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Thursday, April 29, 2010

Prepping for AP Exams

The Physics AP Exam is Monday, May 10, with mechanics at noon, E&M at 2 pm.

A class motto has been: 'A problem a day keeps the 1, 2, 3's away.' You should know by now the topics and types of problems that cause you headaches, so do those problems each day (takes about 20 or so minutes to try the problem and then check the solution) to fix your understanding or to develop the right question to ask me or another student at school, and you will be set for the real thing. There are LOTS of resources and practice problems, so take advantage of them, and just dedicating 20-30 minutes per night gets you away from having to cram all weekend. Try to have some fun with it, as you try to apply all that we have learned through the year to figure out some challenging, neat problems!

Good luck over the next week and a half, and see you at school!!

How to Handle Real Pulley in a Tension Problem

This is our old favorite - a couple masses tied together, string going over a pulley. In the good ol' days, the pulley was frictionless, and we simply could say that the tensions at the two ends of the (massless) string were the same, and set up F = ma for the system to get the acceleration. Worked out well.

But, in reality the pulley accelerates, too. How do we handle this? This means there has to be a net torque on the pulley in order to cause an angular acceleration. The only way for this to happen is if the tensions on the sides of the pulley, due to the hanging masses, are different. Here we see how to deal with this new situation, and apply F = ma on the two blocks, and torque = I*alpha on the pulley. We will assume there is no slipping between the string and the pulley, so we can relate linear motion of the blocks to the rotational motion of the pulley. Check it out.

Wednesday, April 28, 2010

List of How to Videos on the blog

Below are links to specific “How To” videos that are relevant to 3 and 4 Chem-Phys. These show how to access programs perhaps, or how to do certain problems. It has a voice-over and screencast from Doc V’s tablet computer, so it is similar to being in class as we model how to do certain problems or run certain programs. These can be useful if you were gone the day we covered the topic, or need more examples with explanations, or want to review things from class prior to quizzams.

Student Independent Science Research


Air Friction – the math

Binary Orbits

Derivatives! What are they and how to do them.

General Relativity and the Principle of Equivalence - Why does Gravity Bend Light?

Gravitational Potential Energy and Space Launches

Momentum Conservation – Why?

Momentum Conservation - How to do Inelastic Collisions

Moment of Inertia Using the Integral – Disks

Moment of Inertia Using the Integral – Sticks

Parallel Axis Theorem (finding moments of inertia)

Pendulum: Simple Harmonic Motion for small angles

Potential Wells

Quantum Numbers: Using Simple Harmonic Motion to help see where these come from

Rotations: Both Linear and Rotational Motion Simultaneously

Rotations: Collisions and Conservation of Angular Momentum

Rotations: NON-Constant Acceleration

Simple Harmonic Motion: General Derivation of sine, cosine solutions

Simple Harmonic Motion: Solving with specific initial conditions using phase angle

Special Relativity – mass and energy, where E = mc2 comes from

Springs and Energy

Tension problems with rotating pulleys:

Tension problems with systems of objects:


Ammeters and Voltmeters - How they work

Ampere’s law applications

Capacitance – how to find capacitance for the 3 types of capacitors

Capacitor Circuits – How to find stored charge

Electric Circuit analysis

Electromagnetic Induction – how Induced Currents turn on (includes circulating E-field)

Faraday’s law – Changing area with constant B-field

Faraday’s law – Changing B-field with constant area

Gauss’s law with conductors

Gauss’s law with NON-conductors: charge density

Gauss’s law with NON-uniform charge densities

Integration of E-fields to get Potential

LC Circuit – similar to simple harmonic motion

Magnetic Flux – Rectangular loop next to straight wire with current

Magnetic force between two current carrying wires

Mass Spectrometers and Velocity Selectors – magnetic forces, F = qv x B

Point Charge Systems – finding total electric fields and potentials

Projectile Motion of Electric Charges

RC Circuits – Charging Capacitor (series RC)

RC Circuits – Discharging Capacitor (series RC)

RC Circuits – Resistor and capacitor in parallel

RL Circuits – Current as functions of time

Interactive Physics
Interactive Physics is software where you can design and create your own simulations; mostly for mechanics, but there are some E&M topics you can also work on. This is on school computers.

See how to access the radioactivity iLab:

Tuesday, April 27, 2010

How to do the Math of LC Circuits

Interesting things happen when you combine loops of wire and metal plates in series. Inductors and capacitors together create AC currents at tunable frequencies, which in principle is the essence of our wireless society. In classical physics, the easiest way to create electromagnetic radiation is to accelerate (such as by shaking) electric charges, and that is exactly what you have in an AC current. Check this out to remind you of the math, and how LC circuits are the equivalent of simple harmonic motion of masses on springs from mechanics.

Monday, April 26, 2010

How to Find Magnetic Flux - Straight wire next to a loop

This is a classic type of problem, where a current in a straight wire is next to a rectangular loop, and we need to find the total magnetic flux through the loop. We need to break the loop into skinny strips of area, and find the flux through those skinny strips, then add them all up, i.e. integrate! Check it out to remind yourself.

