How to Play Einstein for a Day: Mass and Energy

We know that strange things happen when objects move relative to other objects. Clocks run more slowly, lengths get shorter, and mass increases for the moving object, relative to an observer at rest. These are well confirmed over the last century from countless experiments.  We also know E = mc^2 comes from special relativity. But how did Einstein do it? Where did he get the idea that energy and matter aren't just related, but actually equivalent to each other?

Check this out!!  It is one way of thinking about it, and getting to a result that shows energy and matter are of the form E = mc^2.  It makes use of the binomial approximation when thinking in terms of everyday speeds, which are far slower than light.

How to Find Capacitance for Spherical and Cylindrical Capacitors

There are three shapes of capacitors in practice: parallel-plate, spherical and cylindrical. Conveniently, these are the three shapes we have for Gauss's law applications. We will use Gauss's law to find the E-field in the capacitors, then integrate the fields to get the potential difference across the capacitor, and then use our definition of capacitance, C = Q / V, to get the capacitance expressions. Let's take a look at two of the three, spherical and cylindrical capacitors.

Scientific American's Top 10 Science Stories of 2009

Scientific American has its top 10 science stories of 2009. Perhaps the lead story for 2010 will be the discovery of the Higgs boson at either Fermilab or CERN.

How to Solve a Discharging RC Circuit

Here is an example of how to find and solve the differential equation for a discharging capacitor, as part of an RC circuit. This happens when the battery is disconnected from the charged capacitor...there is nothing left to hold the charge on the capacitor. It uses the stored energy to discharge, and that energy gets burned off by the resistance in the circuit as the charge and current die off exponentially.

How to Solve a Charging RC Circuit

RC circuits make up the next level of sophistication for us when it comes to circuits. We have done plain resistor circuits, plain capacitor circuits, and now RC circuits. Here is an example of how to find the charge as a function of time for a charging capacitor. It involves setting up a first-order differential equation for the circuit, and then solving that equation. In the end, we have exponential increase of charge on the capacitor, and exponential decay of current through the resistor and battery. Keep in mind the key is to write the equation for the circuit, and that depends on Kirchhoff's voltage rule, V = V(resistor) + V(capacitor).

How to Find Charge on Capacitors in Circuits

There are a lot of parallels (ha, ha) between resistor and capacitor circuits. For resistor circuits, we tend to re-draw the circuit as a series circuit in order to find the voltage differences across parallel sets of resistors. This allows us to then find how the current splits in any parallel sets. We do basically the same thing for capacitor circuits, where we re-draw the circuit as a series, find the voltages, and then go and find how the stored charge splits on parallel branches. Here is an example of this process.

How to Start Science Research in High School

Many of our students have an interest in science research, but really do not have a clue as to how to go about the process. Don't feel bad if this is your situation...getting started is the most difficult part for anyone! This video gives a brief tour of my website with a couple useful resources for a student. You can come talk with me any time about any of the information on the site, and to sit down and brain-storm what your interests are and what is doable for a high school student. I strongly encourage you to look at some former student papers, to see the level that can be achieved and to get ideas for your own work. This type of work is absolutely long-term, as science is not done 'overnight.' Check it out, and if it seems like something you are interested in, let's talk!

How to Find the Basics of Binary Orbits

Most stars we see are actually two stars in orbit around each other. Technically, all orbiting systems are binary orbits, so understanding the basics of these is important in physics. Let's see how these work and get the basic concepts down, especially in terms of finding how fast things must be moving in order for these orbits (which we will assume are circular) form.

How to Find Gravitational Potential Energy when Gravity Changes (i.e. large heights)

We look at how to do gravitational energy the 'right way.' This is because gravity really is a non-constant force, and U = mgh is simply an approximation. While it works when the height is small compared to the radius of the earth, it does not work for space launches, orbits, and so on. Let us see what happens and how to do this, starting with Newton's law of gravity and our old gradient friend, F = -dU/dr. We will also see how this gets us escape velocity and Schwartzschild radius (for black holes).

How to Interpret Potential Wells

We have looked at potential wells, which are just graphs of potential energy as a function of position. The classic way of thinking about this is to imagine one object fixed at the origin, and have a second object moving along the x-axis. The graph tells us the potential energy between the two as you move that second object. We look at the potential energy gradient in order to determine the force function, F = -dU/dr, or in other words the force graph is found by plotting the negative slope of the potential energy graph. I outline this by using the 'Nike' problem from an old AP exam. Check it out!