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\begin{document}
\title{Physics, Motion and Forces, 9-12}
\author[ ]{Noah Phipps}
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\date{}
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\tableofcontents
\newpage
\section{Equations of Motion}
\subsection{Scalars and Vectors}
\begin{itemize}
\item Scalar quantities just have magnitude
\begin{itemize}
\item Energy
\item Distance
\item Speed
\item Mass
\end{itemize}
\item Vectors have magnitude and direction
\begin{itemize}
\item Velocity
\item Displacement
\item Acceleration
\item Force
\end{itemize}
\end{itemize}
\subsection{Relative Velocity}
Velocity is relative and must be taken with respect to a frame of reference, although this is typically the surface of the Earth.\\\\ Vectors can be drawn to scale, and must form a closed shape to show the resultant vector. A vector can also be resolved into horizontal and vertical components which can then be considered independently of each other.
\subsection{Acceleration}
\begin{equation*}
a=\frac{\Delta v}{\Delta t}
\end{equation*}
\subsection{Time and Motion Graphs}
\subsubsection{Displacement-Time Graphs}
\begin{itemize}
\item Gradient gives velocity
\item Area irrelevant
\end{itemize}
\subsubsection{Velocity-Time Graphs}
\begin{itemize}
\item Gradient gives acceleration
\item Area gives total displacement
\end{itemize}
\subsection{Equations of Motion}
These can only be used with constant acceleration.
\begin{gather}
v=u+at \\
s=ut+\frac{1}{2}at^2 \\
s=vt-\frac{1}{2}at^2 \\
v^2=u^2+2as \\
s=\frac{u+v}{2}t
\end{gather}
\subsection{Free Fall and Terminal Velocity}
An object is in free fall if the only force acting on it is the gravitational force.
\subsubsection{The Effect of Air Resistance}
\begin{itemize}
\item The atmosphere exerts a drag force, $d$, on any falling object, due to air resistance
\item Air resistance increase with speed
\end{itemize}
\begin{align*}
\text{At low speeds:} & d \propto v \\
\text{At higher speeds:} & d \propto v^2
\end{align*}
\begin{itemize}
\item At terminal velocity, air resistance is equal to the gravitational force: $d=mg$.
\end{itemize}
\subsection{Projectile Motion in Two Dimensions}
\begin{itemize}
\item The horizontal and vertical components are independent of each other
\item Centre of mass is used as a point mass
\item Components must be resolved
\end{itemize}
\newpage
\section{Forces in Balance}
\subsection{Statics}
Statics is the branch of mechanics that deals with forces which are in equilibrium.
\subsubsection{Weight}
\begin{equation} w=mg \end{equation}
Weight is the force that acts on a mass due to a gravitational attraction, and acts through the centre of gravity. The centre of mass is different - a force applied through the centre of mass would cause no rotation.
\subsubsection{Contact Force}
\begin{itemize}
\item The force between two objects that are touching
\item Electromagnetic
\item Repulsive forces between charges
\end{itemize}
\subsubsection{Surface Friction}
\begin{itemize}
\item Friction is a force that opposes motion
\item $F= \mu R$
\item Not dependent on relative speed
\subsubsection{Tension}
An object in tension is subject to the forces stretching it, and exerts an equal but opposite force on the object stretching it.
\subsubsection{Upthrust}
\begin{itemize}
\item Also known as buoyancy
\item Due to pressure at different depths
\item Pressure on bottom of object greater than that from on top
\item \textit{The upthrust on an object in a fluid is equal to the weight of the fluid displaced by the object}
\end{itemize}
\end{itemize}
\subsubsection{Lift}
\begin{itemize}
\item Treated as a force acting upwards
\end{itemize}
\subsubsection{Drag}
\begin{itemize}
\item Opposes motion through any fluid
\item Often referred to as Air Resistance
\end{itemize}
\subsection{Free-Body Diagrams}
\begin{itemize}
\item Bases of arrows should be at origin of forces
\item Forces can be combined to act through the centre of mass
\item Produces resultant force
\item If in equilibrium, the arrows will form a closed polygon of forces
\end{itemize}
\subsection{Turning Effects of Forces}
\begin{equation*}
M=F \times D_{p}
\end{equation*}
\subsubsection{The Principle of Moments}
For an object to be in equilibrium, the clockwise moments at any given point must be equal to the anti-clockwise moments
\subsubsection{Couples and Torques}
\begin{itemize}
\item A torque may oppose or cause rotation - due to friction
\item A pair of couples will cause mo linear acceleration
\end{itemize}
\subsection{Equilibrium}
An object is in equilibrium when:
\begin{itemize}
\item Vector sum of all external forces is zero
\item The sum of moments at any point is zero
\end{itemize}
\newpage
\section{Forces and Motion}
\subsection{Newton's Laws}
\subsubsection{Newton's First Law}
An object will remain at rest, or continue to move with uniform velocity, unless acted on by an external, resultant force
\subsubsection{Newton's Second Law}
The rate of change of momentum of a body is directly proportional to the external resultant force acting on it. The change of momentum takes place in the direction of that force.
\begin{equation*}
F=ma
\end{equation*}
\subsubsection{Newton's Third Law}
When body A exerts a force on body B, then body B exerts an equal but opposite force on body A.
