PHY 131 Formula Sheet

Comprehensive vocabulary flashcards covering key physics equations and concepts across kinematics, dynamics, gravitation, rotation, fluids, waves, and thermodynamics.

Projectile Motion Maximum Height ($\Delta y_{\max}$)

The maximum vertical displacement of a projectile launched with initial velocity $v_0$ at angle $\theta$: $\Delta y_{\max} = \frac{(v_0 \sin(\theta))^2}{2g}$.

Projectile Motion Horizontal Range ($x_R$)

The total horizontal distance traveled by a projectile when landing at the launch height ($y_f = y_i$): $x_R = \frac{v_0^2 \sin(2\theta)}{g}$.

Relative Velocity Addition Formula

The relationship relating the velocity of an object relative to frame $O$ through a moving frame $O'$: $\vec{v}_{\text{w.r.t. } O} = \vec{v}_{\text{w.r.t. } O'} + \vec{v}_{O' \text{ w.r.t. } O}$.

Newton's Law of Universal Gravitation

The attractive gravitational force between two point masses separated by distance $r$: $F_G = \frac{G m_1 m_2}{r^2}$, where $G = 6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$.

Circular Orbital Speed ($v_{\text{circ}}$) and Escape Velocity ($v_{\text{esc}}$) Relation

For a circular orbit of radius $R$ around mass $M$, the orbital speed is $v_{\text{circ}} = \sqrt{\frac{GM}{R}}$, and the escape velocity is $v_{\text{esc}} = \sqrt{2}v_{\text{circ}}$.

Kepler's Third Law for Circular Orbits

The relationship between orbital period $T$ and orbital radius $R$ about a central mass $M$: $\frac{T^2}{R^3} = \frac{4\pi^2}{GM}$.

Universal Gravitational Potential Energy ($U_G$)

The potential energy of two masses $m_1$ and $m_2$ separated by distance $r$: $U_G = -\frac{G m_1 m_2}{r}$.

Work-Energy Theorem

The theorem stating that the net work done on a particle equals the change in its kinetic energy: $W_{\text{net}} = \Delta K$.

Power ($P$)

The instantaneous rate of doing work, defined as $P = \lim_{\Delta t \to 0} \frac{W}{\Delta t} = \vec{F} \cdot \vec{v}$.

Center of Mass Position ($\vec{r}_{\text{CM}}$) and Acceleration ($\vec{a}_{\text{CM}}$) Equations

The mass-weighted coordinates and kinematics of a system: $\vec{r}_{\text{CM}} = \frac{\sum m\vec{r}}{\sum m}$ and $\vec{F}_{\text{net ext}} = m_{\text{tot}}\vec{a}_{\text{CM}}$.

1D Elastic Collision Velocity Condition

The relationship between initial and final velocities in a one-dimensional head-on elastic collision: $v_1 + v'_1 = v_2 + v'_2$.

Completely Inelastic Collision Kinetic Energy Loss ($\Delta K$)

The change in kinetic energy when two masses stick together in a 1D collision: $\Delta K = -\frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}(v_1 - v_2)^2$.

Moment of Inertia of a Thin Rod about Center of Mass

$I_{\text{CM}} = \frac{1}{12}ML^2$, where $M$ is the total mass and $L$ is the length of the rod.

Moment of Inertia of a Thin Hoop about Center of Mass

$I_{\text{CM}} = MR^2$, where $M$ is the total mass and $R$ is the radius of the hoop.

Moment of Inertia of a Solid Disk about Center of Mass

$I_{\text{CM}} = \frac{1}{2}MR^2$, where $M$ is the total mass and $R$ is the radius of the disk.

Moment of Inertia of a Solid Sphere about Center of Mass

$I_{\text{CM}} = \frac{2}{5}MR^2$, where $M$ is the total mass and $R$ is the radius of the sphere.

Parallel-Axis Theorem

A theorem relating the moment of inertia about any axis parallel to a center-of-mass axis: $I = I_{\text{CM}} + Md^2$, where $d$ is the distance between the two axes.

Rolling Without Slipping Conditions

The kinematic relations linking translational motion to rotational motion: arc length $l = R\theta$, linear velocity $v = R\omega$, and linear acceleration $a = R\alpha$.

Rotational Kinetic Energy ($K_{\text{rot}}$) and Torque ($\vec{\tau}$)

The kinetic energy due to rotation is $K_{\text{rot}} = \frac{1}{2}I\omega^2$, and the net torque is $\vec{\tau} = \vec{R} \times \vec{F} = I\vec{\alpha}$.

