Mechanics

Motion

Newton’s Laws of Motion

1st Law

A body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon by a force.

2nd Law

The acceleration of a body is directly proportional to the net force acting on it, and inversely proportional to its mass.

• $F=ma$
• $\vec{\mathbf{F}}=m\vec{\mathbf{a}}$
• $\vec{\mathbf{F}}=\frac{d\vec{\mathbf{p}}}{dt}$
• (rotational form) $\tau =I\alpha =\frac{dL}{dt}$
  • $\tau$ is the torque
  • $I$ is the moment of inertia
  • $\alpha$ is the angular acceleration
  • $L$ is the angular momentum

3rd Law

For every action force ${\vec{F}}_{1\to 2}$ exerted by object 1 on object 2, there is an equal in magnitude and opposite in direction reaction force ${\vec{F}}_{2\to 1}$ exerted by object 2 on object 1. ${\vec{F}}_{1\to 2}=-{\vec{F}}_{2\to 1}$

Linear/Translational quantities

Momentum

• $\vec{p}=m\vec{v}$
  • $\vec{p}$ is the (linear) momentum vector (in $\mathsf{kg}\cdot m/s$)
  • $m$ is the mass (in $\mathsf{kg}$)
  • $\vec{v}$ is the velocity vector (in $m/s$)

Impulse

• If a constant force $\vec{\mathbf{F}}$ acts on an object, the impulse $\vec{\mathbf{J}}$ delivered to the object over a time interval $\Delta t$ is given by $\vec{\mathbf{J}}=\vec{\mathbf{F}}\Delta t$
• (Impulse-Momentum Theorem) $\vec{\mathbf{J}}=\Delta \vec{\mathbf{p}}$ (the impulse is equal to the change in momentum)

Motion in One Dimension

Constant Acceleration

• $t$ is the time duration
• $v_0$ is the initial velocity
• $x_0$ is the initial position
• $a$ is the acceleration
• $x=x_0+v_{0}t+\frac{1}{2}a{t}^2$
• $v=\frac{dx}{dt}=v_0+at$ (velocity as a function of time)
• $t=\frac{v-v_0}{a}$
• $x-x_0=v_{0}t+\frac{1}{2}a{t}^2=v_0\left[\frac{v-v_0}{a}\right]+\frac{1}{2}a{\left[\frac{v-v_0}{a}\right]}^2=\frac{v^2-v_0^2}{2a}$
• $v^2=v_0^2+2a(x-x_0)$
• $v=\sqrt{v_0^2+2a(x-x_0)}$ (velocity as a function of position)

Motion in Multiple Dimensions

todo Ballistic coefficient

% Author: Izaak Neutelings (April 2021)
\usepackage{tikz}
\usepackage{amsmath}
\usepackage{physics}
\usepackage{siunitx}
\usepackage{xcolor}
\usepackage{etoolbox} %ifthen
\usepackage[outline]{contour} % glow around text
\tikzset{>=latex} % for LaTeX arrow head
\usetikzlibrary{angles,quotes,arrows.meta} % for pic
\contourlength{1.0pt}
\colorlet{myblue}{blue!70!black}
\colorlet{mydarkblue}{blue!40!black}
\colorlet{mygreen}{green!50!black}
\colorlet{myred}{red!65!black}
\colorlet{xcol}{blue!85!black}
\colorlet{vcol}{green!70!black}
\colorlet{projcol}{vcol!90!black!60}
\tikzstyle{wave}=[myblue,thick]
\tikzstyle{xline}=[very thick,myblue]
\tikzstyle{vline}=[very thick,mygreen]
\tikzstyle{vector}=[->,very thick,vcol,line cap=round]
\tikzstyle{mydashed}=[green!30!black!90,dash pattern=on 2pt off 2pt,very thin]
\tikzstyle{mymeas}=[{Latex[length=3,width=2]}-{Latex[length=3,width=2]},thin]
\def\tick#1#2{\draw[thick] (#1) ++ (#2:0.05*\ymax) --++ (#2-180:0.1*\ymax)}
 
 
\begin{document}
 
\def\xmax{3.8}
\def\ymax{2.4}
\def\v{1.0}
\def\ang{30}
\def\d{(0.9*\xmax)} % distance landing point
\def\b{tan(30)} % slope at x=0
\def\h{0.6*\ymax} % height h
\def\a{-((\b*\d+\h)/\d^2)} % coefficient
\def\nsamples{100}
 
