Skills

Units

  • You must be able to convert units. For instance you must be able to convert from km/hr to m/s.
  • Dimensional analysis: Suppose a problem involves a mass $m$ [kg], a length $L$ [m], a time $t$ [s], and a force $F$ [N]. You are asked to calculate a velocity. The expressions $3L/t$ and $\pi\frac{Ft}{m}$ could possibly be correct because they have the units of [m/s]. The expressions $3Lt$ and $\pi\frac{F}{m}$ must be incorrect because they do not have the units of [m/s]. Whenever you derive an expression, you should check the units. If the units are wrong, you have made a mistake.

Vectors

You must be able to:

  • add two vectors, $\vec{A}+\vec{B}=(A_x+B_x)\hat{x}+ (A_y+B_y)\hat{y}+ (A_z+B_z)\hat{z}$;
  • determine the length of a vector, $|\vec{A}|=\sqrt{A_x^2+A_y^2+A_z^2}$;
  • determine the unit vector pointing in the same direction as a vector, $\hat{A}=\frac{\vec{A}}{\left|\vec{A}\right|}$;
  • decompose a vector into its $x$-, $y$-, and $z$-components;
  • calculate the inner product of two vectors, $\vec{A}\cdot\vec{B}=\left|\vec{A}\right|\left|\vec{B}\right|\cos(\theta)=A_xB_x+A_yB_y+A_zB_z$;
  • calculate the cross product of two vectors, $\vec{A}\times\vec{B}=(A_yB_z-A_zB_y)\hat{x}+ (A_zB_x-A_xB_z)\hat{y}+ (A_xB_y-A_yB_x)\hat{z}$.

App: Everything about vectors $\vec{A}$ and $\vec{B}$

Forces

You should know the formulas for the gravitational force, the Coulomb force, the spring force (Hooke's law), the Lorentz force, and the drag force. See the formula collection.

Integration and differentiation

    You must know:
  • how to differentiate and integrate the functions $\exp(x)$, $\sin(x)$, $\cos(x)$, and polynomials $x^n$, $1/x^n$;
  • the product rule of differentiation;
  • the quotient rule of differentiation;
  • the chain rule of differentiation.

You can check your work with the numerical integration and differentiation app.

Mechanics of point-like particles

Given the position $\vec{r}$ [m], the velocity $\vec{v}$ [m/s], the acceleration $\vec{a}$ [m/s²], or the force $\vec{F}$ [N] as a function of time for an particle, you must be able to calculate any other of these four quantities by either integrating or differentiating.

App: Numerical integration and differentiation of functions of $t$.

Working with data

Sometimes data will be given in the form of text columns. You should be able to:

  • calculate the mean and standard deviation of any column;
  • multiply all numbers in a column by a value (for instance if a column represents the acceleration of a particle for different times, multiply by the mass will yield the force);
  • plot the data in a column;
  • numerically integrate the data in a column;
  • numerically differentiate the data in a column;
  • convert data between a format that uses a '.' as the decimal mark and a format that uses a ',' as a decimal place.

Apps: Mean and Standard Deviation, Numerical integration and differentiation of functions of $t$, Numerical integration and differentiation of functions of $x$, Decimal point ↔ Comma.

Equations of a single variable

You must be able to solve any equation of a single variable. For two functions $y_1(x)$ and $y_2(x)$, you must be able to say which values of $x$ solve $y_1(x)=y_2(x)$. For some functions, this problem must be solved numerically. For instance you must be able to find the values of $x$ that solve $3x^3-2\sin x =0$.

Apps: Graphical solutions, Numerical integration and differentiation of functions of $x$.

