Kinetics
velocity: $\mathbf{v}=\frac{d\mathbf{r}}{dt}=r'_x \mathbf{i}+r'_y \mathbf{j}+r'_z \mathbf{k}$
acceleration: $\mathbf{a}=\frac{d\mathbf{v}}{dt}=v'_x \mathbf{i}+v'_y \mathbf{j}+v'_z \mathbf{k}$
motion laws:
$v=v_0 +at$
$x-x_0=v_0 t+ \frac{1}{2}at^2$
$v^2-v_0^2=2a(x-x_0)$
$x-x_0=\frac{(v_0+v)t}{2}$
centripedal acceleration: $a=\frac{v^2}{r}=\frac{4\pi ^2 r}{T^2}=4\pi ^2 rf^2=\omega ^2r$
velocity in two frames: $\mathbf{v}_{\mathrm{PA}}=\mathbf{v}_{\mathrm{PB}}+\mathbf{v}_{\mathrm{BA}}$
―same acceleration measured in all frames: $\mathbf{a}_{\mathrm{PA}}=\mathbf{a}_{\mathrm{PB}}$
Kinetics 2
Newton's second law: $\mathbf{F}_\mathrm{T}=m\mathbf{a}$
Newton's third law: $\mathbf{F}_{\textrm{A to B}}=-\mathbf{F}_{\textrm{B to A}}$
simple pulley: $a=\frac{m_1-m_2}{m_1+m_2}g$
drag force (body in fluid): $D=\frac{1}{2}C\rho Av^2$
―terminal speed: $v=\sqrt{\frac{2mg}{C\rho A}}$
―C: drag coefficient, A: effective cross-sectional area, ρ: air density
work: $W=\int \mathbf{F}\cdot d\mathbf{s}=\int_{x_i}^{x_f}F_x dx+\int_{y_i}^{y_f}F_y dy+\int_{z_i}^{z_f}F_z dz$
Hooke's law: $\mathbf{F}=-k\mathbf{x}$
―work: $W=\frac{1}{2}kx_i^2-\frac{1}{2}kx_f^2$
work-kinetic energy theorem: $W=K_f-K_i=\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2$
power: $P=\frac{dW}{dt}=\mathbf{F}\cdot \mathbf{v}$
Kinetics 3
definition of change in potential energy: $\Delta U=-W$
change in mechanical energy: $\Delta K+\Delta U=0$
total mechanical energy: $E=U+K$ (isolated)
U-x graph:
force: $F(x)=-\frac{d}{dx}U(x)$
neutral equilibrium: $E=U$
unstable equilibrium: $U''< 0, U'=0$
stable equilibrium: $U''> 0, U'=0$
total m.energy of block-spring system: $E=\frac{1}{2}kx^2+\frac{1}{2}mv^2$
total m.energy of particle-earth system: $E=mgy+\frac{1}{2}mv^2$
conservation of energy: $\Delta K+\Delta U+\Delta E_{\textrm{internal}}\left ( + \textrm{other forms} \right )=0$ (isolated)
external work done: $W=\Delta K+\Delta U+\Delta E_{\textrm{internal}}$
change in energy: $\Delta E=\Delta K+\Delta U$
loss in mechanical energy: $\Delta E=-fd$
energy loss due to emitted light: $E_x-E_y=hf$
System kinetics
centre of mass: $\mathbf{r}_{\mathrm{CM}}=\frac{1}{M} \sum _i m_i \mathbf{r}_i$
continuous: $\mathbf{r}_{\mathrm{CM}}= \frac{1}{M}\int\mathbf{r}\rho dV= \frac{1}{V}\int\mathbf{r}dV$
relation: $dm=\rho dV$
linear momentum: $\mathbf{p}=m\mathbf{v}$
net force: $\mathbf{F}_\mathrm{T}=\frac{d\mathbf{p}}{dt}$
Newton's second law: $\sum \mathbf{F}_{\textrm{external}}=M\mathbf{a}_{\mathrm{CM}}=\frac{d\mathbf{P}}{dt}$
conservation of linear momentum: $\mathbf{P}=\textrm{constant}$
циолковский rocket formula:
$\mathbf{F}=(\mathbf{v}-\mathbf{u})\frac{dm}{dt}+m\frac{d\mathbf{v}}{dt}$ or $\frac{d}{dt}m\mathbf{v}=\mathbf{F}+\mathbf{u}\frac{dm}{dt}$