 
 
% TRAJECTORY - PARABOLA + breakdown
\begin{tikzpicture}
  \def\v{1.4}
  \def\ang{35}
  \def\h{0.5*\ymax} % height h
  \def\vx{{\v*cos(\ang)}}
  \def\vy{{\v*sin(\ang)}}
  \coordinate (O) at (0,\h);
  \coordinate (Vx) at ({\v*cos(\ang)},\h);
  \coordinate (Vy) at (0,{\h+\v*sin(\ang)});
  \coordinate (V) at ({\v*cos(\ang)},{\h+\v*sin(\ang)});
  
  % AXES & TRAJECTORY
  \draw[->,thick]
    (-0.1*\ymax,0) -- (1.06*\xmax,0) node[right=4,below=-1] {$x$};
  \draw[->,thick]
    (0,-0.1*\ymax) -- (0,\ymax) node[below=4,left=0] {$y$};
  \draw[xline,variable=\t,samples=\nsamples,smooth,domain=0:\d+0.1]
    plot(\t,{\a*\t^2+\b*\t+\h}); %node[right=7,above=-2] {$x=x(t)$};
  
  % VELOCITY VECTOR
  \draw pic["\contour{white}{$\theta$}",draw=white,double=black,double distance=0.4,
            angle radius=13,angle eccentricity=1.4] {angle = Vx--O--V};
  \draw[mydashed]
    (Vx) |- (Vy);
  \draw[<->,projcol,thick]
    (Vy) -- (O) node[scale=0.9,midway,left=-1] {$v_{0y}$}
      -- (Vx) node[scale=0.9,midway,below=-1] {$v_{0x}$};
  \draw[->,vcol,very thick,line cap=round]
    (O) --++ ({\ang}:\v) node[above right=-4] {$\vec{v}_0$};
  \tick{O}{0} node[left] {$y_0$};
  \tick{{\d},0}{90} node[below] {$R$};
  
\end{tikzpicture}
 
 
 
\end{document}
 
 

Projectile Motion

• ${\vec{F}}_{\text{air}}=0$ (neglecting air resistance)
• ${\vec{F}}_{\text{gravity}}=m\vec{g}$ (force due to gravity)
• $\theta$ is the angle of projection
• ${\vec{v}}_0$ is the initial velocity
  • $v_{0x}=v_0\cos (\theta )$ is the initial horizontal velocity
  • $v_{0y}=v_0\sin (\theta )$ is the initial vertical velocity
• ${\vec{r}}_0$ is the initial position
  • $x_0$ is the initial horizontal position (most often $0$)
  • $y_0$ is the initial vertical position
• $\vec{v}$ is the velocity
  • $v_x=v_0\cos (\theta )$ is the horizontal velocity (constant as the initial velocity, no horizontal acceleration)
  • $v_y=v_0\sin (\theta )-gt$ is the vertical velocity
• $\vec{r}$ is the position
  • $x=x_0+v_0\cos (\theta )t$ is the horizontal position
  • $y=y_0+v_0\sin (\theta )t-\frac{1}{2}g{t}^2$ is the vertical position
• $T=\frac{2v_0\sin (\theta )}{g}$ is the time of flight (time to reach the ground)
• $y_{\text{max}}=y_0+\frac{v_0^2{\sin }^2(\theta )}{2g}$ is the maximum height
• $R=x_0+v_0\cos (\theta )T=x_0+\frac{v_0^2\sin (2\theta )}{g}$ is the range (horizontal distance)

