dt}\).
Angular Momentum: \(\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}\). Conserved (\(\vec{L}_i = \vec{L}_f\)) if net external torque is zero.
Rotational Kinetic Energy: \(K_R = \frac{1}{2}I\omega^2\).
Power: \(P = \tau\omega\).
Pure Rolling (No slipping): \(v_{cm} = \omega R\) | \(a_{cm} = \alpha R\). Total \(K = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2\). Friction does zero work.

Rotating Frames & Fictitious Forces

When calculating motion in a rotating (non-inertial) frame, add fictitious forces:

Coriolis Force: \(-2m\vec{\omega} \times \vec{v}'\) (Deflects moving objects; right in Northern Hemisphere).
Centrifugal Force: \(-m\vec{\omega} \times (\vec{\omega} \times \vec{r}')\) (Points outward from rotation axis).

2. Static Equilibrium

For a rigid body to remain completely at rest, two conditions must be met:

Translational Equilibrium: \(\sum \vec{F}_{net} = 0\) (Linear momentum is constant/zero).
Rotational Equilibrium: \(\sum \vec{\tau}_{net} = 0\) (Angular momentum is constant/zero). Rule of thumb: You can calculate torque from ANY arbitrary origin to simplify math.
Center of Gravity (COG): The effective point of gravitational torque. It coincides with the Center of Mass (COM) in uniform gravity.
Concurrent Forces Theorem: If a body is in equilibrium under 3 non-parallel forces, their lines of action must intersect at a single point.

3. Oscillatory Motion (SHM)

Simple Harmonic Motion Basics

Occurs near stable equilibrium points where restoring force is linear.

Equation of Motion: \(\ddot{x} = -\omega^2(x - x_0)\) / \(\frac{d^2x}{dt^2} = -\omega^2 x\) where \(\omega = \sqrt{k/m}\).
Kinematics Solution: \(x(t) = A \cos(\omega t + \phi)\).
Parameters: Period \(T = 2\pi/\omega\) | Frequency \(f = 1/T = \omega/2\pi\) (Hz).
Energy: Total \(E = \frac{1}{2}kA^2\). Kinetic and Potential energies oscillate at \(2\omega\) (twice the frequency of position).

Pendulums

Simple Pendulum: \(\omega = \sqrt{g/L}\).
Physical Pendulum: \(\omega = \sqrt{mgd/I}\) (\(d\) is distance from COM to pivot).
Torsion Pendulum: \(\omega = \sqrt{k/I}\) (where \(k\) is the torsional constant).

Damping & Resonance

Damping Regimes: Controlled by damping constant \(b\):
- Underdamped: System oscillates with exponentially decaying amplitude.
- Critically Damped: Fastest return to equilibrium without oscillating.
- Overdamped: Slow return to equilibrium without oscillating.
Forced Oscillation: Maximum amplitude (Resonance) occurs when driving frequency matches natural frequency (\(\omega_{ext} = \omega_0\)).

Modes of Vibration

For an \(N\)-atom molecule, total degrees of freedom = \(3N\):

Linear molecule: \(3N - 5\) vibrational modes.
Non-linear molecule: \(3N - 6\) vibrational modes.

4. Mechanical Waves

The Wave Equation & Parameters

General Wave Equation: \(\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}\).
Sinusoidal Wave: \(y(x,t) = A \sin(kx - \omega t + \phi)\).
Wave Variables: Wave number \(k = 2\pi/\lambda\) | Angular freq \(\omega = 2\pi/T\).
Phase Velocity: \(v = \omega/k = \lambda/T = \lambda f\).

Wave Speeds in Media

String: \(v = \sqrt{T/\mu}\) (Tension / linear mass density).
Fluids (Sound): \(v = \sqrt{B/\rho}\) (Bulk Modulus / density). Water is highly incompressible, so \(v_{water} \gg v_{air}\).

Interference, Boundaries, & Standing Waves

Boundary Reflection:
- Fixed/Heavy boundary: Pulse reflects inverted.
- Free/Light boundary: Pulse reflects upright.
Interference (Path Difference \(\Delta r\)):
- Constructive (In Phase): \(\Delta r = n\lambda\).
- Destructive (Out of Phase): \(\Delta r = (n + 0.5)\lambda\).
Standing Waves (Fixed Ends): Nodes (zero amplitude) separated by \(\lambda/2\). Harmonic frequencies: \(f_n = n \frac{v}{2L}\).
Beats (Temporal Interference): Beat frequency \(f_{beat} = |f_1 - f_2|\).

Energy & Dispersion

Average Power: \(P_{avg} = \frac{1}{2}\mu v \omega^2 A^2\).
Dispersion: Phase velocity \(v_p = \omega/k\). Group velocity (envelope) \(v_g = d\omega/dk\).

5. Sound & Elasticity

Elasticity Formulas

Linear region Hooke's Law approximation: \(\text{Stress} = \text{Modulus} \times \text{Strain}\).

Young's (Tensile): \(F/A = E(\Delta L/L)\).
Shear: \(F/A = G(\Delta x/L)\).
Bulk (Hydraulic): \(P = B(\Delta V/V)\).

Intensity & Decibels

Intensity: \(I = P/A\). For a spherical point source: \(I = P_s / 4\pi r^2\).
Sound Level (\(\beta\)): \(\beta = 10 \log_{10}(I/I_0)\) in decibels (dB), where \(I_0 = 10^{-12} \text{ W/m}^2\).
- Rule of thumb: Adding 10 dB means Intensity is multiplied by 10.

Doppler Effect & Shock Waves

Doppler Equation: \(f' = f \left( \frac{v \pm v_D}{v \pm v_S} \right)\).
- Sign Convention: Use top signs (\(+\) in num, \(-\) in den) when approaching. Use bottom signs when moving away.
Shock Waves (Mach Cone): Occur when \(v_S > v\). The Mach cone angle is \(\sin\theta = v/v_S\).