Differential equations

Many systems that change as a function of time can be described by differential equations. This includes stock markets, animal populations, vibrations in machines, and the motion of particles. For this course we will focus on differential equations that describe the motion of a particle in one, two, or three dimensions. The force on a particle moving in one-dimension can be described in terms of its position $x$, its velocity $v_x$, and the time $t$. If the force on the particle is known, then Newton's law can be written as a differential equation,

\[ \begin{equation} \large m\frac{d^2x}{dt^2}=F_x(v_x,x,t). \end{equation} \]

Here $F$ is the force and $m$ is the mass. This can also be written as two first order differential equations,

$\large \frac{dx}{dt}=v_x$ and $\large \frac{dv_x}{dt}=F(x,v_x,t)/m.$

Given the initial conditions, the trajectory that the particle follows can be determined by using a differential equation solver. If the force is proportional to $x$ and proportional to $v_x$, then it is a linear differential equation and analytic solutions can be found using the app: analytic solutions to second order linear differential equations. For any force, the solution can be found using numerical solutions to second order differential equations.

A particle moving in three dimensions is described by six variables: $x$, $y$, $z$, $v_x$, $v_y$, and $v_z$. If the force on the object is known in terms of these variables and time, then the motion of the particle can be found using the app: numerical solutions of sixth order differential equations.

Apps: Analytic solutions to second order linear differential equations, Numerical solutions to second order differential equations, Numerical solutions of sixth order differential equations.

Parametric equations

You should be able to use parametric equations describe a curve. For instance, $x=\cos (s)$, $y=\sin (s)$, $s=[0,\pi ]$, describes a half circle and $x=2\cos (s)$, $y=3\sin (s)$, $s=[0,2\pi ]$, describes an ellipse. Here $s$ is the parameter.

Apps: Electric field produced by a uniformly-charged curved line, Biot-Savart law.

Work and Energy

The work performed is defined as,

\[ \begin{equation} \large W=\int\limits_{\vec{r}_1}^{\vec{r}_2} \vec{F}\cdot d\vec{r} = \int\limits_{x_1}^{x_2} F_xdx+\int\limits_{y_1}^{y_2} F_ydy+\int\limits_{z_1}^{z_2} F_zdz \end{equation} \]

This is a line integral. You must be able to perform this integral if the force is known as a function of position (such as this problem) or if the force given in terms of the position, the velocity and the time and the position is given as a function of the time (such as this problem). You must be able to determine the power by differentiating the work or determine the work by integrating the power. You must know what a conservative force is and how to find the potential energy of a conservative force. You must be able to calculate the kinetic energy of a particle, $E_{kin} = \frac{mv^2}{2}$. There is conservation of energy. The work performed equals the change in kinetic energy plus the change in potential energy plus the work performed against any nonconservative (frictional) forces.

Apps: Numerical integration and differentiation of functions of $t$, Numerical integration and differentiation of functions of $x$.

Derivatives of scalar fields and vector fields

A scalar field is a function that assigns a number to every position in space. Temperature, pressure, density, molecular concentration, electrostatic potential, and charge density are scalar fields. A vector field is a function that assigns a vector to every position in space. Electric fields and magnetic fields are vector fields. The gradient of a scalar field $\phi$ is a vector field. Minus the gradient of the pressure points in the direction that the wind blows (high pressure to low pressure). Minus the gradient of temperature points in the direction that the heat flows (high temperature to low temperature). Minus the gradient of the electrostatic potential points in the direction of the electric field (from high potential to low potential).

The divergence of a vector field $\vec{A}$ is a scalar field. The divergence tells us if the vectors in the vector field are moving apart or moving together. Imagine a small sphere about some point in space. If more vectors on the surface of the sphere are pointing outwards, the divergence is positive. If more vectors on the surface of the sphere are pointing inwards, the divergence is negative.

The curl of a vector field describes how the vectors are rotating about a certain point.

The gradient of a scalar field is a vector field,

\begin{equation} \large \nabla \phi = \frac{\partial \phi }{\partial x}\hat{x}+\frac{\partial \phi }{\partial y}\hat{y}+\frac{\partial \phi }{\partial z}\hat{z}. \end{equation}

$\phi(x,y,z)=$  
$\nabla \phi = $ () $\hat{x}$ + () $\hat{y}$ + () $\hat{z}$

Generally, the electrostatic potential depends on $x$, $y$, and $z$. To take the partial derivative $\frac{\partial \phi }{\partial x}$, differentiate with respect to $x$ while treating $y$ and $z$ as constants.