Angular/Rotational quantities

clockwise is negative by convention

Angular Velocity & Acceleration

• $\vec{\omega }=\frac{d\vec{\theta }}{dt}$ is the angular velocity (in $rad/s$)
• $\vec{\alpha }=\frac{d\vec{\omega }}{dt}$ is the angular acceleration (in $rad/{\mathsf{s}}^2$)
• For a point $P$ at a distance $r$ from the axis of rotation
  • $\vec{\mathbf{a}}=\overrightarrow{{\mathbf{a}}_{\text{t}}}+\overrightarrow{{\mathbf{a}}_{\text{c}}}$
  • $a=\sqrt{a_{\text{t}}^2+a_{\text{c}}^2}$
    • $a_{\text{t}}=r\alpha$ is the tangential acceleration (in $m/{\mathsf{s}}^2$)
    • $a_{\text{c}}=\frac{v^2}{r}=\frac{r{\omega }^2}{r}=r{\omega }^2$ is the centripetal acceleration (in $m/{\mathsf{s}}^2$)
    • $v=r\omega$ is the tangential velocity (in $m/s$)
• For constant angular acceleration
  • $\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$ is the angular position
  • $\omega ={\omega }_0+\alpha t$ is the angular velocity
  • ${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$ is the angular velocity as a function of position

Moment of Inertia

• moment of inertia I = angular momentum L / angular velocity ω
• todo Moments of inertia for various objects of uniform composition, each with mass M.

Angular Momentum

• $\vec{\mathbf{L}}=\vec{\mathbf{r}}\times \vec{\mathbf{p}}$ is the angular momentum (or moment of momentum) vector (in $\mathsf{kg}\cdot {\mathsf{m}}^2/s$)
  • $\vec{\mathbf{r}}$ is the position vector (from the pivot point to the point of application of the force) (in $\mathsf{m}$)
  • $\vec{\mathbf{p}}$ is the momentum vector (in $\mathsf{kg}\cdot m/s$)
• $\vec{\mathbf{L}}=I\vec{\omega }$ is the angular momentum (in $\mathsf{kg}\cdot {\mathsf{m}}^2/s$)
  • $I$ is the moment of inertia (in $\mathsf{kg}\cdot {\mathsf{m}}^2$)
  • $\vec{\omega }$ is the angular velocity (in $rad/s$)
• todo
  • (Bozeman Science) Angular Momentum
  • (Bozeman Science) Conservation of Angular Momentum
  • Giancoli, 8–8 Angular Momentum and Its Conservation

Torque (Moment of Force)

\usepackage{tikz}
\usepackage[outline]{contour} % Glow around text
\usetikzlibrary{calc,angles,quotes} % For pic and angle
\tikzset{>=latex} % LaTeX arrow head
\contourlength{1.1pt}
 
\newcommand{\vb}[1]{\vec{\mathbf{#1}}}
 
% Color definitions
\colorlet{xcol}{blue!98!black}
\colorlet{xcoldark}{blue!50!black}
\colorlet{vcol}{green!70!black}
\colorlet{myred}{red!80!black}
\colorlet{mypurple}{blue!60!red!80}
\colorlet{acol}{red!50!blue!80!black!80}
 
% TikZ styles
\tikzstyle{rvec}=[->,xcol,very thick]
\tikzstyle{force}=[->,myred,very thick]
\tikzstyle{mass}=[line width=0.6,red!30!black,fill=red!40!black!10,rounded corners=1,
                  top color=red!40!black!20,bottom color=red!40!black!10]
 
% TikZ pictures
\tikzset{
  pics/Tin/.style={
    code={
      \def\R{0.12}
      \draw[pic actions,line width=0.6,#1,fill=white] (0,0) circle (\R) 
        (-135:.75*\R) -- (45:.75*\R) (-45:.75*\R) -- (135:.75*\R);
  }},
  pics/Tout/.style={
    code={
      \def\R{0.12}
      \draw[pic actions,line width=0.6,#1,fill=white] (0,0) circle (\R);
      \fill[pic actions,#1] (0,0) circle (0.3*\R);
  }},
  pics/Tin/.default=mypurple,
  pics/Tout/.default=mypurple,
}
 