v: rocket's velocity in earth, u: fuel's velocity in earth
change in translational k. energy: $\Delta K_{\mathrm{CM}}=F_{\mathrm{ext}}d_{\mathrm{CM}}$
König's theorem: $K=K_{\textrm{related to CM}}+\frac{1}{2}mv_{\mathrm{CM}}^2$
Collision
impulse: $\mathbf{J}=\Delta \mathbf{p}=\int \mathbf{F}dt$
elastic collision:
$v_1'=\frac{m_1-m_2}{m_1+m_2}v_1+\frac{2m_2}{m_1+m_2}v_2$
$v_2'=\frac{m_2-m_1}{m_1+m_2}v_2+\frac{2m_1}{m_1+m_2}v_1$
$v_\mathrm{CM}=\frac{P}{m_1+m_2}$
complete inelastic collision: $m_1v_1+m_2v_2=(m_1+m_2)V_{\mathrm{(CM)}}$
reactions: $Q=-\Delta mc^2$
+Q: exothermic, -Q: endothermic
Rotation
angular position: $\theta =s/r$
angular velocity: $\omega =d\theta /dt$
angular acceleration: $\alpha =d\omega /dt$
motion laws:
$\omega =\omega _0+\alpha t$
$\theta =\omega _0t+\frac{1}{2}\alpha t^2$
$\omega ^2-\omega _0^2=2\alpha \theta $
$\theta =\frac{(\omega _0+\omega )t}{2}$
linear-angular relation (non-uniform cir.
motion): $s=\theta r$, $v=\omega r$
―tangent acceleration: $a_t=\alpha r$
―radial acceleration: $a_r=\frac{v^2}{r}=\omega ^2r$
rotational inertia: $I=\sum _i m_i r_i^2$
continuous: $I=\int r^2dm$
total rotational inertia: $I_{\mathrm{whole}}=\sum _i I_i$ (all to one axis)
parallel-axis theorem: $I=I_{\mathrm{CM}}+Mh^2$
perpendicular-axis theorem: $I_P=I_x+I_y$ (no thickness)
rotational kinetic energy: $K=\frac{1}{2}I\omega ^2$
torque: $\boldsymbol{\tau } =\mathbf{r}\times \mathbf{F}$
Newton's second law: $\tau _\mathrm{T}=I\alpha $
work: $W=\int F_t rd\theta =\int \tau d\theta $
power: $P=\frac{dW }{dt}=\tau \omega $
work-kinetic energy theorem: $W=\Delta K=\frac{1}{2}I\omega _f^2-\frac{1}{2}I\omega _i^2$
Rotational inertias
hoop, central axis: $I=MR^2$
hoop, diameter: $I=\frac{1}{2}MR^2$
annular cylinder, central axis: $\frac{1}{2}M\left ( R_1^2+R_2^2 \right )$
annular cylinder, central diameter,$\frac{1}{4}M\left ( R_1^2+R_2^2 \right )$
solid cylinder/disk, central axis: $\frac{1}{2}MR^2$
solid cylinder/disk, central diameter: $I=\frac{1}{4}MR^2+\frac{1}{12}ML^2$
rod, centre of length: $I=\frac{1}{12}ML^2$
rod, one end: $I=\frac{1}{3}ML^2$
triangle, parallel to base a (whose height is h, through CM): $I=\frac{1}{18}Mh^2$
solid sphere, diameter: $I=\frac{2}{5}MR^2$
spherical shell, diameter: $I=\frac{2}{3}MR^2$
slab, centre: $I=\frac{1}{12}M(a^2+b^2)$
slab, along edge:$I=\frac{1}{3}Ma^2$
Rolling
CM displacement/distance rolled: $x_{\mathrm{CM}}=\theta R$, $v_{\mathrm{CM}}=\omega R$
kinetic energy: $K=K_{\mathrm{rot}}+K_{\mathrm{tra}}=\frac{1}{2}I_{\mathrm{CM}}\omega ^2+\frac{1}{2}mv_{\mathrm{CM}}^2$
ideal yoyo: $a=-g(\frac{1}{1+I/MR^2})$
angular momentum: $\boldsymbol{\ell}=\mathbf{r}\times \mathbf{p}=\mathbf{r}\times m\mathbf{v}$
―rigid, fixed axis: $L=I\omega $
angular impulse: $\Delta \boldsymbol{L}=\int \boldsymbol{\tau }dt$
