How to Find Terminal Velocity of Conducting Loop Falling into B-field

Here is a different type of induction problem. This has to do with the long aluminum tube we have in the lab, and even though it is non-magnetic, a falling magnet in the tube falls with a terminal velocity. Huh?! The reason for this is as the magnet moves, it is changing flux in the loop. This induces a voltage (Faraday) and therefore a current (Ohm). In a tube these are called eddy currents.

But those currents then feel a force since they are in a magnetic field. This is F = Il x B. The force is upward, trying to stop the motion and therefore stopping the change in flux (Lenz). The mathematics turn out to be identical to that of a sky diver with air friction! We will get an exponential solution, and a terminal velocity. Check it out!

How to do Air Friction on a Sky Diver

Here is a case where air friction acts on a falling object, such as a sky diver. This is one of the trickier math problems we will do in physics, as it involves calculus (anything with air friction will, since it is a non-constant force: f = -kv). We specifically want to solve for the velocity as a function of time for the sky diver. Check this out to get a feel for how Newton's 2nd law sets up the equation, and then we do almost all algebra with a step of calculus to solve for velocity. Note that terminal velocity is the speed you reach when air friction matches the strength of gravity, and the person falls with a constant speed at that point. Also note that we do a very simplified model of air friction. Other factors we do not worry about here include the shape of the object, air density that varies with altitude, wind, air temperature that varies, the material of the object, the gaseous composition, and so on (for us, all this information is contained in the constant, k).

How to Solve a Charging RC Circuit

RC circuits make up the next level of sophistication for us when it comes to circuits. We have done plain resistor circuits, plain capacitor circuits, and now RC circuits. Here is an example of how to find the charge as a function of time for a charging capacitor. It involves setting up a first-order differential equation for the circuit, and then solving that equation. In the end, we have exponential increase of charge on the capacitor, and exponential decay of current through the resistor and battery. Keep in mind the key is to write the equation for the circuit, and that depends on Kirchhoff's voltage rule, V = V(resistor) + V(capacitor).

How To Deal With Air Friction Mathematically - Hockey Puck

Air friction is a different creature compared to static or kinetic friction. Static and kinetic frictions are forces between two solid surfaces, whereas air friction is between a solid surface and a fluid. Something like water friction behaves similar to air friction, where these friction forces depend on how fast you try to move through the fluid (think about how it is actually tougher to try and run in water than to walk in water). Check out how to handle this fluid friction mathematically...it is certainly more involved than dealing with static or kinetic friction, which we treat as constant forces. Air friction is non-constant, and calculus must be used to find an exponential behavior with time.