Newton's Laws & Momentum

$\vec{F}_{net} = m\vec{a} \quad [M^1L^1T^{-2}] \quad \text{(N)}$

$\vec{F}_{net} = \frac{d\vec{p}}{dt} \quad \text{where} \quad \vec{p} = m\vec{v} \quad [M^1L^1T^{-1}] \quad \text{(kg m/s)}$

$\vec{J} = \vec{F}\,\Delta t = \Delta\vec{p} \quad [M^1L^1T^{-1}] \quad \text{(N·s)}$

$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \quad \text{(conservation of momentum, } F_{ext}=0\text{)}$

Apparent Weight in Lift

$W'_{up} = m(g + a) \quad [M^1L^1T^{-2}] \quad \text{(N)}$

$W'_{down} = m(g - a) \quad [M^1L^1T^{-2}] \quad \text{(N)}$

$W'_{freefall} = 0 \quad (a = g)$

Atwood Machine

$a = \frac{(m_1 - m_2)\,g}{m_1 + m_2} \quad [M^0L^1T^{-2}] \quad \text{(m/s}^2\text{)}$

$T = \frac{2m_1 m_2\,g}{m_1 + m_2} \quad [M^1L^1T^{-2}] \quad \text{(N)} \quad ; \quad m_2 g < T < m_1 g$

Friction

$f_s \leq \mu_s N \quad ; \quad f_k = \mu_k N \quad ; \quad f_r = \mu_r N \quad [\text{dimensionless} \times M^1L^1T^{-2}]$

$\tan\theta_{repose} = \mu_s \quad \text{(angle of repose, dimensionless)}$

Circular Motion

$F_c = \frac{mv^2}{r} \quad [M^1L^1T^{-2}] \quad \text{(N)}$

$v_{max,\,level} = \sqrt{\mu r g} \quad [M^0L^1T^{-1}] \quad \text{(m/s)}$

$\tan\theta_{bank} = \frac{v^2}{rg} \quad \text{(no friction, dimensionless)}$

Inclined Plane Components

$N = mg\cos\theta \quad ; \quad F_{\parallel} = mg\sin\theta \quad [M^1L^1T^{-2}]$