Newton's second law: $\boldsymbol{\tau }_\mathrm{T}=\frac{d\boldsymbol{L}}{dt}$
conservation of angular momentum: $\boldsymbol{L}=\textrm{constant}$
Elasticity
static equilibrium: $\mathbf{P}=0,\boldsymbol{L}=0$
requirements of equilibrium: $\sum \mathbf{F}_{\mathrm{ext}}=0,\sum \boldsymbol{\tau }_{\mathrm{ext}}=0$
tensile stress: $\frac{F}{A}=E\frac{\Delta L}{L}$, E: Young's modulus
sheering stress: $\frac{F}{A}=G\frac{\Delta x}{L}$, G: sheer modulus
hydraulic compression: $p=B\frac{\Delta V}{V}$, B: Bulk modulus
Oscillation
simple harmonic motion (F=-mω2x):
$x(t)=x_m \mathrm{cos}(\omega t+\phi )$, $\omega =\frac{2\pi }{T}=2\pi f$
$v(t)=-\omega x_m \mathrm{sin}(\omega t+\phi )$
$a(t)=-\omega ^2 \, x(t)$
linear oscillator
definition: $\frac{d^2x}{dt^2}+\omega ^2x=0$
angular frequency: $\omega =\sqrt{\frac{k}{m}}$
period: $T=2 \pi \sqrt{\frac{m}{k}}$
potential energy: $U(t)=\frac{1}{2}kx_m^2\mathrm{cos}^2(\omega t+ \phi)$
kinetic energy: $K(t)=\frac{1}{2}kx_m^2\mathrm{sin}^2(\omega t+ \phi)$
total energy: $E=\frac{1}{2}kx_m^2$
simple pendulum
period: $T=2 \pi \sqrt{\frac{L}{g}}$
restoring force: $F\approx -(\frac{mg}{L})s$
torsion pendulum
period: $T=2 \pi \sqrt{\frac{I}{\kappa }}$
restoring torque: $\tau =-\kappa \theta $
physical pendulum
period: $T=2 \pi \sqrt{\frac{I}{mgh}}$
restoring torque: $\tau =-(mg\mathrm{sin}\theta )h$
damped simple harmonic motion
damping force: $F_d=-bv$, b:damping constant
definition: $\frac{d^2x}{dt^2}+\frac{b}{m}\frac{dx}{dt}+\frac{k}{m}x=0$
solution: $x(t)=x_me^{-bt/2m}\mathrm{cos}(\omega _dt+ \phi)$, $\omega _d=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}$
total energy: $E(t)\approx \frac{1}{2}kx_m^2e^{-bt/m}$
Gravitation
Newton's law of gravitation: $F=\frac{GMm}{r^2}$
gravitational constant G=6.67*10-11N·m2/kg2
differential: $dF=\frac{Gm_1}{r^2}dm$
gravitational field: $g=\frac{GM}{r^2}$
gravitational potential energy: $U=\int_{\infty }^{r}\frac{GMm}{x^2}dx=-\frac{GMm}{r}$
escape speed: $v=\sqrt{\frac{2GM}{r}}$
orbits:
path of planet: $r=\frac{p}{1+e\mathrm{cos}\theta }$, $p=\frac{L^2}{GMm^2}$, $e=\sqrt{1+\frac{2EL^2}{G^2M^2m^3}}$
net angular momentum: $L=mr^2\dot{\theta }$
net mechanical energy: $E=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta} ^2)-\frac{GMm}{r}=\frac{1}{2}m\dot{r}^2+(\frac{L^2}{2mr^2}-\frac{GMm}{r})$
total energy of satellite-earth ellipse system: $E=-K=-\frac{GMm}{2r}$
―periphelion: $r_1=a-c$
―aphelion: $r_2=a+c$, $a=\frac{r_1+r_2}{2}=-\frac{GMm}{2E}$, $c=\frac{r_2-r_1}{2}$, $b=\sqrt{a^2-c^2}=\sqrt{r_1r_2}=\frac{L}{\sqrt{-2mE}}$
total energy of satellite-earth hyperbola system: $E=\frac{GMm}{2r}$
total energy of satellite-earth parabola system: $E=0$
law of periods: $\frac{T^3}{r^3}=\frac{4 \pi ^2}{GM}$