\newcommand\rightAngle[4]{
  \pgfmathanglebetweenpoints{\pgfpointanchor{#2}{center}}{\pgfpointanchor{#3}{center}}
  \coordinate (tmpRA) at ($(#2)+(\pgfmathresult+45:#4)$);
  \draw[white,line width=0.7] ($(#2)!(tmpRA)!(#1)$) -- (tmpRA) -- ($(#2)!(tmpRA)!(#3)$);
  \draw[xcoldark] ($(#2)!(tmpRA)!(#1)$) -- (tmpRA) -- ($(#2)!(tmpRA)!(#3)$);
}
 
% BICYCLE WHEEL
\def\r{0.16} % Axis radius
\def\Ri{1.18} % Wheel rims inside
\def\Rr{1.30} % Wheel rims outside
\def\Rt{1.45} % Wheel tire
 
% TORQUE perpendicular and angle
\begin{document}
\def\R{1.6} % Wheel rims inside
 
\begin{tikzpicture}
  \def\ang{43} % Angle position
  \def\angF{8} % Angle force
  \def\F{1.1}  % Force size
  \coordinate (O) at (0,0);
  \coordinate (R) at (\ang:\R);
  \coordinate (RT) at (90+\angF:{\R*sin(\ang-\angF)});
  \coordinate (R') at (2*\ang-180-\angF:\R);
  \coordinate (F) at ($(R)+(\angF:\F)$);
  \coordinate (FT) at ($(R)+(\ang-90:{\F*sin(\ang-\angF)})$);
  \clip (-1.2*\Rr,-1.17*\Rr) rectangle (2.04*\Rr,1.54*\Rr);
  
  \rightAngle{R}{RT}{O}{0.40}
  \rightAngle{R}{FT}{F}{0.35}
  
  \draw[line width=0.8,dashed,white] (R) -- (RT) (R) --++ (\ang:0.4*\R) coordinate (RE);
  \draw[line width=0.5,dashed,xcol] (R) -- (RT) --++ (180+\angF:0.3) (R) --++ (\ang:0.5*\R);
  \draw[force] (R) -- (F) node[right=-2] {$\vb{F}$};
  \draw[force,myred!80!black!60] (R) -- (FT) node[below right=-3] {$\vb{F}_{\perp}$};
  \pic[scale=1] at (R) {Tin};
  \draw[dashed,red!20!black] (F) -- (FT);
  \node[mypurple,above=2] at (R) {$\vb{\tau}$};
  \draw[rvec,xcol!90!black!50] (O) -- (RT) node[midway,above=3,left=-2] {\contour{white}{$\vb{r}_{\perp}$}};
  \draw[rvec] (O) -- (\ang:0.95*\R) node[midway,below=2,right=1] {\contour{white}{$\vb{r}$}};
  \draw pic["$\theta$",xcoldark,draw=xcoldark,angle radius=14,angle eccentricity=1.4] {angle=F--R--RE};
  \draw pic[thick,draw=white,angle radius=14,angle eccentricity=1.4] {angle=RT--R--O};
  \draw pic["$\theta$",xcoldark,draw=xcoldark,angle radius=14,angle eccentricity=1.4] {angle=RT--R--O};
\end{tikzpicture}
 
 
 
 
 
 
 
