How to Find the Tension in a Pendulum - Circular Motion

Here is a case where we have vertical circular motion, a pendulum or swing, where it is a bit more complicated than horizontal circular motion. For a pendulum, the tension and gravity components vary with the angle, so we have non-constant forces and acceleration. This could be tough to solve with Newton's laws. However, with energy we can find speeds more easily, and this will allow us to find the tension in the string for any angle of its motion. Check it out.

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...

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.