\subsection{Conservation of Linear Momentum}
\begin{equation*}
\rho = mv
\end{equation*}
\textit{The total linear momentum of a system does notchange, provided that no net external force is acting}
\subsection{Impulse}
\begin{itemize}
\item Impulse is change in momentum
\item $I = \Delta t \times F$
\item Ns\textsuperscript{-1} or kgms\textsuperscript{-1}
\item The impulse of a force is the area below the curve of a force-time graph
\end{itemize}
\subsection{Collisions}
Momentum is always conserved in collisions.
\subsubsection{Elastic Collisions}
\begin{itemize}
\item KE is conserved
\end{itemize}
\subsubsection{Inelastic Collisions}
\begin{itemize}
\item KE is not conserved
\end{itemize}
\subsubsection{Explosions}
\begin{itemize}
\item KE is not conserved
\end{itemize}
\subsection{Work, Energy \& Power}
\subsubsection{Work}
\begin{align*}
\text{Work done is equal to the change in energy}\\
\text{Work}=\text{Force}\times \text{Distance}\\
\end{align*}
Work is a scalar quantity. The total word done by a force is given by the area under a force-displacement graph.
\subsubsection{Energy}
The total amount of energy in any isolated system is constant.
\begin{gather*}
E_k =\frac{1}{2}mv^2\\
E_p =mg\Delta h
\end{gather*}
\subsubsection{Power}
Power is the rate at which energy is transferred.
\begin{equation*}
P=\frac{\Delta W}{\Delta t}
\end{equation*}
Also, for vehicles with an engine power of $P_0$,
\begin{equation*}
P_0 = Fv \end{equation*}
\subsection{Efficiency}
\begin{equation*}
\text{Efficiency}=\frac{\text{Useful}}{\text{Total}}
\end{equation*}
\newpage
\section{The Strength of Materials}
\subsection{Density, Stress, and Strain}
\subsubsection{Density}
\begin{equation*}
\text{Density}=\frac{\text{mass}}{\text{volume}}
\end{equation*}
\subsubsection{Stress}
Stress is the applied tensile force per unit cross sectional area:
\begin{equation*}
\sigma = \frac{F}{A}
\end{equation*}
The largest stress that can be applied to an object before it breaks is its Ultimate Tensile Stress (UTS).
\subsubsection{Strain}
Tensile strain is the extension of a material per unit length
\begin{equation*}
\epsilon = \frac{\Delta l }{l}
\end{equation*}
Strain has no units.
\subsubsection{Properties involving Stress and Strain}
\begin{itemize}
\item An \textit{ELASTIC} material will return to its original length after a force is removed.
\item A \textit{PLASTIC} material will not
\item Above the \textit{YIELD STRESS}, a material will begin to rapidly deform, as it has passed its \textit{ELASTIC LIMIT}
\item A \textit{DUCTILE} material can have a large plastic deformation before it will break
\item A \textit{TOUGH} material will absorb a lot of energy before it breaks
\item A \textit{BRITTLE} material will show hardly any plastic deformation before it breaks
\end{itemize}
\subsection{Springs}
The amount of strain caused by a given stress depends on the \textit{stiffness} of a material.
\begin{gather*}
F \propto \Delta l\\
F = k \Delta l
\end{gather*}
The extension of a spring is proportional to the force exerted upon it, until the elastic limit - Hooke's law.\\
As $k=\frac{F}{\Delta l}$, it is the gradient of a force-extension graph.
\subsubsection{Strain Energy}
The energy in a spring is equal to the work done is stretching it:
\begin{gather*}
E=\frac{1}{2}k(\Delta l)^2 \\
E= \frac{1}{2}F \Delta l
\end{gather*}
This is equal to the area under a force-extension graph.
\subsection{Materials in tension}
\begin{gather*}
\text{Young Modulus}=\frac{\text{Tensile Stress}}{\text{Tensile Strain}}\\
E=\frac{\sigma}{\epsilon}
\end{gather*}
This is equal to the gradient of a stress-strain graph in its linear section.
\subsection{Energy Stored in Stretched Materials}
\begin{gather*}
E=\frac{1}{2}k(\Delta l)^2 \\
E= \frac{1}{2}F \Delta l
\end{gather*}
This gives the energy stored pet unit volume, which is the work done in stretching the material. It is also a measure of a materials toughness.\\\\A hysteresis loop represents energy lost in the stretching and relaxing of a material, to internal energy.
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