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.

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.