Mechanics

| Linear | Rotational |

| -------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------- |

| Displacement $\vec x$ | Angle $\vec \theta$ |

| Mass $m$ | Moment of inertia $I=\alpha MR^2$ where $M$ is mass and $R$ is the largest distance a particle is from axis or $I=\sum m_ir_i^2$ for $i$th particle [$kg\cdot m^2$] |

| Velocity $\vec v={\vec{\Delta x}\over\Delta t}$ [$\frac{m}s$] | Angular velocity $\vec \omega=\frac{\Delta\vec \theta}{\Delta t}$ [$\frac{\text{rad}}{s}$] $v_t= \vec \omega r$ ([$v_t$=tangential velocity] must be in rads) |

| Momentum $\vec p=m\vec v$ [$kg\cdot m\over s$] | Angular Momentum $\vec L=I\vec \omega$ [$kg\cdot m^2\over s$] |

| Impulse $\vec{\Delta p}=\vec{p_f}-\vec{p_0}$ [ditto] $\vec{\Delta p}$ is also written as $J$ | Twirl $\vec{\Delta L}=\vec{L_f}-\vec{L_i}$ - change in angular momentum |

| Force $\vec F=\frac{\vec{\Delta p}}{\Delta t}$ - average impulse over time [$N$ or $kg\cdot m\over s^2$] | Torque $\vec \tau=\frac{\vec{\Delta L}}{\Delta t}$ - average twirling over time [$N\cdot m$ or $kg\cdot m^2\over s^2$] |

| Kinetic energy $K=\frac12mv^2$ | Rotational kinetic energy $K_{rot}=\frac12I\omega^2$ |

• $L=mvr$ derivation

• Centripetal acceleration $a_c = \frac{v^2}r$

• Kinematics

  • $x_f - x_i = \int_{t_0}^{t_f} v(t) dt$

  • $v_f^2 = v_i^2 + 2a\Delta x$

    • TODO: derive

  • $K = \frac12 mv^2 + \frac12 I\omega^2$

• Polar coordinates

  • Acceleration in polar coordinates $\frac{d^2 \vec{r}}{dt^2} = \left(\ddot{r} - r\dot{\theta}^2\right) \hat{r} + \left(2\dot{r}\dot{\theta} + r\ddot{\theta}\right) \hat{\theta}$

• $K = \frac12 mv^2 + \frac12 I\omega ^2$

• Simple harmonic motion

  • If satisfied

    • There is a balancing force proportional to how much displacement

      • Hooke's law $F_x = -kx$

  • Then you know

    • $x(t) = A \cos(\omega t)$ - displacement

    • Angular velocity $\omega = \sqrt{\frac k m}$ (from setting $\rm KE_{max}=PE_{max}$)

    • More that's obvious

      • $v(t) = -A\omega\sin(\omega t)$

      • $v_{\text{max}} = A\omega$ - magnitude of velocity when $x = 0$

      • $T = {2\pi \over \omega}$

• Momentum (conserved in collisions)

• $\Sigma E_i+W=\Sigma E_f$ - initial energy of system + work done on system = final energy of system

  • energy [$J$]

    • $K$ - kinetic energy (movement)

      • Translational: $K=\frac12m|\vec v|^2$

      • Rotational: $K=\frac12I\omega^2$

    • $V$ - potential energy (interaction)

      • Gravitational: $V_G(r)=\frac{-Gm_1m_2}{r}$

        • where universal gravitational constant $\rm G=6.67\times10^{-11}\frac{N\cdot m^2}{kg^2}$

        • Simplification for on Earth: $V_G(z)=m|\vec{g}|z$

          • $z$ is height

          • $|\vec{g}|$ is $\rm 9.8 \frac{m}{s^2}$

      • Electrostatic: $V_e=\frac1{4\pi\epsilon_0}\frac{q_1q_2}r$

        • Coulomb’s constant $\frac1{4\pi\epsilon_0}=9\times10^9 \frac{\rm N~m^2}{\rm C^2}$

          

      • Spring: $V_s=\frac12kx^2$

        • $k$ is spring constant (specific to an individual spring)

          • low value - easily moved

          • high value - could not move (spring at end of train station)

        • $x$ displacement of spring

    • $U$ - internal energy (usually does not change that much)

      • nuclear energy

      • chemical energy

      • thermal energy

  • work

    • $W=\vec{F} \cdot \vec{d}$

    • Watt (W) is $kg\cdot m^2\over s^3$ or $\frac{J}s$

    • Work is zero if there is no displacement/movement

  • power (derivative of work)

    • $P=\frac{\Delta E}{\Delta t}=\frac{W}{\Delta t}=F\cdot v$

    • watt (W)=J/s

    • horsepower (hp)=746 W

  • Forces

    • $|\vec{F}_{SF}|\leq\mu_{s}|\vec{F}_N|\approx|\vec{F}_{\text{SF,max}}|$

      • Static friction

      • $\mu_s$ depends on materials in contact

    • $|\vec{F}_{KF}|\approx\mu_{k}|\vec{F}_n|$

      • Kinetic friction

    • $|\vec{F}_D|\approx\frac12C\rho A|\vec v|^2$

      • Drag

    • ${F}_{sp}=-kx$

      • Spring force (AKA Hooke’s law)

Derivations

• Kinematics

  • $x_f - x_i = \int_{t_0}^{t_f} v(t) dt$

    $= \int_{t_0}^{t_f} a t~dt$ for constant acceleration

    $= [\frac12 at^2]^{t_f}_{t_0}$

    $= \frac12 at_f^2$

• Simple harmonic motion

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