536 lines
16 KiB
TeX
Executable File
536 lines
16 KiB
TeX
Executable File
% use [nosolutions] flag to hide solutions.
|
|
% use [solutions] flag to show solutions.
|
|
\documentclass[
|
|
solutions
|
|
]{../../resources/ormc_handout}
|
|
\usepackage{../../resources/macros}
|
|
|
|
|
|
|
|
\uptitlel{Intermediate 2}
|
|
\uptitler{\smallurl{}}
|
|
\title{Newton's Laws of Motion}
|
|
\subtitle{
|
|
Prepared by Mark on \today \\
|
|
Based on a handout by Oleg Gleizer
|
|
}
|
|
|
|
\begin{document}
|
|
|
|
\maketitle
|
|
|
|
\section{Newton's First Law}
|
|
If the net force acting on an object is zero, the velocity of that object does not change. \\
|
|
Conversely, if the velocity of an object doesn't change, the net force acting on it is zero.
|
|
|
|
\medskip
|
|
|
|
In the context of vectors, the ``net force'' is the sum of all the force vectors acting on the object. ``Speed'' is the length (or \textit{magnitude}) of the velocity vector.
|
|
|
|
|
|
\problem{}
|
|
There are no forces acting on the object
|
|
$A$ below. The current velocity of the object,
|
|
in meters per second, is represented by the vector $\overrightarrow{v}$.
|
|
Draw the position of the object two seconds later.
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-11,-7) grid (1,1);
|
|
\draw [line width = 1.5pt, ->] (-9,0) -- (-5,-3);
|
|
\coordinate [label = above:{$\overrightarrow{v}$}] (v) at (-6.9,-1.5);
|
|
\filldraw (-9,0) circle (3pt);
|
|
\coordinate [label = left:{$A$}] (a) at (-9,0);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
The sides of the grid squares
|
|
on the picture above are one metre long.
|
|
What is the speed of the object?
|
|
\vfill
|
|
|
|
What distance would the object cover
|
|
in two seconds? \\
|
|
\vfill
|
|
|
|
\pagebreak
|
|
|
|
\problem{}
|
|
There are no forces acting on the object
|
|
$A$ below.
|
|
The current velocity of this object,
|
|
in miles per hour,
|
|
is represented by the vector $\overrightarrow{v}$.
|
|
Draw the position of the object half an hour later.
|
|
\vspace{20pt}
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-11,-7) grid (1,1);
|
|
\draw [line width = 1.5pt, ->] (-9,0) -- (-1,-6);
|
|
\coordinate [label = above:{$\overrightarrow{v}$}] (v) at (-4.8,-3);
|
|
\filldraw (-9,0) circle (3pt);
|
|
\coordinate [label = left:{$A$}] (a) at (-9,0);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
The sides of the grid squares
|
|
on the picture above are ten miles long.
|
|
What is the speed of this object?
|
|
|
|
\vfill
|
|
|
|
What distance would the object cover
|
|
in half an hour? \\
|
|
|
|
\vfill
|
|
|
|
What distance would the object cover
|
|
in three hours? \\
|
|
|
|
\vfill
|
|
|
|
\pagebreak
|
|
|
|
\definition{Acceleration}
|
|
\textit{Acceleration} is the rate at which velocity changes.
|
|
Let's represent acceleration by the vector $\overrightarrow{a}$.
|
|
|
|
If $\overrightarrow{a}$ does not change over time, then the speed
|
|
of an object at time $t$ is given by the following equation:
|
|
|
|
\begin{equation} \label{eq:ac}
|
|
\overrightarrow{v_t} = \overrightarrow{v_0} +
|
|
t \overrightarrow{a}.
|
|
\end{equation}
|
|
|
|
\problem{}
|
|
It takes a minivan seven seconds to accelerate from 0 to 60 miles per hour. Find its acceleration in meters per second squared. \\
|
|
\hint{1 mile $\approx$ 1600 meters}
|
|
|
|
\vfill
|
|
|
|
Note that in the previous problem, motion is one-dimensional (it happens on a straight line). In this case, both velocity and acceleration are one-dimensional vectors---in other words, (real) numbers! \\
|
|
|
|
In general, velocity and acceleration are \textit{not} numbers, but vectors. You'll see this in the next few problems.
|
|
|
|
\pagebreak
|
|
|
|
|
|
Now, let's try a few examples with vectors: \\
|
|
\medskip
|
|
Consider an object, currently at $P_0$, moving along the vector $\overrightarrow{v_0}$.
