Physics 1 Equations

The following are the elements used in creating the Physics 1 Equation Sheet (pdf). The equation sheet was assembled with Inkscape.

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MathKinematicsGalilean RelativityCircular MotionDynamicsWork and EnergyMomentumRotational DynamicsStatics and ElasticityGravitationFluidsOscillationsWavesSymbols and UnitsConstants


Math

\[ \vec{A} = \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix} = A_x \hat{\imath} + A_y \hat{\jmath} + A_z \hat{k} = A\hat{u} \] \[ \hat{u} = \frac{\vec{A}}{ A } \] \[ c\vec{A}+d\vec{B} = \begin{bmatrix}cA_x+dB_x \\ cA_y+dB_y\\ cA_z+dB_z\end{bmatrix} \] \[ \vec{A} \cdot \vec{B} = A_x B_x +A_y B_y + A_z B_z = A B \cos \theta \]

\[ \vec{A}\times\vec{B}=\left (AB \sin \theta \right ) \hat{u}_{\perp} \] \[ \vec{A}\times\vec{B} =\left (A_yB_z-A_zB_y \right )\hat{\imath}+ \left (A_zB_x-A_xB_z \right )\hat{\jmath}+ \left (A_xB_y-A_yB_x \right ) \hat{k} \] \[ \hat{u}=\hat{\imath} \cos \theta_x + \hat{\jmath} \sin \theta_x \] \[ \sin \theta_x = \cos \theta_y \]

Kinematics

\[ \vec{r}(t) = x(t)\hat{\imath} + y(t)\hat{\jmath} + z(t)\hat{k} \] \[ \vec{v} = \frac{d}{dt} \vec{r}(t) \] \[ \vec{a} = \frac{d}{dt} \vec{v}(t) \] \[ \vec{r}(t) = \vec{r}_0 + \int_{t_0}^t \vec{v}(t') \; dt' \] \[ \vec{v}(t) = \vec{v}_0 + \int_{t_0}^t \vec{a}(t') \; dt' \] \[ \left\{\begin{matrix} \vec{v}_0 = \vec{v}(t_0) \\ \vec{r}_0 = \vec{r}(t_0) \end{matrix}\right. \]

\[ \vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \tfrac{1}{2}\vec{a}t^2 \] \[ \vec{v}(t) = \vec{v}_0 + \vec{a}t \] \[ v^2_s = v_{0s}^2 + 2a_s \Delta s \]

\[ \begin{bmatrix}x(t)\\y(t)\end{bmatrix} = \begin{bmatrix}x_0\\y_0\end{bmatrix} + t \begin{bmatrix}v_{0x}\\v_{0y}\end{bmatrix} + \tfrac{1}{2}t^2 \begin{bmatrix}0\\-g\end{bmatrix} \]

\[ \begin{bmatrix}v_x(t)\\v_y(t)\end{bmatrix} = \begin{bmatrix}v_{0x}\\v_{0y}\end{bmatrix} + t \begin{bmatrix}0\\-g\end{bmatrix} \]

Galilean Relativity

\(\vec{r}_{AB}\) and \(\vec{v}_{AB}\) represent the location and velocity of \(A\) relative to \(B\)

\[ \vec{r}_{BA} = -\vec{r}_{AB}\] \[\vec{r}_{AB} = \vec{r}_{AC} + \vec{r}_{CB} \] \[\vec{v}_{AB} = - \vec{v}_{BA} \] \[\vec{v}_{AB} = \vec{v}_{AC} + \vec{v}_{CB} \]

Circular Motion

\[ \vec{r}(t) = r \left [\hat{\imath} \cos \theta + \hat{\jmath} \sin \theta \right ] \] \[ \omega = \frac{d \theta}{dt} = \frac{v}{r} \] \[ \alpha = \frac{d \omega}{dt} = \frac{a_t}{r} \] \[ a_c = \frac{v^2}{r} = \omega^2 r \] \[ a_t = \frac{d \left |\vec{v} \right |}{dt} \] \[ \theta(t) = \theta_0 + \int_{t_0}^{t} \omega(t') dt' \] \[ \omega(t) = \omega_0 + \int_{t_0}^{t} \alpha(t') dt' \] \[\left\{\begin{matrix} \theta_0 = \theta(t_0) \\ \omega_0 = \omega(t_0) \end{matrix}\right. \]

\[ \vec{a} = \vec{a}_c + \vec{a}_t \] \[ v = \frac{2\pi r}{T} \] \[ \theta = \frac{s}{r} \]