% CENTER OF MASS 1D
\begin{tikzpicture}
  \def\L{4.2} % length
  \def\w{1.3} % base width
  \def\h{1.0} % base height
  \def\F{0.8} % force magnitude
  \coordinate (O) at (0,0);
  \coordinate (M1) at (-0.55*\L,0.04*\h);
  \coordinate (M2) at ( 0.45*\L,0.04*\h);
  \coordinate (T1) at (-0.60*\L,0.1*\h);
  \coordinate (T2) at ( 0.50*\L,0.1*\h);
  \draw[thin,brown!40!black,fill=brown!80!black,rounded corners=0.5] (M1) --++ (\L,0) |-++ (-\L,-0.10*\h) -- cycle;
  \draw[mass] (M1) rectangle++ (0.7,0.5) node[midway] {$m_1$};
  \draw[mass] (M2) rectangle++ (-0.8,0.6) node[midway] {$m_2$};
  \draw[rvec] (O)++(-0.03,0.06) --++ (-0.55*\L+0.6,0) node[midway,above=-2] {$\vb{r}_1$};
  \draw[rvec] (O)++(0.03,0.06) --++ (0.45*\L-0.7,0) node[midway,above=-2] {$\vb{r}_2$};
  \draw[force] (M1)++(0.35,0.08*\h) --++ (0,-0.8*\F) node[above=2,left=0] {$m_1\vb{g}$};
  \draw[force] (M2)++(-0.4,0.08*\h) --++ (0,-\F) node[above=2,right=0] {$m_2\vb{g}$};
  \draw[force] (O) --++ (0,1.7*\F) node[above=-2] {$\vb{F}_\mathrm{N}$}; %$-(m_1+m_2)\vb{g}$
  \pic[scale=1] at (T1) {Tout};
  \node[mypurple,left=1] at (T1) {$\vb*{\tau}_1$};
  \pic[scale=1] at (T2) {Tin};
  \node[mypurple,right=2] at (T2) {$\vb*{\tau}_2$};
  \draw[thick,rounded corners=4,blue!20!black,
        top color=blue!40!black!50,bottom color=blue!40!black!15,shading angle=20]
    (-\w/2,-\h) -- (O) -- (\w/2,-\h) -- cycle;
  \draw[->] (M2)++(45:0.25*\L) arc(-10:80:0.12*\L) node[left=-1,scale=0.8] {$+\theta$};
\end{tikzpicture}
 
 
 
 
\end{document}

• The axis of rotation (or pivot point) $O$ is the point about which the object rotates

• $\vec{\tau }=\vec{\mathbf{r}}\times \vec{\mathbf{F}}$ is the torque or moment of force vector (in $\mathsf{N}\cdot \mathsf{m}$) (cross product)

• $\vec{\mathbf{r}}$ is the position vector (from the pivot point to the point of application of the force) (in $\mathsf{m}$)

• $\vec{\mathbf{F}}$ is the force vector (in $\mathsf{N}$)

• $\theta$ is the angle between $\vec{\mathbf{r}}$ and $\vec{\mathbf{F}}$

• $\tau ,r,F$ are the magnitudes of the vectors

• $\tau =rF\sin \theta =r{F}_{\perp }=r_{\perp }F$ is the magnitude of the torque

  • $F_{\perp }=F\sin \theta$ is the component of the force perpendicular to the position vector
  • The lever arm (or moment arm) $r_{\perp }=r\sin \theta =\frac{|\mathbf{r}\times \mathbf{F}|}{|\mathbf{F}|}$ is the perpendicular distance from the axis of rotation to the line along which the force acts

• $\alpha =\frac{\tau }{I}$ is the angular acceleration (in $rad/{\mathsf{s}}^2$)

• todo https://youtu.be/5Zrphnd_0VI?list=PLllVwaZQkS2rxqMXTH-cdE0LIX9Zi_oS1

Simple Harmonic Motion

• A displacement function $x(t)$ is said to describe simple harmonic motion iff it satisfies the differential equation $\frac{d^{2}x}{d{t}^2}=-{\omega }^{2}x$
• $x(t)=A\cos (\omega t+\varphi )$ is the displacement from the equilibrium position
  • $\varphi$ is the phase angle (in $\mathsf{rad}$)
  • $A$ is the amplitude (the maximum displacement from the equilibrium)
  • $\omega$ is the angular frequency (in ${\mathsf{s}}^{-1}$)
• $F=-kx$ is the restoring force (Hooke’s Law)
• $k$ is the spring constant (related to the stiffness of the spring) (in $\mathsf{N}\cdot {\mathsf{m}}^{-1}$)
• $T=\frac{2\pi }{\omega }$ is the period (in $\mathsf{s}$)
• $v=\omega \sqrt{A^2-x^2}$ is the velocity as a function of position
• $v_{\text{max}}=\omega A$ is the maximum velocity
• $m$ is the mass of the oscillating body (in $\mathsf{kg}$)
• $E=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2$ is the total mechanical energy (in $\mathsf{J}$)
  • $\frac{1}{2}m{v}^2$ is the kinetic energy (in $\mathsf{J}$) (it’s total energy in the moment of equilibrium, $x=0$)
  • $\frac{1}{2}k{x}^2$ is the elastic potential energy (in $\mathsf{J}$) (it’s total energy in the monent of turning point, $v=0$)

