PHYS 218
Motion
(1)
\begin{align} \vec{v}_{\text{avg}} = {\vec{r}_2-\vec{r}_1 \over t_2 - t_1} \end{align}

Equation for average velocity

(2)
\begin{align} \vec{v} = \frac{d\vec{r}}{dt} \end{align}

Equation for velocity

(3)
\begin{align} \vec{a}_{\text{avg}} = {\vec{v}_2-\vec{v}_1 \over t_2-t_1} \end{align}

Equation for average acceleration

(4)
\begin{align} \vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2} \end{align}

Acceleration

(5)
\begin{align} \vec{r} = \vec{r}_0 + \vec{v}_0t + \frac{1}{2}\vec{a}t^2 \end{align}

Position as a function of time given constant acceleration

(6)
\begin{align} \vec{r}(t) = \vec{r}_0 + \int_0^t\vec{v}(t)\; dt \end{align}

Position as a function of time

(7)
\begin{align} \vec{v} = \vec{v}_0 + \vec{a}t \end{align}

Velocity as a function of time given constant acceleration

(8)
\begin{align} \vec{v}(t) = \vec{v}_0+\int_0^t\vec{a}(t)\; dt \end{align}

Velocity as a function of time

(9)
\begin{align} v_f^2 = v_0^2+2\vec{a} \cdot (\vec{r}-\vec{r}_0) \end{align}

Final velocity squared as a function of position given constant acceleration

(10)
\begin{align} y = y_0+x\tan\theta-\frac{gx^2}{2v^2\cos^2\theta} \end{align}

Two dimensional projectile motion

(11)
\begin{align} 2(\vec{r}-\vec{r}_0) = t(\vec{v}_f+\vec{v}_0) \end{align}

Position and velocity relation given constant acceleration

(12)
\begin{align} a_\text{rad} = \frac{v^2}{R} \end{align}

Radial acceleration

(13)
\begin{align} v = \frac{2\pi R}{T} = 2\pi Rf \end{align}

Velocity and Period relation for circular motion

Force
(14)
\begin{align} \sum \vec{F} = 0 \end{align}

Sum of forces for a body in equilibrium; Newton's first law

(15)
\begin{align} \sum \vec{F} = m\vec{a} \end{align}

Sum of forces on an accelerating object; Newton's second law

(16)
\begin{align} \vec{w} = -mg\hat{j} \end{align}

Weight vector; for magnitude, ignore the negative and j

(17)
\begin{align} \vec{F}_\text{B on A} = -\vec{F}_\text{A on B} \end{align}

Newton's third law

(18)
\begin{align} F_x = F\cos\theta \end{align}

Force component in the x direction given an angle from the +x-axis

(19)
\begin{align} F_y = F\sin\theta \end{align}

Force component in the y direction given an angle from the +x-axis

(20)
\begin{align} \text{f}_s = \mu_sN \end{align}

Static frictional force given the Normal force

(21)
\begin{align} \text{f}_k = \mu_kN \end{align}

Kinetic frictional force given the Normal force

(22)
\begin{align} \vec{F}_s = k\vec{r} \end{align}

Force a spring exerts across a distance r

Energy
(23)
\begin{align} W = \vec{F}\cdot\vec{d} = Fd\cos\phi \end{align}

Work, where Force is constant

(24)
\begin{align} W = \int_0^x\vec{F}\cdot d\vec{r} \end{align}

Work where Force is not constant

(25)
\begin{align} K = \frac{1}{2}mv^2 \end{align}

Kinetic Energy

(26)
\begin{align} W_\text{tot} = \Delta K \end{align}

Work-Energy Theorem

(27)
\begin{align} P_\text{avg} = \frac{\Delta W}{\Delta t} \end{align}

The average power of given work over a period of time

(28)
\begin{align} P = \frac{dW}{dt} = \vec{F}\cdot\vec{v} \end{align}

General equation for power

(29)
\begin{equation} U_g = mgy \end{equation}

Gravitational Potential Energy

(30)
\begin{align} U_s = \frac{1}{2}kx^2 \end{align}