Dynamics

\[\vec{F}_{\rm{net}} = m \vec{a} \] \[\vec{F}_{\rm{net}} = \textstyle{\sum_{i}} \vec{F}_i \] \[\vec{F}_{12} = -\vec{F}_{21} \] \[\vec{F}_c = m \vec{a}_c \] \[\vec{F}_{G} = m \vec{g} \] \[F_{\rm{Sp}} = -k (x-x_0) \] \[f_s \leq f_{s,\rm{max}} = \mu_s n \] \[f_k = \mu_k n \] \[F_D = \tfrac{1}{2} C \rho A v^2 \]

Free-Body Diagrams:

  1. identify system boundary
  2. draw FBD w/ only external forces (direct contact or field force)
  3. draw \(\vec{F}_{\rm{net}}\) vector below the FBD
  4. construct equation \(F_{\rm{net}} =ma\) for each coordinate direction
  5. repeat as needed for another system

Work and Energy

\[ W = \int_A^B \vec{F} \cdot d\vec{s} = F \Delta s \cos \theta \] \[ \overbrace{\rm{if} \; \vec{F} \; \rm{and} \; d\vec{s} \; \rm{are}\; \rm{constant}} \] \[ E_{\rm{mech}} = K + U \] \[ K = \tfrac{1}{2}mv^2 \] \[ \Delta U = - W_{\rm{int}} = -\int_A^B \vec{F}_{\rm{int}} \cdot d\vec{s} \] \[ F_s = -\frac{dU}{ds} \] \[ U_G = mgy \] \[ U_{\rm{Sp}} = \tfrac{1}{2}ks^2 \] \[ P = \frac{dW}{dt} = \vec{F} \cdot \vec{v} \] \[ W_{\rm{ext}} = \Delta E_{\rm{sys}} = \Delta (K + U + E_{\rm{th}}) \]

if \(W_{\rm{ext}} = 0\) and no friction loss then \(\Delta K + \Delta U =0\)

Momentum

\[\vec{p} = m \vec{v} \] \[\vec{P} = \sum_{i}\vec{p}_i \] \[\vec{P} = M \vec{v}_{\rm{cm}} \] \[v_{2i} = 0 \] \[v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_{1i} \] \[v_{2f} = \frac{2 m_1 }{m_1 + m_2} v_{1i} \] \[\vec{v}_{\rm{cm}} = \frac{d}{dt} \vec{r}_{\rm{cm}} \] \[\vec{F}_{\rm{net}} = \frac{d \vec{P}}{dt} \] \[K = \frac{p^2}{2m} \] \[\Delta \vec{P} = \vec{J} = \int \vec{F}_{\rm{ext}} \; dt = \vec{F}_{\rm{ave}} \Delta t \] \[M = \textstyle{\sum_{i}} m_i \] \[x_{\rm{cm}} = \frac{\textstyle{\sum_{i}} m_i x_i }{M} = \frac{1}{M} \int x \; dm \] \[\vec{r}_{\rm{cm}} = \frac{1}{M} \int \vec{r} \; dm \] \[F_{\rm{thrust}} = Ru \]

if \(\vec{F}_{\rm{ext}} = 0\) then \(\Delta \vec{P} = 0\)

if \(\vec{F}_{\rm{ext}} \neq 0\) then \(\Delta \vec{P} = \vec{J}\)

Rotational Dynamics

\[ \vec{\tau} = \vec{r} \times \vec{F} \]

\[ \tau = r F \sin \phi = \pm Fd \] \[ I_{\rm{particle}} = m r^2 \] \[ I_{\rm{disk}} = \tfrac{1}{2} m r^2 \] \[ I_{\rm{para}} = I_{\rm{cm}} + md^2 \] \[ \vec{L} = \vec{r} \times \vec{p} \] \[ \vec{L} = I \vec{\omega} \] \[ \vec{\tau} = I \vec{\alpha} \] \[ \vec{\tau} = \frac{d \vec{L}}{dt} \] \[ \Omega = \frac{mgd}{I \omega } \] \[ K_{\rm{total}} = K_{\rm{trans}} + K_{\rm{rot}} \] \[ K_{\rm{rot}} = \tfrac{1}{2} I \omega^2 \]

Statics and Elasticity

\[ \begin{eqnarray*} \vec{F}_{\rm{net}} =0 \\ \vec{\tau}_{\rm{net}} =0 \\ \frac{F}{A} = Y \frac{\Delta L}{L} \\ p = -B \frac{\Delta V}{V} \\ \end{eqnarray*} \]