Angular Momentum of a Rigid Object ($\vec{L}_{\text{tot}}$) and Conservation Law

Given rotation about a symmetry axis, $\vec{L}_{\text{tot}} = I\vec{\omega}$; when the net external torque is zero ($\vec{\tau}_{\text{net ext}} = \vec{0}$), $\vec{L}_{\text{tot}}$ is conserved.

Simple Harmonic Motion Differential Equation

The second-order ordinary differential equation describing simple harmonic motion: $\frac{d^2x}{dt^2} + \omega^2x = 0$, with general displacement solution $x = A\cos(\omega t + \phi)$.

Angular Frequency ($\omega$) of Spring-Mass and Simple Pendulum Systems

For a spring-mass system, $\omega = \sqrt{\frac{k}{m}}$; for a simple pendulum of length $l$, $\omega = \sqrt{\frac{g}{l}}$.

Archimedes' Principle (Buoyant Force, $F_B$)

The upward buoyant force exerted on a body partially or totally submerged in a fluid: $F_B = \rho_{\text{fluid}}V_{\text{displaced}}g$.

Barometric Altitude Formula (Earth's Atmosphere)

The hydrostatic pressure variation with altitude $z$: $p = p_0 \exp\left(-\frac{\rho_0 g z}{p_0}\right)$.

Continuity Equation for Incompressible Fluids

The expression for mass conservation in flowing fluids through varying cross-sectional areas: $Av = \text{const.}$.

Bernoulli's Equation

The conservation of mechanical energy per unit volume along a streamline for an ideal fluid: $p + \frac{1}{2}\rho v^2 + \rho gz = \text{const.}$.

One-Dimensional Traveling Wave Equation

The mathematical expression for a sinusoidal wave: $D(x,t) = A\sin(kx \mp \omega t)$, where the minus sign ($-$) denotes propagation to the right and the plus sign ($+$) denotes propagation to the left.

Wave Number ($k$)

The spatial frequency of a wave defined as $k = \frac{2\pi}{\lambda}$, where $\lambda$ is the wavelength.

Standing Wave Function

The mathematical form of a standing wave resulting from interference: $D(x,t) = 2A\sin(kx)\cos(\omega t)$.

Volumetric and Linear Thermal Expansion Coefficients

The temperature-dependent volume expansion $V = V_0(1 + \beta\Delta T)$, where the volumetric coefficient relates to the linear expansion coefficient as $\beta \simeq 3\alpha$.

First Law of Thermodynamics

The fundamental conservation of energy principle for thermodynamic systems: $\Delta E_{\text{int}} = Q - W$, where $Q$ is the heat added to the gas and $W$ is the work done by the gas.

Molar Heat Capacity at Constant Volume ($C_V$) and Degrees of Freedom

$C_V = \frac{n_{\text{d.o.f.}}}{2}R$, where $n_{\text{d.o.f.}} = 3$ for a monatomic ideal gas and $n_{\text{d.o.f.}} = 5$ for a diatomic ideal gas.

Specific Heat Relation ($C_P$) and Adiabatic Index ($\gamma$)

For an ideal gas, the molar heat capacity at constant pressure is $C_P = C_V + R$, and the ratio of specific heats is $\gamma = \frac{C_P}{C_V}$.

Isochoric Process ($V = \text{const.}$)

A quasi-static constant-volume process where $\frac{P}{T} = \text{const.}$, work done by the gas is $W = 0$, and $Q = \Delta E_{\text{int}} = nC_V(T_f - T_i)$.

Isobaric Process ($P = \text{const.}$)

A quasi-static constant-pressure process where $\frac{V}{T} = \text{const.}$, work done by the gas is $W = P(V_f - V_i)$, and heat added is $Q = nC_P(T_f - T_i)$.

Isothermal Process ($T = \text{const.}$)

A quasi-static constant-temperature process where $PV = \text{const.}$ and heat added equals work done by the gas: $Q = W = nRT\ln\left(\frac{V_f}{V_i}\right)$.

Adiabatic Process ($Q = 0$)

A quasi-static process with no heat exchange ($Q = 0$) characterized by $PV^{\gamma} = \text{const.}$ and work done by the gas $W = -nC_V(T_f - T_i)$.

Heat Engine Thermal Efficiency ($e$) and Carnot Efficiency

The efficiency of a heat engine is $e = \frac{W}{Q_H} = 1 - \frac{|Q_L|}{Q_H}$, with maximum Carnot efficiency given by $e = 1 - \frac{T_L}{T_H}$.

Coefficient of Performance (Refrigerator / Air Conditioner)

The measure of cooling efficiency defined by $\text{COP} = \frac{Q_L}{|W|}$.

Coefficient of Performance (Heat Pump)

The measure of heating efficiency defined by $\text{COP} = \frac{|Q_H|}{|W|}$.