The divergence of a vector field is a scalar field,

\begin{equation} \large \nabla \cdot \vec{A} = \left( \frac{\partial }{\partial x}\hat{x} +\frac{\partial }{\partial y}\hat{y} +\frac{\partial }{\partial z}\hat{z}\right)\cdot \vec{A}=\frac{\partial A_x}{\partial x} +\frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}. \end{equation}

$\vec{A}(x,y,z)=$  $\hat{x}$ +  $\hat{y}$ +  $\hat{z}$  
$\nabla\cdot \vec{A} = $ () + () + ()

The curl of a vector field is a vector field,

\begin{equation} \large \nabla\times\vec{A}=\left(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\right)\hat{x}+ \left(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}\right)\hat{y}+ \left(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\right)\hat{z}. \end{equation}

$\vec{A}(x,y,z)=$  $\hat{x}$ +  $\hat{y}$ +  $\hat{z}$  
$\nabla\times \vec{A} = $ ()$\hat{x}$ + ()$\hat{y}$ + ()$\hat{z}$

Electrostatics

Electrostatics describes the relationship between the charge distribution, the electric field, and the electrostatic potential. Given a collection of point charges, you must be able to calculate the corresponding electric field and electrostatic potential. The formulas for this are in the formula collection. You can use an app to calculate the electric field and the electrostatic potential produced by a collection of point charges.

Superposition is a central concept in the Electricity section. Superposition says that if you know the electric fields of two charge distributions, the electric field of the sum of the two charge distributions is the vector sum of their two electric fields. The electric field caused by two point charges is the vector sum of the two electric fields produced by the two point charges individually. The electric field produced by a line of charge can be determined by puting many point charges in a line and the electric field produced by a plane of charge can be determined by puting many point charges in a plane.

If there are very many point charges, it becomes more practical to use the charge density $\rho$ [C/m³] to describe the distribution of the charges. Doing the calculation in the direction $\varphi(\vec{r}) \rightarrow \vec{E}(\vec{r}) \rightarrow \rho(\vec{r})$ is straightforward. Just use the definition of gradient and divergence, $\vec{E}(\vec{r})=-\nabla \varphi(\vec{r})$ and $\nabla\cdot\vec{E}(\vec{r})=\frac{\rho(\vec{r})}{\epsilon_r\epsilon_0}$. Calculating in the other direction, $\rho(\vec{r}) \rightarrow \vec{E}(\vec{r}) \rightarrow \varphi(\vec{r})$, is more difficult because the following integrals must be solved,

\[ \begin{equation} \large \vec{E}(\vec{r})= \int\frac{\rho(\vec{r})}{4\pi \epsilon_r \epsilon_0}\frac{\vec{r}}{ |\vec{r}|^3}d^3r \hspace{1cm}\text{[V/m]}, \end{equation} \]

and,

\[ \begin{equation} \large \varphi (\vec{r})=\int \frac{\rho(\vec{r})}{4\pi \epsilon_0 |\vec{r}|}d^3r\hspace{1cm}\text{[V]}. \end{equation} \]

If the electrostatic potential or the electric field only depend on one variable $x$ then the integrals can be done numerically using the Numerical integration app: $E(x) =\frac{1}{ \epsilon_r \epsilon_0} \int \rho(x)dx$ and $\varphi (x)=-\int E(x)dx$.

Solutions in two and three dimensions are known for some simple geometries such as: an infinitely long line of uniformly distributed charge, an infinitely long cylinder of uniformly distributed charge, an infinitely long cylindrical shell of uniformly distributed charge, an infinitely large plane of uniformly distributed charge, a sphere of uniformly distributed charge, and a spherical shell of uniformly distributed charge. The expressions you need to know are given in the formula collection.