|
|
|
|
As before, let $\overrightarrow{a}$ represent the acceleration of the object. This could be caused by gravity, current, or any other constant force.
|
|
|
|
\medskip
|
|
|
|
One second later, the object will be at $P_1$, and has the velocity vector $\overrightarrow{v_1} = \overrightarrow{v_0} + \overrightarrow{a}$.
|
|
|
|
\medskip
|
|
|
|
Two seconds later, the object will be positioned at $P_2$ and will have the velocity vector $\overrightarrow{v_2} = \overrightarrow{v_0} + 2\overrightarrow{a}$.
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-19,-15) grid (1,1);
|
|
|
|
\draw [line width = 1.5pt, ->] (-18,-1) -- (-16,-4);
|
|
\coordinate [label = left:{$\overrightarrow{v_0}$}]
|
|
(v0) at (-17.1,-2.6);
|
|
\draw [line width = 1.5pt, ->] (-18,-1) -- (-15,-2);
|
|
\coordinate [label = above:{$\overrightarrow{a}$}]
|
|
(a) at (-16.4,-1.5);
|
|
\filldraw (-18,-1) circle (3pt);
|
|
\coordinate [label = above:{$P_0$}] (p0) at (-18,-1);
|
|
|
|
\filldraw (-14.5,-4.5) circle (3pt);
|
|
\coordinate [label = right:{$P_1$}] (p1) at (-14.45,-4.4);
|
|
\draw [line width = 1.5pt, ->] (-14.5,-4.5) -- (-9.5,-8.5);
|
|
\coordinate [label = right:{$\overrightarrow{v_1}$}]
|
|
(v1) at (-11.75,-6.5);
|
|
\draw [->] (-14.5,-4.5) -- (-12.5,-7.5);
|
|
\coordinate [label = left:{$\overrightarrow{v_0}$}]
|
|
(v01) at (-13.6,-6);
|
|
\draw (-12.5,-7.5) -- (-9.5,-8.5);
|
|
\coordinate [label = left:{$\overrightarrow{a}$}]
|
|
(a1) at (-11,-8.3);
|
|
|
|
\filldraw (-8,-9) circle (3pt);
|
|
\coordinate [label = above:{$P_2$}] (b) at (-8,-9);
|
|
\draw [->] (-8,-9) -- (-6,-12);
|
|
\coordinate [label = left:{$\overrightarrow{v_0}$}]
|
|
(v02) at (-7,-10.5);
|
|
\draw (-6,-12) -- (0,-14);
|
|
\coordinate [label = left:{$2\overrightarrow{a}$}]
|
|
(a1) at (-3.5,-13);
|
|
\draw [line width = 1.5pt, ->] (-8,-9) -- (0,-14);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
$t$ seconds later, the object
|
|
will have the velocity
|
|
\[
|
|
\overrightarrow{v_t} = \overrightarrow{v_0} + t \overrightarrow{a}
|
|
\]
|
|
|
|
Note that the word {\it acceleration} has two different meanings.
|
|
One is a vector, as in the example above.
|
|
The other meaning is the length
|
|
of the vector that represents the magnitude
|
|
of the velocity change. In this case,
|
|
we do not put an arrow above the letter
|
|
$a$ representing acceleration.
|
|
In other words, $a = |\overrightarrow{a}|$.
|
|
Similarly, speed is the length
|
|
of the velocity vector,
|
|
$v = |\overrightarrow{v}|$.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
|
|
\example{}<accel_example>
|
|
Moving with a constant acceleration,
|
|
an object moves from point $A$ to point $B$
|
|
in one second. The velocities of the motion,
|
|
in metres per second, are represented
|
|
by the vectors $\overrightarrow{v_0}$ and
|
|
$\overrightarrow{v_1}$.
