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.

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.

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!

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!

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.

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.

How to solve simple harmonic motion problems using initial conditions

We have seen how the general solution for simple harmonic motion involves sines or cosines. These functions are periodic functions, and it makes sense that they are used to solve period motion problems. But what about problems where we are given initial conditions (i.e. at t = 0)? How do we find specific solutions to specific problems? Well, here is an example of how to do this. We will have a case where a mass oscillating on a spring has BOTH initial position and initial velocity. We introduce the general solution that involves a phase angle. Check it out.

How to find solutions for simple harmonic motion (SHM)

Simple harmonic motion is a subset of periodic motion, defined as motion that depends on displacement. Springs certainly follow this definition since Hooke's law gives us F = -kx. This picture of a mass oscillating at the end of a spring also makes use of energy, U = .5kx^2. But our goal is to understand the motion as a function of time, as well as a function of position. Bringing time into the picture is the issue, and it turns out we need a second order differential equation. This video outlines how to get a solution for position of the mass on a spring as functions of time. Turns out these require sine or cosine functions. Other videos will focus on other details and specific problems to show a general solution we can use.

How to handle rotations with Rolling without Slipping

This is a classic rotations problem, where a ball rolls down a hill without any slipping. Because both linear and rotational motions happen simultaneously, we need to solve both motions simultaneously with F = ma and t = I(alpha). This would be true for any sort of 'rolling without slipping' problem, as well. Check it out!

How to Apply Conservation of Angular Momentum to Rotational Motion and Collisions

Here is a sliding block and hanging rod, where the block collides and sticks to the rod. Angular momentum is needed, and here is a case where conservation of angular momentum is used to figure out the initial speed of the block before the collision occurs. Take a look to see the general, symbolic setup for such a problem, which will be similar to a ballistic pendulum from linear momentum days. I hope this helps.