How to Use the Parallel Axis Theorem to Find Moments of Inertia

The parallel axis theorem is a neat shortcut that allows us to find moments of inertia for objects when the axis of rotation is somewhere other than the center of mass of the object. If you know the inertia for objects when going through the center of mass, you can quickly find the new value of I for any axis that is parallel to the center of mass axis and displaced by some distance from the center of mass, d. The theorem says I_new = I_cm + Md^2. We do not have to use the integral to apply the theorem, which is why it is such a nice shortcut.

This video shows a couple quick examples of how to apply the theorem. Hope it helps!

How to do Rotational Motion for a rotating, falling bar - NON-constant acceleration

Check out an example of a NON-constant angular acceleration problem, where a bar starting in static equilibrium (up = down, cw = ccw) goes into non-equilibrium and accelerates. You can see how torque = I*(alpha) gives us the angular acceleration of the bar at any given angle it has rotated through, and also how to use rotational energy and energy conservation to determine the angular speed at any given angle.

One thing to keep in mind as far as linear acceleration and linear speed is that each point of the bar has different values for these quantities, as determined by a = R*alpha and v = R*omega. Check it out...

Rotational Motion - New Concepts

For the 3 Chem-Phys sections, we are into rotational motion. This is typically a challenging topic because it is brand new. Keep in mind what makes it new revolves (ha, ha) around 3 new concepts:
- Torque
- Moment of inertia
- Angular momentum

Torques are produced by forces, and specifically those forces that cause a change in rotational motion. In other words, torques produce angular accelerations (analogous to forces causing linear accelerations). Mathematically, individual forces cause a torque = F(r)[sin(theta)]. Torque is a cross product vector, t = r x F.

Moment of inertia is analogous to mass in linear motion. It is a 'resistance to a change in rotational motion.' The higher the inertia, the smaller the angular acceleration from the same torque. Together, torque, t, and moment of inertia, I, are related through the 2nd law for rotations:
t = I(alpha)

The moment of inertia has units of kg m^2, and numerically tells us about the distribution of mass about the axis of rotation of the object or system.

Angular momentum, L, is also a cross product vector, L = r x p. The direction is found with the curly RHR, as we do in class. Remember the conservation of linear momentum? Momentum is conserved for a system if no external forces act on the system. Here is the analogy: angular momentum is conserved for a system if there is no external torques acting on the system. Individual objects can have angular impulse in collisions, but for the system it is conserved. We will get into this in a big way, and I'll soon have some how to videos up for rotations.

Let's have some fun with it!

How to Apply Ampere's Law

Ampere's law is to magnetic fields as Gauss's law is to electric fields. We only use it in 3 cases, just like Gauss, and it even looks similar to Gauss's law, only it is a 1-D integral compared to a 2-D integral. A line integral, or to some a path integral, basically means we are looking for the magnetic field times the length of the path the B-field follows. This can work for us with long, straight wires with current, a solenoid, and a toroid. Check out how to apply Ampere's law in 2 of the 3 cases, those being a straight wire and toroid.

How to Find Magnetic Forces Between Current Carrying Wires

Many electronic devices have parallel wires with currents flowing. Now, each current produces magnetic fields that circulate around the current, and these magnetic fields interact with the other current to produce a force, due to F = Il x B. Check out this video to see how to combine a couple concepts - Ampere's law determines the strength of the magnetic field from one of the currents, and then this goes into the force equation to determine the strength of the force. The right hand rule will help determine the direction of the force. Net result is that currents in the same direction attract, and in opposite directions repel. Hope this helps!

Why is Momentum Conserved for Colliding Objects?

A brief explanation of why momentum is conserved when multiple objects collide. It is important to distinguish between impulse, or a change of an individual's momentum, and conservation of momentum, which is true for a system that has no external forces acting on the system. When combined with the 3rd law of motion, for every action there is an equal and opposite reaction, impulse and the 3rd law show that the system's impulse is 0...momentum does not change for the system if all we have are the internal forces between the objects.

Impulse: Golf Club Hitting Ball

This is happening at 70,000 frames per second (a bit quicker than the 30 fps of a standard camcorder)at 150 mph. Enjoy! Check out the complete deformation of the ball, which is normally rigid and quite hard.

Congratulations to Aaron - Intel National Semifinalist

Congratulations go to Aaron Damashek for being named an Intel Science Talent Search National Semifinalist! His work on the finding stable planetary orbits in binary star systems, and then examining climate changes through computer simulations, earned him this honor. Finalists are named on Wednesday, Jan. 27. Finalists then compete for a top prize of $100,000 in college scholarships in this top science contest for high school students.

For any student interested in doing independent science research, see Doc V and we can try to find a project that fits your interests and timetable. It is a truly unique experience while still in high school!

How to do RC Circuit with R and C in parallel

Here is a case where we have an RC circuit, but with a resistor and the capacitor in parallel with each other. This is tricky mathematically, but we can do it conceptually and only worry about numbers when t = 0 and after 'a long time.' Let's take a look.

Importance of Imagery for Memory & Learning

Thanks to the Drs. Eide for a post on imagery studies and how they play a role in memory and learning. If you reflect on instances when there is some physical activity or complex calculation or cognitive exercise you need to do, can you remember a time when you tried to 'see' yourself doing it ahead of time? It may seem to be an instinctive process or action, but I certainly have imagined doing a tough calculation prior to a math test, or have caught myself imagining myself playing a tough trumpet lick on a bus as we drove to a music contest. Professional musicians and athletes often refer to this mental practice since they are on the road so often, without the ability to physically practice like they are used to doing. Read a good article on this topic here.

Mental imagery is something that can help build memory for particular actions or cognitive activities, largely because neuroimaging experiments show as much as 90% of the neurons that are used in the actual, physical activity are firing in mental imagery exercises. To the brain, imagery is not so different from the real thing. Imagery can help us with the following:

- not only visualizing what the activity is, but also gaining spatial, auditory and kinesthetic information and practice and memory for that activity;
- helps with activities with high levels of organization, multi-steps, and decision making;
- positive imagery has a positive effect on real performance results: for example, golfers do 30% better on putting when positively imagining sinking putts, and 20% worse when imagining missing putts;

For readers, 60% of 5th graders report naturally using some imagery during 'think aloud' breaks in reading stories. It appears to be a natural reaction, even for children, to try and 'see' the scenes that words are trying to convey in order to develop memories of a story that we, ourselves, are not part of in reality. Humans are more visual creatures, as I like to tell my own students, and it is important to remind and also teach students how to visualize physical events and experiences. In fact, in problem solving in physics, I try and teach as an essential part of every single problem to draw a picture and mentally 'see' what is happening in the problem. We use a technique that requires making pictures and labeling all forces on the picture, and then use the picture to actually set up the math (for F = ma problems). So science and imagery are naturally connected, just as reading, writing and imagery are connected. Memory improves when visualization and imagery are used for stories or for how physical events play out in reality. The experimental finding that a good majority of the brain used for the physical activity is used in imagery, too, begins to explain why this process works.

Imagery is used extensively in elementary grades, and the combination of mental imagery with drawing pictures and other hands-on, physical activities makes for a powerful way of building memory and learning. We tend to actually decrease the use of imagery techniques as students progress into higher grades. Perhaps imagery is used most extensively in science classes by the time students get to high school, but it seems as if the use of imagery and hands on activities decreases significantly in literature and history/social studies classes, at least via anecdotal evidence and through conversations with students. Perhaps this is something educators need to consider more in practice.