|
|
Find the acceleration vector. \\
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-18,-8) grid (1,1);
|
|
\draw [line width = 1.5pt, ->] (-17,-1) -- (-13,-4);
|
|
\coordinate [label = above:{$\overrightarrow{v_0}$}]
|
|
(v0) at (-14.9,-2.5);
|
|
\filldraw (-17,-1) circle (3pt);
|
|
\coordinate [label = left:{$A$}] (a) at (-17,-1);
|
|
\filldraw (-10,-3) circle (3pt);
|
|
\coordinate [label = above right:{$B$}] (b) at (-10,-3.1);
|
|
\draw [line width = 1.5pt, ->] (-10,-3) -- (0,-4);
|
|
\coordinate [label = above:{$\overrightarrow{v_1}$}]
|
|
(v1) at (-5,-3.5);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
According to formula on the previous page,
|
|
$\overrightarrow{v_1} = \overrightarrow{v_0} + (1\text{ sec}) \times \overrightarrow{a}$. \\
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-18,-8) grid (1,1);
|
|
\draw [line width = 1.5pt, ->] (-17,-1) -- (-13,-4);
|
|
\coordinate [label = above:{$\overrightarrow{v_0}$}]
|
|
(v0) at (-14.9,-2.5);
|
|
\filldraw (-17,-1) circle (3pt);
|
|
\coordinate [label = left:{$A$}] (a) at (-17,-1);
|
|
\filldraw (-10,-3) circle (3pt);
|
|
\coordinate [label = above right:{$B$}] (b) at (-10,-3.1);
|
|
\draw [line width = 1.5pt, ->] (-10,-3) -- (0,-4);
|
|
\coordinate [label = above:{$\overrightarrow{v_1}$}]
|
|
(v1) at (-5,-3.5);
|
|
\draw [line width = 1.5pt, ->] (-10,-3) -- (-6,-6);
|
|
\coordinate [label = below:{$\overrightarrow{v_0}$}]
|
|
(v01) at (-8.2,-4.5);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
\problem{}
|
|
Complete \ref{accel_example}: \\
|
|
Draw the acceleration vector on the above diagram.
|
|
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\problem{}
|
|
At an initial moment in time,
|
|
the object at point $A$
|
|
is moving with the velocity vector
|
|
$\overrightarrow{v_0}$,
|
|
measured in metres per second.
|
|
The constant acceleration
|
|
acting on the object
|
|
is represented by the vector
|
|
$\overrightarrow{a}$,
|
|
measured in meters per second squared.
|
|
Two seconds later, the object is located
|
|
at point $B$. Draw its velocity vector
|
|
at that moment.
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-19,-15) grid (1,1);
|
|
\draw [line width = 1.5pt, ->] (-18,-1) -- (-16,-4);
|
|
\coordinate [label = left:{$\overrightarrow{v_0}$}]
|
|
(v0) at (-17.1,-2.6);
|
|
\draw [line width = 1.5pt, ->] (-18,-1) -- (-15,-2);
|
|
\coordinate [label = above:{$\overrightarrow{a}$}]
|
|
(a) at (-16.4,-1.5);
|
|
\filldraw (-18,-1) circle (3pt);
|
|
\coordinate [label = left:{$A$}] (a) at (-18,-1);
|
|
\filldraw (-8,-9) circle (3pt);
|
|
\coordinate [label = left:{$B$}] (b) at (-8,-9);
|
|
%\draw [line width = 1.5pt, ->] (-8,-9) -- (0,-14);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
\vspace{12pt}
|
|
|
|
|
|
The side length of the grid squares
|
|
on the picture above is one metre.
|
|
Find the speed of the object when it is
|
|
at point $B$.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\section{Newton's Second Law}
|
|
|
|
The second law is simple:
|
|
\[
|
|
\overrightarrow{F} = m \times \overrightarrow{a}
|
|
\]
|
|
|
|
Here, $\overrightarrow{F}$ is the net force acting on an object with mass $m$, and $\overrightarrow{a}$ is the acceleration the object experiences as a result of this action. Mass is a measure of an object's \textit{inertia}: the heavier an object is, the more effort it takes to change its velocity. \\
|
|
|
|
In civilized countries, mass is measured in grams and force is measured in \textit{newtons}. One newton is the force it takes to accelerate 1 kg of mass to 1 meter per second. In other words,
|
|
|
|
\[
|
|
1\ N = (1\ kg) (1\ \frac{m}{s})
|
|
\]
|
|
|
|
\problem{}
|
|
The {\it Millennium Falcon}, at point $A$ at the moment,
|
|
is trying to escape from the Death Star, which is trying
|
|
to arrest the ship using its attracting beam.
|
|
The thrust of the Falcon's engines,
|
|
200,000 kN in total, is represented
|
|
by the vector $\overrightarrow{T}$.
|
|
The force of the beam is represented
|
|
by the vector $\overrightarrow{B}$.
|
|
In addition, a nearby star
|
|
exerts a gravitational force of
|
|
$\overrightarrow{G}$ on the ship.