Mass-Spring System

• $T=2\pi \sqrt{\frac{m}{k}}$
• $v_{\text{max}}=A\sqrt{\frac{k}{m}}$
• $v_{\text{max}}=\pm A\sqrt{\frac{k}{m}}$ is the maximum velocity

Pendulum

• The oscillating body is the pendulum bob
• $T=2\pi \sqrt{\frac{L}{g}}$
• $L$ is the length of the pendulum
• $g$ is the acceleration due to gravity
• $\frac{mg}{L}$

Force

• Non-contact Forces
  • Gravitational Force
  • Electromagnetism
    • electricity
    • magnetism
  • strong nuclear force
  • weak nuclear force
• Contact Forces
  • Normal Force

Conservation

• conservative forces
  • gravitational force
  • elastic force
  • electrostatic force
  • magnetic force
• non-conservative forces
  • friction
  • air resistance
  • tension
  • normal force
  • applied force
  • spring force

Mechanical Equilibrium

todo

Gravity

Newton’s Law of Universal Gravitation

$F=G\frac{m_1{m}_2}{r^2}$

• ${\mathbf{F}}_{12}=-F{\hat{\mathbf{r}}}_{12}=F{\hat{\mathbf{r}}}_{21}=-{\mathbf{F}}_{21}\text{(vector form)}$
• $F=|{\mathbf{F}}_{12}|=|{\mathbf{F}}_{21}|$ is the magnitude of gravitational force (${\mathbf{F}}_{12}$ is the force exerted by $m_1$ on $m_2$, and vice versa)
• $m_1$ and $m_2$ are the (center of) mass of the two objects
• ${\mathbf{r}}_{12}=-{\mathbf{r}}_{21}$ is the position vector from $m_1$ to $m_2$,
  • $r=|{\mathbf{r}}_{12}|=|{\mathbf{r}}_{21}|$ the distance between the two objects
• ${\hat{\mathbf{r}}}_{12}=\frac{{\mathbf{r}}_{12}}{r}$ is the unit vector in the direction of ${\mathbf{r}}_{12}$ (and vice versa)
• $G=6.67430(15)\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ is the gravitational constant (dim. ${\mathsf{M}}^{-1}{\mathsf{L}}^3{\mathsf{T}}^{-2}$)

Mechanical Advantage

• The mechanical advantage of a machine is the ratio of the output force to the input force $\mathrm{MA}=\frac{F_{\text{out}}}{F_{\text{in}}}$

• A simple machine is a mechanical device that changes the magnitude of a force (i.e. the $\mathrm{MA}$ is not $1$), or the direction of a force

• Lever

  • $F_{\text{effort}}{d}_{\text{effort}}=F_{\text{load}}{d}_{\text{load}}$ (Law of the Lever)
  • $F_{\text{effort}}$ and $F_{\text{load}}$ are the effort and load forces
  • $d_{\text{effort}}$ and $d_{\text{load}}$ are the effort and load distances from the fulcrum
  • Class 1 Lever: fulcrum between the effort and load
  • Class 2 Lever: load between the fulcrum and the effort
  • Class 3 Lever: effort between the fulcrum and the load

• MA of bicycletodos

  • $\text{MA}=\frac{d_{\text{in}}}{d_{\text{out}}}=\frac{F_{\text{out}}}{F_{\text{in}}}=\frac{r_{\text{sprocket}}}{r_{\text{chainring}}}\cdot \frac{r_{\text{crank}}}{r_{\text{wheel}}}$

Center of Mass (CM)

todo super simple physics page 86