Physics 151

\[|\vec{A}|=\sqrt{A_x^2+A_y^2+A_z^2}\] \(A_x\) = x-component of \(\vec{A}\) \(A_y\) = y-component of \(\vec{A}\) \(A_z\) = z-component of \(\vec{A}\) \[\vec{A}=A_x\hat{\imath}+A_y\hat{\jmath}+A_z\hat{k}\] \(\hat{\imath}\) = unit vector in x \(\hat{\jmath}\) = unit vector in y \(\hat{k}\) = unit vector in z \[\hat{A}=\vec{A}/|\vec{A}|\] \(\hat{A}\) = direction of \(\vec{A}\) with magnitude 1 \[\vec{A}\cdot\vec{B}=AB\cos\theta\] \(A\) = magnitude of \(\vec{A}\) \(B\) = magnitude of \(\vec{B}\) \(\theta\) = angle between \(\vec{A}\) and \(\vec{B}\) \[\vec{A}\times\vec{B}=AB\sin\theta\,\hat{n}\] \(\hat{n}\) = unit vector perpendicular to plane of \(\vec{A},\vec{B}\)

\[\bar{v}=\dfrac{\Delta x}{\Delta t}\] \(\Delta x\) = displacement \(\Delta t\) = time interval \[v=\dfrac{dx}{dt}\] \[\bar{a}=\dfrac{\Delta v}{\Delta t}\] \[a=\dfrac{dv}{dt}\] \[v=v_0+at\] \[x=x_0+v_0t+\tfrac{1}{2}at^{2}\] \[v^{2}=v_0^{2}+2a(x-x_0)\] \(x_0\) = initial position \(v_0\) = initial velocity \(a\) = acceleration \(t\) = time \[y=y_0+v_0t-\tfrac{1}{2}gt^{2}\] \(y_0\) = initial height \(g\) = gravitational acceleration

\[x=(v_0\cos\theta)t\] \[y=(v_0\sin\theta)t-\tfrac{1}{2}gt^{2}\] \(v_0\) = launch speed \(\theta\) = launch angle \[v_x = v_0 \cos\theta\] \[v_y = v_0 \sin\theta - gt\] \(v_x\) = horizontal velocity \(v_y\) = vertical velocity \[\vec{v}_{A/C}=\vec{v}_{A/B}+\vec{v}_{B/C}\]

\[a_c=\dfrac{v^{2}}{r}=\omega^{2}r\] \(a_c\) = centripetal acceleration \(v\) = linear speed \(r\) = radius \(\omega\) = angular velocity Relation: \(v=\omega r\)

An object at rest stays at rest, and an object in motion continues with constant velocity if no net external force acts. \[\vec{F}_{\text{net}}=m\vec{a}\] \(\vec{F}_{\text{net}}\) = net force \(m\) = mass \(\vec{a}\) = acceleration For every action there is an equal and opposite reaction. \[\vec{F}_{A\to B}=-\vec{F}_{B\to A}\]

\[W=mg\] \(W\) = weight \(m\) = mass \(g\) = gravitational acceleration \[N=mg\] \(N\) = normal force \[N=mg\cos\theta\] \(\theta\) = incline angle \[F_{\parallel}=mg\sin\theta\] \[F_{\perp}=mg\cos\theta\] \(F_{\parallel}\) = downslope component \(F_{\perp}\) = perpendicular component Force transmitted along a rope or string

\[f_s\le \mu_s N\] \(f_s\) = static friction \(\mu_s\) = static coefficient \(N\) = normal force \[f_k=\mu_k N\] \(f_k\) = kinetic friction \(\mu_k\) = kinetic coefficient

\[f_d=bv\] \(f_d\) = drag force \(b\) = linear drag constant \[f_d=\tfrac{1}{2}C\rho A v^{2}\] \(C\) = drag coefficient \(\rho\) = fluid density \(A\) = cross-sectional area \[v_t=\sqrt{\dfrac{2mg}{C\rho A}}\] \(v_t\) = terminal speed