Rolling Object going up a FrictionLESS Hill

Rotations can be challenging, but over time hopefully things start to click a bit more and make a little more sense. Here is one of those strange examples of a 'what if' situation: a ball rolling without slipping on a flat surface, but then going up a frictionLESS incline. What would this look like? Are there any differences from what we are used to seeing, which is something rolling without slipping even up the incline.

In everyday life, we are used to seeing something roll up a hill and both stop moving linearly as well as stop spinning at its maximum height. Would this happen if there is no friction on the hill?

The answer is NO...at the maximum height, the linear motion stops, that much we know. But here is the kicker: without friction between the ball and the surface it sits on, there is no torque. Without torque, there is no change in the rotational motion of the object. In other words, the ball will still be spinning with the same angular velocity as it had at the bottom of the hill! It would look weird.

What's more, the ball would then start to move back down the hill, since a component of gravity produces a net force. As it accelerates linearly down the hill, the spin is opposite what we would normally see since that original spin has not at all changed! Very odd! Check out this in a bit more detail.

Moment of Inertia when there are Holes in the Object - Yikes!

Moments of inertia are essential to rotating objects. This quantity replaces mass in our equations that we are used to for linear motions, and it represents the distribution of mass about the axis of rotation. For a point mass, for instance, I = mr^2, where r is the distance from the axis to the mass. I have videos for sticks and disks, using the integral version for I.

Then there is the parallel-axis theorem, which is for finding the moment of inertia for an object that rotates about an axis that is not its center of mass. Imagine spinning a disk around a point on its edge! This theorem says I = I_cm + md^2, where I_cm is the usual moment of inertia for its axis at the center of mass, and d is the distance from the center of mass to the new axis of rotation.

Here, we take it one step further. What if the object has a hole drilled out, so some of the mass is missing, and the object still rotates? Huh?!?! Check it out. These are nice examples of rotations, moments of inertia, and the parallel-axis theorem for an unusual situation.

History of the Universe in 18 Minutes!

Here's an interesting and well done TED talk by David Christian, as he takes us through the evolution of the universe in 18 minutes. Keep in mind that this has as a key idea that what started off as something very simple, an early universe of just energy and very little matter and no atoms (in the first few hundred thousand years of time), which then had huge complexity develop up to the present.

Quantum Biology

As nanotechnology advances, and biological studies reduce down to molecular and even atomic levels, that fuzzy world of quantum mechanics may be more relevant for biological systems than previously thought. Quantum biology and biophysics are seemingly growing fields of study, and make for a neat area of research for those who are interested in both disciplines.

For example, think about animals (and even microorganisms) that can sense and navigate via magnetic fields. How does that work? Or what are the quantum features of photosynthesis and many other life processes that take place in the atomic and molecular realms? Or how does smell work, when certain molecules are released into the air? What are the physical processes at small size scales when electrochemical signals more through neural networks in the body? What is consciousness? A review of a new book on this topic can be found here.

Advanced Topics for Olympiad

Here are some quick links to videos about topics that are more likely to make appearances on the Olympiad exams than on the AP.
- air friction, horizontal
- air friction, with a second applied force (think sky diver)
- Gauss's law for gravity
- Gauss's law for gravity (inside earth)
- Non-gauss cases for gravity
- Momentum, ballistic pendulum type cases
- Moment of inertia, sticks
- Moment of inertia, disks
- Moment of inertia, where axis of rotation is not center of mass (parallel axis theorem)
- Pendulum, finding tension as it swings
- Pendulum, small-angle approximation for SHM
- SHM for springs, general
- SHM for springs, initial conditions (i.e. phase angle)
- SHM, spring and stick system oscillating through small angle
- Springs in series and parallel

Go to video page of this blog for numerous videos on all aspects of rotations.

Capacitors in Series and Parallel

Similar to resistors, we can connect multiple capacitors in series and parallel. Also, we must commonly find the total capacitance, measured in Farads, for series and parallel. But what are these rules, and where do they come from?

It turns out, Kirchhoff's rules, just like we do for resistors in series and parallel.

In series, the voltage of the battery = sum of voltages of components.
In parallel, the total charge being stored by the circuit = sum of charges stored on branches.

Check out this video to see we end up with the two same rules for capacitors, only they are flip-flopped compared to resistors: in parallel we just add capacitors, and in series we have the reciprocal rule.

Resistors in Series and Parallel

Resistor circuits are the starting point for learning about circuits. Resistance is a natural feature of all everyday circuits, and is purposely used to regulate how much current is in a circuit. Generally, there are two main ways of combining multiple resistors in a circuit, series and parallel. One feature of circuit analysis is to try and find the total, or effective, resistance, in order to find total currents.

Two fundamental rules we will use over and over in circuit analysis are Kirchhoff's rules. For series, the rule is whatever voltage is put into a circuit, it is all used up by the time you make it around. FOr parallel, whatever your total current is coming into a junction, the sum of the currents in individual branches must equal the total current that came in (i.e. you don't lose current). The resistor rules for series and parallel follow directly from Kirchhoff's rules.

Check it out!

"Interstellar" has a Stellar look at Black Holes!

Thanks to Noah S. for giving me the link to a Wired article and video about the new movie Interstellar. Kip Thorne, a world famous expert on black holes and general relativity, consulted with the movie makers and their special effects and graphics teams to produce the best simulations of what Einstein's field equations from general relativity produce for black holes and worm holes. The graphics are stunning, and Thorne expects to get several research publications out of this because of the details they learned by seeing what a black hole should really look like! Please check it out, and be ready to be amazed!