PHYS 20300 Formulas

Dr. James Hedberg

Updated: January 2025

Kinematics

\begin{equation} \begin{aligned} & \overline{\mathbf{v}} = \frac{\Delta \mathbf{x}}{\Delta t} \\[5pt] & {\mathbf{v}} =\lim_{\Delta t \rightarrow 0} \frac{\Delta \mathbf{x}}{\Delta t} = \frac{d\mathbf{x}}{dt}\\[5pt] & \overline{\mathbf{a}} = \frac{\Delta \mathbf{v}}{\Delta t} \\[5pt] & {\mathbf{a}} =\lim_{\Delta t \rightarrow 0} \frac{\Delta \mathbf{v}}{\Delta t} =\frac{d\mathbf{v}}{dt}\\[5pt] & \mathbf{v} = \mathbf{v}_0 + \mathbf{a}t \\[5pt] & \mathbf{x} = \frac{1}{2}(\mathbf{v}_0+\mathbf{v})t \\[5pt] & \mathbf{x} = \mathbf{x}_0 + \mathbf{v}_0 t + \frac{1}{2}\mathbf{a}t^2 \\[5pt] &v^2 = v_0^2 + 2ax \end{aligned} \end{equation}

Forces

\begin{equation} \begin{aligned} & \sum \mathbf{F} = \mathbf{F}_\textrm{net}= m \mathbf{a} \\[5pt] & F_\textrm{G} = G \frac{m_1 m_2}{r^2}\\[5pt] & g = \frac{G M_E}{r^2_E} = 9.8 \;\rm m/s^2 \approx 10 \; \rm m/s^2\\[5pt] & f_s^\textrm{max} = \mu_s F_N \\[5pt] & f_k = \mu_k F_N \\[5pt] & F_D = \frac{1}{2}C \rho A v^2 \end{aligned} \end{equation}

Circular Motion

\begin{equation} \begin{aligned} & v = \frac{2 \pi r}{T} \\[5pt] &a_c = \frac{v^2}{r} \\[5pt] & v = \sqrt{\frac{G M_E}{r} }\\ \end{aligned} \end{equation}

Work and Energy

\begin{equation} \begin{aligned} & W = \mathbf{F} \cdot \mathbf{s} \\[5pt] & W = \int_{x_i}^{x_f} F(x)\; dx\\[5pt] & KE = \frac{1}{2} mv^2\\[5pt] & W = KE_f - KE_i \\[5pt] & W_\textrm{grav} = mg(h_0-h_f) \\[5pt] & PE = mgh \\[5pt] & W_\textrm{nc} = E_f - E_0 \\[5pt] & \overline{P} = \frac{\Delta E}{\Delta t} \\[5pt] & \overline{P} = \mathbf{F} \cdot \mathbf{v}\\[5pt] & F_x = -\frac{d U }{dx} \\ \end{aligned} \end{equation}

Rotational Motion

\begin{equation} \begin{aligned} & \theta = \frac{s}{r} \\[5pt] & \overline{\omega} = \frac{\Delta \theta}{\Delta t} \\[5pt] & \overline{\alpha} = \frac{\Delta \omega}{\Delta t} \\[5pt] & \theta = \frac{1}{2}(\omega_0+\omega)t \\[5pt] & \theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2 \\[5pt] & \omega^2 = \omega_0^2 + 2\alpha \theta \\[5pt] & v_T = r \omega \\[5pt] & a_T = r \alpha \\[5pt] & a_c = r \omega^2 \\[5pt] \end{aligned} \end{equation}

Rotational Dynamics

\begin{equation} \begin{aligned} & \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = r F \sin \phi \\[5pt] & \tau = I \alpha \\[5pt] & \tau = \frac{d \mathbf{L}}{dt} \\[5pt] & I = \int r^2 dm \\[5pt] & I = I_\textrm{com}+mh^2\\[5pt] & KE_\textrm{rot} = \frac{1}{2}I \omega ^2 \\[5pt] & L = I \omega \\[5pt] & \mathbf{L} = \mathbf{r} \times \mathbf{p} \end{aligned} \end{equation}

Simple Harmonic Motion

\begin{equation} \begin{aligned} & x(t) = A \cos (\omega t) \\ & v(t) = - A \omega \sin (\omega t) \\ & a(t) = - A \omega^2 \cos (\omega t) \\ & f = \frac{1}{T} \\ & \omega = 2 \pi f \\ & \omega = \sqrt{\frac{k}{m}} \\ & \mathbf{F}_\textrm{sp} = -k\mathbf{x} \\[5pt] & U_\textrm{sp} = \frac{1}{2}kx^2 \\[5pt] & T_\textrm{pend} = 2 \pi \sqrt{ \frac{L}{g}} \end{aligned} \end{equation}

Momentum and Impulse

\begin{equation} \begin{aligned} & \mathbf{J} = \overline{\mathbf{F}} \Delta t \\ & \mathbf{p} = m \mathbf{v} \\ & \mathbf{J} = \Delta \mathbf{p} \end{aligned} \end{equation}

Fluids

\begin{equation} \begin{aligned} & \rho = \frac{m}{V} \\ & P = \frac{F}{A} \\ & P_2 = P_1 + \rho g h \\ & F_B = W_\textrm{fluid} \\ & \rho_1 A_1 v_1 = \rho_2 A_2 v_2 \\ & P_1 + \frac{1}{2} \rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g y_2 \\ \end{aligned} \end{equation}

Temperature and Heat

\begin{equation} \begin{aligned} &T = T_C + 273.15 \\ &T_C = (T_F-32)* \frac{5}{9} \\ & \Delta L = \alpha L_0 \Delta T \\ & \Delta V = \beta V_0 \Delta T \\ & Q = cm \Delta T \\ & Q = mL_F \; \textrm{or} \; mL_V \\ & Q = \frac{ k A \Delta T \; t }{L} \end{aligned} \end{equation}

Thermodynamics

\begin{equation} \begin{aligned} & W = -\int P dV \\ & PV = nRT = Nk_B T \\ & U_\textrm{th-IG} = \frac{3}{2}Nk_B T \\ & \Delta U = U_f - U_i = Q + W \\ \end{aligned} \end{equation}

\begin{equation} \begin{aligned} & V_\textrm{sphere} = \frac{4}{3} \pi r^3 \\ & 1 \textrm{mL} = 1 \textrm{cc} = 1 \textrm{gram} \\ & \textrm{Mass of the Earth} (M_E) = 5.97 \times 10^{24} \; \textrm{kg} \end{aligned} \end{equation}

\begin{equation} \begin{aligned} & \textrm{Radius of the Earth} (R_E) = 6,371 \; \textrm{km}\\ & \rho_\textrm{air} = 1.28 \; \rm kg/m^3 \\ & \rho_\textrm{water} = 1000 \; \rm kg/m^3 \\ & 1 \; \rm atm = 101,325 \; Pa \\ \end{aligned} \end{equation}