|
|
Draw the vector of the net force
|
|
acting on the vessel. \\
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-16,-9) grid (1,1);
|
|
\draw [line width = 1.5pt, ->] (-8,-5) -- (-12,-5);
|
|
\coordinate [label = below:{$\overrightarrow{T}$}] (v) at (-10,-5.1);
|
|
\filldraw (-8,-5) circle (3pt);
|
|
\coordinate [label = below:{$A$}] (a) at (-8,-5.1);
|
|
\draw [line width = 1.5pt, ->] (-8,-5) -- (-3,-6);
|
|
\coordinate [label = above:{$\overrightarrow{B}$}] (b) at (-5.4,-5.5);
|
|
\draw [line width = 1.5pt, ->] (-8,-5) -- (-9,-2);
|
|
\coordinate [label = left:{$\overrightarrow{G}$}] (g) at (-8.6,-3.5);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
The mass of the ship
|
|
is 400,000 kg. What acceleration,
|
|
in metres per second squared,
|
|
does the ship experience?
|
|
\vfill
|
|
|
|
|
|
|
|
The purpose of this daring mission
|
|
was to find out the force
|
|
that the the attracting beam exerts.
|
|
However, Han Solo is not particularly
|
|
good with vectors. Please help him complete the mission.
|
|
\vfill
|
|
\pagebreak
|
|
|
|
|
|
\section{Newton's Third Law}
|
|
|
|
Newton's third law also concerns forces. It states that \textit{every action has an equal and opposite reaction}.
|
|
|
|
In other words, this means that when one object exerts force on another, the second simultaneously exerts a force equal in magnitude and opposite in direction to the force exerted on it by the first.
|
|
|
|
\medskip
|
|
|
|
In fact, we saw this in our first handout!
|
|
|
|
\begin{tcolorbox}[
|
|
colback=white,
|
|
colframe=gray!75!black,
|
|
title={ Handout 1, Page 7 }
|
|
]
|
|
|
|
Here is an important example of an inverse vector. When you stand still, the floor pushes you up with the force opposite to the force of the gravitational pull, a.k.a. \textit{weight}.
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\begin{normalsize}
|
|
\draw (0,0) -- (6,0);
|
|
\draw (2.5,0) -- (3,1);
|
|
\draw (3.5,0) -- (3,1);
|
|
\draw (3,1) -- (3,2);
|
|
\draw (3,2.3) circle (.3);
|
|
\draw (2.6,1.7) -- (3.4,1.7);
|
|
\draw [->, line width = 2pt] (2.5,0) -- (2.5,-1);
|
|
\coordinate [label = below: {weight}]
|
|
(g) at (2.5,-1.1);
|
|
\draw [->, line width = 2pt] (3.5,0) -- (3.5,1);
|
|
\coordinate [label = right: {floor reaction}]
|
|
(w) at (3.7,1);
|
|
\end{normalsize}
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
The two opposing vectors add up to the zero vector, and therefore you don't move.
|
|
|
|
\end{tcolorbox}
|
|
|
|
\problem{}
|
|
In a second, the truck and car on the picture below will collide in a crash test.
|
|
The weight of the truck is 20 metric tons. The weight of the car is 2 metric tons.