How to do Electrical Projectiles

When a charge flies between the plates of a parallel-plate capacitor that is charged up, the particle will essentially be in a uniform electric field. This means there is a constant electric force, F = qE, and therefore we have a condition similar to a ball rolling off a table. The charge will move in a parabolic path as if it were a has a constant horizontal speed, and a constant vertical acceleration. Check this out to remind yourself how to do the mechanics of a projectile.

How to think about mass spectrometers

We know that electric charges moving in magnetic fields feel a force, called the Lorentz force, which is a cross product: F = qv x B.

This force is always perpendicular to the motion and B-field, and because of this particles get pushed into circular paths. This means the centripetal force is determined by the magnetic force. No work is done, as the energy is unchanged, but just the direction of the particle is changed.

In order to get a mass spectrometer to work, we also need to know the velocity of the particles. We can use electric fields to create a velocity selector. Keep in mind there is a good ActivPhysics simulation for mass spectrometers, 13.10, you may want to check out, too.

How to Analyze RL Circuits

An inductor is like a small solenoid in a circuit. It behaves like any loop of wire with currents, and follows the rules of em induction, such as Lenz's law. Conceptually, inductors resist changes in magnetic flux. This means they fight batteries when first connected, and therefore prevent current from flowing initially, and after a long time become nothing more than wires in a circuit, with steady current flowing.

The voltage across an inductor was derived from Faraday's law to be V = -L di/dt. Inductors with current flowing around the loops of wire also has a B-field in the tube, and this field stores energy, U = (1/2)Li^2. Check out this video to see how to do the mathematical derivations of current as a function of time when inductors are in series with a resistor.

Saturday, April 24, 2010

How to find electric fields and potentials for systems of point charges

One of the most essential and basic systems we study includes systems of stationary point charges. These bring out the essence of what charges do, which is produce a vector quantity, electric field, and a scalar quantity, electric potential.

We need to remember that finding electric fields includes us forgetting about the sign of a charge when calculating the fields. Being a vector, let the picture tell you whether we are dealing with positive or negative directions of components. But for potentials, which are scalars, we DO need to include the signs of charges numerically since positive charges produce positive voltage, and negative charges produce negative voltage, and we just add up the values.

How to do Gauss's law with NON-uniform charge density inside a non-conducting material

The vast majority of Gauss's law problems we do deal with uniform charge densities for non-conductors/insulators. These are the cases where there is charge inside the material, and therefore an electric field, and our job is to find the electric field inside. When the charge density, rho, is constant/uniform, the classic result is the field is linear with r. That is the result whether we have a sphere or a cylinder, and would be the same for gravity, electric fields, or magnetic fields (using Ampere's law).

But what about NON-uniform charge density, where rho depends on radius, r? What do we do with Gauss's law to find the electric fields inside these type of materials and objects? This video is an example of how to handle it. The gist is we need to set up an integral where we add the charges within skinny, hollow spheres of charge. Each little sphere has its own charge density value, and so the charge of each hollow shell is rho x dV. The trick is the dV = (4*pi*r^2)(dr), at least for a sphere. The dr is the small thickness of the hollow shell. The same idea holds for cylinders, where dV = (2*pi*L*r)(dr). I hope this helps!

How to find the Moment of Inertia for a Solid Disk

We have explicitly found the moments of inertia for point masses and also for sticks, and we have even defined the parallel-axis theorem. But what about more complicated objects, such as solid disks or cylinders that spin or roll? We have been given the inertia expression of (1/2)MR^2, and have used it quite a bit in problems, but where in the world does it come from? How do we use the integral definition of inertia to solve this? That is what this video is about, as a number of people have asked about where the inertia expressions for rolling objects come from. I hope this helps and makes sense.

How to Calculate Moments of Inertia with Integral - Sticks

When it comes to finding moments of inertia, the one thing we can find exactly is the inertia of a point mass, I = mr^2. Here, m is the mass, and r is the distance from the mass to the axis of rotation. But we run into some amount of difficulty when we have real objects that rotate and are made of countless point masses, i.e. atoms. How can we get the total moment of inertia?

We have an integral definition for I. What is is really telling us to do is break up the object into a bunch of small pieces of mass, dm. This is like saying break it up into a bunch of point masses, find each individual inertia, and then add (integrate) them all up to get the total. I will show how to do this with a stick in this video, which is the main case we would ever need to use the integral for class. Keep in mind that the other inertias we use for disks, balls, and so on, are found with this integral, too. I hope this helps. Keep in mind that I also have another video that shows how to use the parallel-axis theorem, which is a way around having to do the integration if you happen to know the moment of inertia of an object with the axis through the center of mass of the object.