Apps: Electric field caused by a collection of point charges, Electric field produced by a uniformly-charged curved line, Electric field caused by a collection of uniformly-charged parallel lines.

Magnetostatics

Magnetostatics describes the magnetic field generated by constant electrical currents. If the current distribution is known, the magnetic field can be calculated using the Biot-Savart law,

\begin{equation} \large \vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\int\frac{I d\vec{r}_{wire} \times (\vec{r}-\vec{r}_{wire})}{|\vec{r}-\vec{r}_{wire}|^3}\hspace{1cm}\text{[T]}. \end{equation}

In general, this integral must be done numerically but there are two special cases where the integral can be performed: an infinitely long straight wire and a solenoid. The formulas for these two cases are in the formula collection.

Once the magnetic field is known, it can be used to calculate the force on a current carrying wire. This force can be calculated by summing the Lorentz force law $\vec{F}=q\vec{v}\times\vec{B}$ for every particle with charge $q$ and velocity $\vec{v}$ that make up the current. Doing this results in the formulas for the force on a wire found in the formula collection.

If the magnetic field is known, the current density $\vec{J}$ can be determined by Ampère's law, $\nabla\times\vec{B}=\mu_0 \vec{J}$. This equation can be rewritten in another form that relates the line integral of the magnetic field going once around a loop $C$ to the current $I_{enc}$ passing through the loop,

\begin{equation} \large \oint\limits_{C}\vec{B}\cdot d\vec{l}=\mu_0 I_{enc}. \end{equation}

App: Biot-Savart law.

Oscillations

All of the oscillations we consider can be described by differential equations and you should be able to apply the differential equation solvers determine the oscillations.

It is often useful to use complex numbers to describe oscillations. A point moving in a circle in the complex plane has a real part that executes sinusoidal oscillations so when we see a sinusoidal oscillation we can always imagine that it corresponds to circular motion in the complex plane. Mathematically, this is expressed by Euler's formula, $e^{i\omega t} = \cos(\omega t) + i \sin(\omega t)$. You must know how to use complex numbers to calculate a resonance curve.

Partial differential equations

Partial differential equations involve the derivatives of space and time. They describe phenomena such as the weather which depends on position and on time. One important partial differential equation you should be familiar with is the wave equation,

\begin{equation} \large \frac{\partial ^2 u}{\partial t^2} = c^2\frac{\partial ^2 u}{\partial x^2}. \end{equation}

Harmonic solution to the wave equation have the form, $u=A\cos(kx-\omega t)$. You must be able to differentiate this to find $\frac{\partial ^2 u}{\partial t^2}$ and $\frac{\partial ^2 u}{\partial x^2}$.

The superposition of waves

Waves depend on position and on time. If there is more than one wave source, you should be able to add the waves together to determine the resulting wave. To calculate the time dependent interference pattern created several waves, just add the real parts together as is done in the app: the interference of surface waves. Snell's law, which describes how light rays are bend at an interface, was determined by adding together plane waves.

To calculate the time independent intensity pattern, it is usual to use a complex scalar field to describe each wave. A complex scalar field assigns a complex number to every point in space. The real part of the complex number describes the oscillation at each point in space. The complex scalar fields of the waves from all sources are added together and then the intensity is calculated: $I\propto\mathcal{A}^*\mathcal{A}$, where $\mathcal{A}$ is the sum of the complex fields. This is how the intensity pattern was calculated for a single slit, two narrow slits, many narrow slits, and two slits with a finite width.

Optics

You must be able to calculate the path that a light ray follows as it passes through an optical system with lenses and mirrors. Snell's law describes how light rays are bent at an interface. The thin lens equation describes how light is bent by a thin lens.

Apps: Refraction, Refraction at a spherical interface (1), Refraction at a spherical interface (2), Real and virtual images, Thin lens equation, Thick lens, Optical instruments, Ray tracing with the transfer matrix method.