Gravitation

\[ \begin{eqnarray*} F_G = G \frac{m_1 m_2}{r^2} \\ \vec{F}_G = G \frac{m_1 m_2}{r^2} \hat{r}\\ g = \frac{GM}{r^2} \\ U_G = - G \frac{m_1 m_2}{r} \\ v_{\rm{esc}} = \sqrt{ \frac{2GM}{r}} \\ E_{\rm{mech}} = K+U_G \\ \end{eqnarray*} \]

for circular orbits:

\[ \begin{eqnarray*} \frac{G M}{r^2} = \frac{v^2}{r} \\ T^2 = \left (\frac{4 \pi^2}{GM} \right )r^3 \\ E_{\rm{mech}} = \tfrac{1}{2} U_G \\ \end{eqnarray*} \]

bound orbit: \(E_{\rm{mech}} < 0\)
unbound orbit: \(E_{\rm{mech}} > 0\)

Fluids

\[ \begin{eqnarray*} p = \frac{F}{A} \\ \rho = \frac{m}{V} \\ p = p_0 + \rho g d \\ F_B = \rho V_{\rm{disp}} g \\ \frac{ V_{\rm{disp}}}{V_o} = \frac{\rho_o}{\rho_f} \; \rm{floating} \\ Q = \frac{dV}{dt} = A \overline{v} \\ p_1 + \tfrac{1}{2} \rho v_1^2 + \rho g y_1 = p_2 + \tfrac{1}{2} \rho v_2^2 + \rho g y_2 \\ p + \tfrac{1}{2} \rho v^2 + \rho g y = \rm{constant} \\ \end{eqnarray*} \]

Oscillations

\[ \begin{eqnarray*} \ddot{s}(t) = - \omega^2 s(t) \\ f = \frac{1}{T} \\ \omega = 2 \pi f = \frac{2 \pi}{T} \\ x(t)= A \cos (\omega t + \phi) \\ v(t) = -\omega A \sin (\omega t + \phi) \\ a(t) = -\omega^2 A \cos (\omega t + \phi) \\ \omega = \sqrt{\frac{k}{m}} \\ \omega = \sqrt{\frac{g}{L}} \\ \omega = \sqrt{ \frac{mgL}{I} } \\ E = K + U = \tfrac{1}{2} k A^2 = \tfrac{1}{2} m v_{\rm{max}}^2 \\ E(t) = E_0 e^{-bt/m} = E_0 e^{-t/\tau} \; \rm{damped} \\ \end{eqnarray*} \]

Waves

\[v^2 \frac{\partial^2 y(x,t)}{\partial x^2} = \frac{\partial^2 y(x,t)}{\partial t^2} \] \[y(x,t) = A \sin(kx\mp \omega t + \phi ) \] \[k = \frac{2 \pi}{\lambda} \] \[v= f \lambda = \frac{ \omega}{k} \] \[v = \sqrt{\frac{F_T}{\mu}} \] \[n \frac{\lambda}{2} = L \]

Symbols and Units

name symbol units
acceleration \(a\) m/s\(^2\)
area \(A\) m\(^2\)
amplitude \(A\) m
bulk modulus \(B\) N/m\(^2\)
damping constant \(b\) kg/s
moment arm \(d\) m
distance \(d\) m
energy \(E\) J
force \(f,F\) N
frequency \(f\) Hz, s\(^{-1}\)
gravity field strength \(g\) N/kg
moment of inertia \(I\) kg\(\cdot\)m\(^2\)
impulse \(J\) N\(\cdot\)s
kinetic energy \(K\) J
spring constant \(k\) N/m
wave number \(k\) rad/m
angular momentum \(L\) kg\(\cdot\)m\(^2\)/s
length \(L\) m
mass \(m,M\) kg
normal force \(n\) N
mode of standing wave \(n\) (none)
pressure \(p\) Pa
power \(P\) W, J/s
momentum \(p,P\) kg\(\cdot\)m/s
position vector \(\vec{r}\) m
radius, distance \(r,R\) m
path length \(s\) m
time \(t\) s
period \(T\) s
unit vector \(\hat{u}\) (none)
rocket exhaust speed \(u\) m/s
potential energy \(U\) J
velocity \(v\) m/s
work \(W\) J
\(x\)-position \(x\) m
\(y\)-position \(y\) m
\(z\)-position \(z\) m
angular acceleration \(\alpha\) rad/s\(^2\)
angle \(\theta\) radians
wavelength \(\lambda\) m
linear mass density \(\mu\) kg/m
kinetic friction coeff \(\mu_k\) (none)
static friction coeff \(\mu_s\) (none)
density \(\rho\) kg/m\(^{3}\)
torque \(\tau\) N\(\cdot\)m
time constant \(\tau\) s
angle \(\phi\) radians
angular velocity \(\omega\) rad/s
angular frequency \(\omega\) rad/s
gyro precession freq \(\Omega\) rad/s

Constants

gravity field on Earth: \(g = 9.81\) N/kg

universal gravity const: \(G = 6.67\times 10^{-11}\) N\(\cdot\)m\(^2\)/kg\(^2\)


Last modified: Thu December 14 2023, 02:02 PM.