Spring Potential Energy

(31)
\begin{align} W_g = -\Delta U_g \end{align}

Work done by gravity

(32)
\begin{align} \vec{F} = -\vec{\nabla}U \end{align}

Force as a gradient of potential energy

(33)
\begin{equation} E = K + U \end{equation}

Mechanical energy of a system

(34)
\begin{align} F = -\frac{dU}{dx} \end{align}

Force from potential energy in one dimension

Momentum
(35)
\begin{align} \vec{p} = m\vec{v} \end{align}

Definition of Momentum

(36)
\begin{align} \sum \vec{F} = \frac{d\vec{p}}{dt} \end{align}

Force from momentum

(37)
\begin{align} \vec{J} = \vec{F}\Delta t \end{align}

Impulse with constant force

(38)
\begin{align} \vec{J} = \int_{t_0}^{t_f} \sum \vec{F} \; dt \end{align}

Definition of Impulse

(39)
\begin{align} \vec{J} = \Delta \vec{p} \end{align}

Impulse-momentum Theorem

Rotation
(40)
\begin{align} \vec{\omega}_{\text{avg}} = {\vec{\theta}_2-\vec{\theta}_1 \over t_2 - t_1} \end{align}

Equation for average angular velocity

(41)
\begin{align} \vec{\omega} = \frac{d\vec{\theta}}{dt} \end{align}

Equation for angular velocity

(42)
\begin{align} \vec{\alpha}_{\text{avg}} = {\vec{\omega}_2-\vec{\omega}_1 \over t_2-t_1} \end{align}

Equation for average angular acceleration

(43)
\begin{align} \vec{\alpha} = \frac{d\vec{\omega}}{dt} = \frac{d^2\vec{\theta}}{dt^2} \end{align}

Angular Acceleration

(44)
\begin{align} \vec{\theta} = \vec{\theta}_0 + \vec{\omega}_0t + \frac{1}{2} \vec{\alpha} t^2 \end{align}

Angular position given constant angular acceleration

(45)
\begin{align} \vec{\theta}(t) = \vec{\theta}_0 + \int_0^t \vec{\omega}(t) \; dt \end{align}

Angular position as a function of time

(46)
\begin{align} \vec{\omega} = \vec{\omega}_0+\vec{\alpha}t \end{align}

Angular velocity given constant angular acceleration

(47)
\begin{align} \vec{\omega} = \vec{\omega}_0+\int_0^t \vec{\alpha}(t) \; dt \end{align}

Angular velocity as a function of time

(48)
\begin{align} \omega_f^2 = \omega_0^2+2\alpha(\vec{\theta}-\vec{\theta}_0) \end{align}

Final angular velocity squared as a function of angular position

(49)
\begin{align} s = r\theta \end{align}

Arc length

(50)
\begin{align} v = r\omega \end{align}

Tangential Velocity

(51)
\begin{align} a_{\text{tan}} = r\alpha \end{align}

Tangential Acceleration

(52)
\begin{align} a_{\text{rad}} = \omega^2r \end{align}

Radial Acceleration

(53)
\begin{align} I = \sum_i m_ir_i^2 \end{align}

Moment of Inertia

(54)
\begin{align} I = \int r^2\rho \; dV \end{align}

Moment of Inertia of a continuous mass

(55)
\begin{equation} I_P = I_{cm} + md^2 \end{equation}

Moment of Inertia at a point distance d from the center of mass

(56)
\begin{align} K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \end{align}

Kinetic Energy of a body with rotational and translational motion

(57)
\begin{align} \tau = rF\sin\phi \end{align}

Torque

(58)
\begin{align} \vec{\tau} = \vec{r} \times \vec{F} \end{align}

Torque as a Vector

(59)
\begin{align} \sum \vec{\tau} = I\vec{\alpha} \end{align}

The sum of torques on an object

(60)
\begin{align} W = \tau\Delta\theta \end{align}

Work done by constant torque over an angle

(61)
\begin{align} W = \int_{\theta_0}^{\theta_f} \vec{\tau} \cdot d\vec{\theta} \end{align}