Monday, April 12, 2010

How to Find Circulating Induced Electric fields when there is dB/dt

We have seen that electric currents create magnetic fields that circulate around the moving charges. This is the essence of Biot-Savart and Ampere's laws. But in certain cases of electromagnetic induction, magnetic fields can vary with time. The easiest example is simply moving a magnet relative to a solenoid or loop of wire. The trouble is, when one considers the physical reason for the induced currents that we find, there is no magnetic force on the charges of the wire, since the wire is at rest (i.e. qv x B = 0). So how does a current begin?

Think of a moving charge. At some fixed point in space, a moving charge would mean that the E-field at the point is changing...think dE/dt. What is the result of this changing E-field? A circulating magnetic field! Could it be that a changing magnetic field then induces a circulating electric field around the magnetic field? Absolutely! And we even know how to mathematically handle a circulating field from Ampere's law.

Turns out that whenever there is a changing magnetic field, dB/dt, an E-field is induced that circulates around the magnetic field! It is basically Ampere's law for electric fields, and therefore it is actually an electric force, F = qE, that pushes the current in the circuit. This video walks through the details of how induced currents physically form.

Sunday, April 11, 2010

How to do Faraday's law for Changing Areas of a Circuit

Here is an example of electromagnetic induction and Faraday's law for a constant B-field and a changing area. A conducting hoop/circuit moves into a B-field, and we determine the induced voltage (i.e. emf) and current. I'll make mention of two different magnetic forces that are relevant here: first, F = qv x B is the force that physically gets the current started since a conductor with free charges is moving through a B-field; second, once that current is turned on, F = Il x B turns on to try and slow the circuit down (magnetic brake). Lenz's law is also discussed.

One other aspect of this is the determination of the velocity of the circuit as a function of time. The magnetic braking force is analyzed with Newton's 2nd law, and we get a similar result as we did in mechanics with air friction, where the force is exponential in time. I hope this helps!

Saturday, April 10, 2010

How to Use Faraday's law for cases where B-field Changes

Faraday discovered that any change in magnetic flux causes induced voltage (i.e. electromotive force, or emf) in a closed conducting circuit. Because there is a voltage, this means an electric current is also induced. Faraday's law, or
induced voltage = -d(flux)/dt, allows us to figure out how much voltage is induced. Ohm's law, i = emf/resistance, allows us to figure out how much current turns on, and Lenz's law tells us the direction of the induced current flow.

Lenz's law is "Nature abhors change," or also we could say, "Get the (change in) flux outta here!" All the induced effects fight the change in flux.

Faraday's law helps explain how generators, electric motors, transformers, the ring launcher, credit card scanners, magnetic brakes, and other devices work, so it is tremendously important for everyday life applications. I hope this video helps!

Friday, April 2, 2010

Where do those Quantum Numbers come from? Using Simple Harmonic Motion to gain some insight...

In Chemistry, you learn about electron configurations, which involves learning the rules for 4 quantum numbers. Three of these numbers are integers. But students tend to be mystified by where these suddenly and almost magically appear. Why integers? Why the values that you are forced to memorize?

It all starts with the heart and soul of quantum mechanics, which is the Schrodinger equation. This is the F = ma of quantum land. For this case, our system is an electron oscillating back and forth between two walls. It is a nice, neat 1-D system. When we see what the Schrodinger equation looks like in this case, it will be identical to what we get for a mass on a spring oscillating back and forth. Since we know the solution of simple harmonic motion is a sine or cosine, then the solution of what turns out to be the wave function for our electron is also a sine or cosine. We will see that the electron will be restricted in its energy, that it will have restricted energies determined by an integer that we get as part of our solution! It is a quantum number!

These quantum numbers are simply part of solutions to complicated equations you get from the Schrodinger equation. An exact solution exists for a hydrogen atom, for example. You get three integers for an electron in an atomic orbital because it is a 3-D system rather than the 1-D system we deal with here. But the idea is the same. Integers are natural parts of these solutions, and they then are part of energy solutions for atoms and particles, which tells us that the energies of the electron have specific, allowed values, and not a continuum of energy that we see for a superball bouncing between two walls in a big, macro-world. The micro-world follows a separate set of rules in quantum mechanics. I hope this helps.

By the way, for an example of how the quantum numbers play out for the periodic table, check out some rules of the game.

Thursday, April 1, 2010

How to get Simple Harmonic Motion Solution for a Pendulum

A pendulum is technically NOT simple harmonic motion like a spring. SHM is defined when a force is proportional to the displacement of the object, just like a spring has F proportional to x. A pendulum is close, but the restoring force, being the tangential component of gravity, is proportional to sin(theta). Check out this video to see what we mean by the small angle approximation, or sin(theta) ~ theta (in radians) when theta is small, or about 10-degrees or smaller. This approximation works well, and you should check it on your calculator to prove it to yourself if you are not familiar with this. So for small angles, a pendulum is mathematically the same as an oscillating spring, and therefore is SHM and has a known solution of sine or cosine of wt plus a phase angle.