Physics GRE Prep - Classical Mechanics

Question	Answer
Blocks on Ramps: Force Parallel and Force Perpendicular	$F_{g,par} = mg \sin \theta$ $F_{g,perp} = mg \cos \theta$
Force of Friction (Normal Force)	$F_{f} = \mu mg \cos \theta$ $N = mg \cos \theta$
Kinematic Equations	$x(t) = v_{0x}t + x_0$ $y(t) = -\frac{1}{2}gt^2 + v_{0y}t + y_0$ Only use kinematics if you need to know the explicit time dependence of a system. Energy considerations are faster otherwise.
Initial and Final Velocity Equation	$v_f^2-v_i^2 = 2a\Delta x$
Radial Acceleration for Uniform Circular Motion	$a = \frac{v^2}{r}$
Centripetal Force for Uniform Circular Motion	$F = \frac{mv^2}{r}$ Does not tell you what kind of force acting on body, just that a body moving in uniform circular motion has this force.
Types of Energy	Translational Kinetic: $\frac{1}{2}mv^2$ Rotational Kinetic: $\frac{1}{2}I\omega^2$ Gravitational Potential: $mgh$ Spring Potential: $\frac{1}{2}kx^2$
Potential Energy Line Integral	$\Delta U = - \int_{a}^{b} \textbf{F} \cdot d\textbf{l}$ For any conservative force - Integral is indirection of the force vector.
Gravitational Force and Gravitational Potential	$\textbf{F}_{grav} = \frac{Gm_1m_2}{r^2} \hat{r}$ $U(r) = \frac{GmM}{r}$
Potential Energy and Force Relationship	$\textbf{F} = -\nabla U$
Linear Velocity of Rolling without Slipping	$v = R\omega$ Questions regarding which arrives with slower linear velocity are NOT questions of which arrive first. Need kinematics. Ex. energy expression for a sphere with mass m and radius r. $mgh = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{5}mr^2)\omega^2$ REMEMBER: $\omega = \frac{v}{r}$
Work Energy Theorem - In terms of Kinetic Energy - In terms of Force	$E_{initial} + W_{other} = E_{final}$ $W = \Delta KE$ $W = \int \textbf{F} \cdot d\textbf{l}$
Linear Collisions Conservation of Momentum Two balls collide, M on m with M scattering at $\theta$ with initial velocity V with the final velocity equal for both at v. What is the scattering angle of $\phi$ of m?	$MV = Mv\cos\theta + mv\cos\phi$ (parallel to initial) $0 = Mv\sin\theta + mv\sin\phi$ (perpendicular to initial) $\phi = \sin^{-1}(\frac{-M}{m}\sin\theta)$ Using limiting cases, triaging is essential. Finding phi analytically takes time.
Angular momentum of a point particle and for an extended body.	$\vec{L} = \vec{r} x \vec{p}$ $\vec{L} = I \vec{\omega}$
Torque (Angular Force)	$\vec{\tau} = \vec{r} x \vec{F}$
Scalar Analogues of $\vec{p} = m\vec{v}$ and $\vec{F} = \frac{d\vec{p}}{dt}$	Angular Momentum: $L = I\omega$ Torque: $\tau = \frac{dL}{dt}$ The vector angular momentum $\vec{L}$ is generally parallel to $\vec{\omega}$ with the angular momentum determined by the right hand rule.
Moment of Inertia (General and Extended Objects)	$I = mr^2$ $I = \int r^2 dm$ for dm = $\rho dV$ and $\rho = Ar^3$ Compute the integral to find the constant A and then insert into the extended objects integral. Ex. $\rho(x) = Ax^2$ $M = \int_{0}^{L} \rho(x)dx = \frac{1}{3} AL^3 \rightarrow A = \frac{3M}{L^3}$
Parallel Axis Theorem	$I = I_{CM} + Mr^2$ For penny through center, $I = \frac{1}{2}MR^2$ For penny axis through edge, $I = \frac{1}{2}MR^2 + MR^2$
Center of Mass Displacement from Origin	$\vec{r}_{CM} = \frac{\int \vec{r} dm}{M}$ Must solve three integrals, one for each dimension. Remember to select a volume integral and corresponding coordinate system that capitalizes on symmetry. For a system of point masses: $\vec{r}_{CM} = \frac{\Sigma_i \vec{r_i} m_i}{M}$
Lagrangian Function	Scalar Function $L(q, \dot{q}, t) = T - U$ for T kinetic energy, U potential energy, and q the generalized coordinate describing degrees of freedom.
How do you compute the correct expression for T in the Lagrangian?	1. Write down expressions for Cartesian coordinates in terms of chosen coordinates q. 2. Differentiate x,y,z to get $\dot{x}, \dot{y}, \dot{z}$ 3. Form kinetic cartesian energy, $T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)$ for point particle, $T = \frac{1}{2}I\omega^2$ for an extended object. For extended object, must express $\omega$ in terms of $\dot{q}$ , but this is easy as the two are generally equal. See page 20 in notebook for excellent example.
Euler-Lagrange Equations	$\frac{d}{dt}\frac{\delta L}{\delta\dot{q}} = \frac{\delta L}{\delta q}$ One equation for each degree of freedom, q. Remember that $\frac{d}{dt}$ is a total time derivative, not a partial derivative. I.e. $\frac{d}{dt}m\dot{x} = m\ddot{x}$
Momentum Conjugate to q (Lagrangian)	$\frac{\delta L}{\delta \dot{q}}$ Iff the Lagrangian is independent of a coordinate q, the corresponding conjugate momentum quantity, $\frac{\delta L}{\delta \dot{q}}$ , is conserved. Quantities whose time derivatives are zero are conserved quantities.
Hamiltonian	$H(p,q) = \Sigma_i p_i \dot{q_i} - L$ for $p_i = \frac{\delta L}{\delta \dot{q_i}}$ If the potential energy is not explicitly dependent on velocities of time, H = T + U Iff the Hamiltonian is independent of a coordinate 1, the corresponding conjugate momentum p is conserved.
Hamiltonian for Particle Moving in One Dimension	$H = T + U = \frac{p_x^2}{2m} + U(x)$
Hamiltonian for Particle Moving in 2D Polar	$H = \frac{p_r^2}{2m} + \frac{p_{\theta}^2}{2mr^2}$
Hamilton's Equations	Scalar function encoding equations of motion: $\dot{p} = -\frac{\delta H}{\delta q}$ $\dot{q} = \frac{\delta H}{\delta p}$
Lagrangian for a Simple Pendulum	Take corresponding derivatives - e.g. $\dot{x} = l\cos\theta\dot{\theta}\cos{\phi} - l\sin\theta\sin\phi\dot{\phi}$ Plug the $\dot{x},\dot{y},\dot{z}$ into the expression for $T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)$ Simplify and put into L = T - U for $U = mgz = - mgl\cos\theta$ Image: facc5beb-5569-442b-b2cf-0f1820177822 (image/jpg)
Conservation of Conjugate Momentum Example	For the Lagrangian independent of $\phi$ , $p_{\phi}$ is conserved. Therefore the following quantity is conserved: $p_{\phi} = \frac{\delta L}{\delta \dot{\phi}}$

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Physics GRE Prep - Classical Mechanics

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	Created by Hayden Tornabene almost 9 years ago