Work done by torque over an angle

(62)
\begin{align} P = \tau\omega \end{align}

Power of a rotating object with torque

(63)
\begin{align} P = \frac{dW}{dt} = \vec{\tau}\cdot\vec{\omega} \end{align}

Power of rotating object

(64)
\begin{align} L = mvr\sin\phi \end{align}

Angular momentum

(65)
\begin{align} \vec{L} = \vec{r} \times \vec{p} \end{align}

Angular momentum as a vector

(66)
\begin{align} \vec{L} = I\vec{\omega} \end{align}

Angular momentum in terms of angular velocity

(67)
\begin{align} \vec{\tau} = \frac{d\vec{L}}{dt} \end{align}

Torque given angular momentum

Gravitation
(68)
\begin{align} F_g = \frac{Gm_1m_2}{r^2} \end{align}

Magnitude of Gravitational Force

(69)
\begin{align} \vec{F}_g = -\frac{Gm_1m_2}{r^2}\hat{r} \end{align}

Gravitational Force

(70)
\begin{align} g = \frac{GM}{R^2} \end{align}

Gravitational Acceleration at the surface of a planet

(71)
\begin{align} U_g = -\frac{Gm_1m_2}{r} \end{align}

Potential energy due to gravity

(72)
\begin{align} W_g = \int^{r_2}_{r_1}\vec{F}_g\cdot d\vec{r} \end{align}

Work due to gravity

(73)
\begin{align} v_{\text{orb}} = \sqrt{\frac{GM}{r}} \end{align}

Orbital velocity in circular orbit

(74)
\begin{align} v_{\text{orb}} = \sqrt{GM\left(\frac{2}{r}-\frac{1}{a}\right)} \end{align}

Orbital velocity in an elliptical orbit

(75)
\begin{align} v_{\text{esc}} = \sqrt{\frac{2GM}{R}} \end{align}

Escape velocity of a planetary body

(76)
\begin{align} v_{\text{esc}} = R\sqrt{\frac{8\pi G\rho}{3}} \end{align}

Escape velocity of a planetary body

(77)
\begin{align} T = \frac{2\pi a^{3/2}}{\sqrt{GM}} \end{align}

Orbital Period of an elliptical orbit

(78)
\begin{align} \frac{dA}{dt} = \frac{1}{2}r^2\frac{d\theta}{dt} \end{align}

Sector velocity in an elliptical orbit

(79)
\begin{align} R_S = \frac{2GM}{c^2} \end{align}

Schwarzschild Radius

Periodic Motion
(80)
\begin{equation} T = f^{-1} \end{equation}

Relationship between Period and Frequency

(81)
\begin{align} \omega = 2\pi f = \sqrt{\frac{k}{m}} \end{align}

Angular frequency

(82)
\begin{align} f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \end{align}

Frequency

(83)
\begin{align} x(t) = A\cos(\omega t + \phi) \end{align}

Displacement of an object in simple harmonic motion

(84)
\begin{align} \frac{d^2x}{dt^2}+\frac{k}{m}x = 0 \end{align}

Equation for Orthogonal Simple Harmonic Motion

(85)
\begin{align} E_{\text{tot}} = \frac{1}{2}kA^2 \end{align}

Energy of object in simple harmonic motion

(86)
\begin{align} \omega = \sqrt{\frac{\kappa}{I}} \end{align}

Angular Frequency

(87)
\begin{align} \theta = \Theta\cos(\omega t+\phi) \end{align}

Angular displacement in angular simple harmonic motion

(88)
\begin{align} \frac{d^2\theta}{dt^2} + \frac{\kappa}{I}\theta = 0 \end{align}

Equation for angular simple harmonic motion

(89)
\begin{align} T = 2\pi\sqrt{\frac{I}{mgd}} \end{align}

Period of a physical pendulum

(90)
\begin{align} x = Ae^{-\frac{b}{2m}t}\cos(\omega t+\phi) \end{align}

Damped Oscillation

Periodic Waves
(91)
\begin{align} v = \lambda f \end{align}

Wave Propagation Speed

(92)
\begin{align} v = \sqrt{\frac{F}{\mu}} \end{align}

Transverse Wave Velocity on a string

(93)
\begin{align} k = \frac{2\pi}{\lambda} \end{align}

The Wave Number

(94)
\begin{align} y = A\cos(kx-\omega t) \end{align}

The Wave Function

(95)
\begin{align} \frac{\partial^2y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2y}{\partial t^2} \end{align}

Differential Wave Equation

(96)
\begin{align} v_y = \omega A \sin(kx-\omega t) \end{align}

Transverse Velocity

(97)
\begin{align} v_y = \frac{\partial y}{\partial t} \end{align}

Transverse Velocity

(98)
\begin{align} a_y = -\omega^2A\cos(kx-\omega t) \end{align}

Transverse Acceleration

(99)
\begin{align} a_y = \frac{\partial^2y}{\partial t^2} \end{align}

Transverse Acceleration

(100)
\begin{align} P_{\text{avg}} = \frac{1}{2}\sqrt{\mu F}\omega^2A^2 \end{align}

Average power of a wave

(101)
\begin{align} P = -F\frac{\partial y}{\partial x}\frac{\partial y}{\partial t} \end{align}

Power of a wave

(102)
\begin{align} y = (A_{\text{SW}}\sin kx)\sin \omega t \end{align}

Wave equation for a standing wave

(103)
\begin{align} f_n = n\frac{v}{2L} \end{align}

Nth harmonic of a standing wave

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