|
|
Find the ratio of the accelerations,
|
|
$a_t$ (acceleration of the truck) and
|
|
$a_c$ (that of the car),
|
|
the vehicles will undergo. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\draw [gray!50!black] (-7,0) -- (7,0);
|
|
|
|
\filldraw [gray!80!blue] (-3.6,1.5) -- (-4.4,1.5) --
|
|
(-4.4,2) -- (-6.8,2) -- (-6.8,.3) -- (-3.6,.3) --
|
|
(-3.6,1.5);
|
|
\filldraw (-6,.3) circle (.3);
|
|
\filldraw [gray] (-6,.3) circle (.15);
|
|
\filldraw (-4,.3) circle (.3);
|
|
\filldraw [gray] (-4,.3) circle (.15);
|
|
|
|
\filldraw [red!80!white] (4,.2) -- (6.6,.2) -- (6.55,.6) --
|
|
(6.2,.6) -- (6.1,.8) -- (4.9,.8) -- (4.7,.6)--
|
|
(4.2,.6) -- (4,.45) -- (4,.2);
|
|
\filldraw (6,.2) circle (.2);
|
|
\filldraw [gray] (6,.2) circle (.1);
|
|
\filldraw (4.6,.2) circle (.2);
|
|
\filldraw [gray] (4.6,.2) circle (.1);
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\medskip
|
|
|
|
\begin{huge}
|
|
$a_t \div a_c =$
|
|
\end{huge}
|
|
|
|
\vfill
|
|
\newpage
|
|
|
|
\section{Bonus Problems}
|
|
|
|
\problem{}
|
|
Use vector algebra to prove
|
|
that diagonals of a parallelogram
|
|
in the Euclidean plane
|
|
split each other in halves. \\
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}
|
|
\begin{normalsize}
|
|
\draw (0,0) -- (4,-2);
|
|
\coordinate [label = left: {$A$}] (a) at (0,0);
|
|
\coordinate [label = right: {$B$}] (b) at (4,-2);
|
|
\coordinate [label = above: {$\overrightarrow{v}$}] (v1) at (2,-.9);
|
|
\draw (-1,-3) -- (3,-5);
|
|
\coordinate [label = left: {$C$}] (c) at (-1,-3);
|
|
\coordinate [label = right: {$D$}] (d) at (3,-5);
|
|
\coordinate [label = below: {$\overrightarrow{v}$}] (v2) at (1,-4.1);
|
|
\draw (-1,-3) -- (0,0);
|
|
\draw (3,-5) -- (4,-2);
|
|
\coordinate [label = left: {$\overrightarrow{w}$}] (w1) at (-.55,-1.5);
|
|
\coordinate [label = right: {$\overrightarrow{w}$}] (w2) at (3.55,-3.5);
|
|
\draw (0,0) -- (3,-5);
|
|
\draw (-1,-3) -- (4,-2);
|
|
\coordinate [label = above: {$M$}] (m) at (1.7,-2.5);
|
|
\end{normalsize}
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
\vfill
|
|
\newpage
|
|
|
|
\problem{}
|
|
Use vector algebra to prove that
|
|
all the three medians of a triangle
|
|
in the Euclidean plane intersect
|
|
at one point that splits each of the medians
|
|
in the ratio 2:1, counting from the vertices. \\
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture}
|
|
\draw (0,0) -- (6,-1);
|
|
\draw (0,0) -- (3,-5);
|
|
\draw (3,-5) -- (6,-1);
|
|
\draw [color = white] (3,-5) -- (9,-6);
|
|
\draw [color = white] (6,-1) -- (9,-6);
|
|
\coordinate [label = left: {$A$}] (a) at (0,0);
|
|
\coordinate [label = right: {$B$}] (b) at (6,-1);
|
|
\coordinate [label = below: {$C$}] (c) at (3,-5);
|
|
\coordinate [label = above: {$\overrightarrow{v}$}]
|
|
(v) at (3,-.45);
|
|
\coordinate [label = left: {$\overrightarrow{w}$}]
|
|
(w) at (1.45,-2.5);
|
|
\filldraw (4.5,-3) circle (1.5pt);
|
|
\coordinate [label = right: {$M_A$}] (ma) at (4.65,-2.85);
|
|
\draw (0,0) -- (4.5,-3);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
|
|
\vfill
|
|
\newpage
|
|
|
|
\problem{}
|
|
Find the area of an equilateral triangle
|
|
with the side length $a$.
|
|
\vfill
|
|
|
|
\problem{}
|
|
Does there exist an equilateral triangle
|
|
in the Euclidean plane
|
|
such that all of its vertices
|
|
have integral coordinates?
|
|
Why or why not? \\
|
|
|
|
\begin{center}
|
|
\begin{normalsize}
|
|
\begin{tikzpicture} [scale=.7]
|
|
\draw[step=1cm, gray, very thin] (-15,-9) grid (1,1);
|
|
\draw [line width = 1.5pt, ->] (-7,-9) -- (-7,1);
|
|
\coordinate [label = above:{$y$}] (y) at (-7,1);
|
|
\draw [line width = 1.5pt, ->] (-15,-4) -- (1,-4);
|
|
\coordinate [label = right:{$x$}] (x) at (1,-4);
|
|
\coordinate [label = below:{$1$}] (ox) at (-6,-4);
|
|
\coordinate [label = left:{$1$}] (oy) at (-7,-3);
|
|
\end{tikzpicture}
|
|
\end{normalsize}
|
|
\end{center}
|
|
\vfill
|
|
|
|
\end{document} |