Equation Sheet – Unit 1 through Unit 15

Beta Keramati

Kinematics Equations

$Constant\ Velocity\ :\ a=0$	$Constant\ Acceleration$
$x =vt + x_0$	$\bar{v} = \frac{v_0 + v_f}{2}$
	$v =at + v_0$
	$x = \frac{1}{2}at^2 + v_0 t + x_0$
	$v^2 - v_0^2 = 2a \Delta x$

$|\vec a_c |= \frac{v^2}{r}$ Centripetal Acceleration

$\vec a = \frac{\sum \vec F}{m}$

$\boxed{F=G\frac{m_1 m_2}{r^2}}$ Newton’s Universal Law of Gravitation

$|\vec f_k|=\mu_k|\vec N|$ Kinetic Friction Force

$0<|\vec f_s|<|\vec f_s|_{max}$ Range Of Static Friction Force

$|\vec f_s|_{max}=\mu_s|\vec N|$ Maximum Static Friction Force

$F_{spring}=-Kx$ Hooke’s Law

$W=FdCos\theta$ Work done by a constant force

$W_{net}=\Delta KE$ is the same as $W_{ncf}=\Delta KE + \Delta U$

If $W_{ncf}=0$ Then $\Delta KE + \Delta U=0$

$\Delta KE + \Delta U=0$ is the same as $KE_i+U_i =KE_f + U_f$

$KE =\frac{1}{2}mv^2$

$U_{grav}=mgy$

$U_{elastic} =\frac{1}{2}Kx^2$

$\vec P = m \vec v$ Linear Momentum

$\Delta \vec P=(\sum \vec F)(\Delta t )$ Impulse-Momentum Equation

$\Delta \vec P_1+\Delta \vec P_2=0$ is the same as $\vec P_{1i}+ \vec P_{2i}=\vec P_{1f}+\vec P_{2f}$ is the same as $\vec P_{(system)i} =\vec P_{(system)f}$ is the same as $\Delta \vec P_{(system)}=0$

Conservation Of Linear Momentum

$x_{cm}=\frac{\sum{m_ix_i}}{M}$ X-Coordinate Of Center Of Mass

$y_{cm}=\frac{\sum{m_iy_i}}{M}$ Y-Coordinate Of Center Of Mass

$\bar \omega=\frac{\Delta \theta}{\Delta t}$ Average Angular Velocity

$\omega=\frac{1}{r}v\ ,\ v=r\omega$

$I=\sum {m_i {r_i }^2}$ Moment Of Inertia

$KE=\frac{1}{2} I \omega^2$ Rotational Kinetic Energy

$I=I_{cm}+Mh^2$ parallel axis theorem

$v_{cm}=R\omega$ Rolling without slipping

$\bar \alpha=\frac{\Delta \omega}{\Delta t}$ Average Angular Acceleration

$\bar \alpha=\frac{1}{r}\bar a_t\ ,\ \bar a_t = r\bar \alpha$


Rotational	Translational	Relationship (r=radius)
$\theta$	$s$	$\theta=\frac{s}{r}$
$\omega$	$v_t$	$\omega=\frac{v_t}{r}$
$\alpha$	$a_t$ (due to speeding up/slowing down)	$\alpha=\frac{a_t}{r}$
	$a_c$ (due to change in direction of velocity)	$a_c=\frac{v_t^2}{r}$


Translational Motion with Constant Acceleration	Rotational Motion with Constant Angular Acceleration
$\bar{v} = \frac{v_0 + v_f}{2}$	$\bar{\omega} = \frac{\omega_0 + \omega_f}{2}$
$v =at + v_0$	$\omega =\alpha t + \omega_0$
$x = \frac{1}{2}at^2 + v_0 t + x_0$	$\theta = \frac{1}{2}\alpha t^2 + \omega_0 t + \theta_0$
$v^2 - v_0^2 = 2a \Delta x$	$\omega^2 - \omega_0^2 = 2\alpha \Delta \theta$