Rotational Kinematics

Circular Motion
Unsurprisingly, circular motion is when something moves in a circle. To model this, we'll use polar coordinates instead of Cartesian coordinates — read up on these if you haven't already. Displacement in a circle, with polar coordinates, is easily represented as , the change in angle (USE RADIANS!!). What's the linear distance of the arc that the object traces out? Since it's an arc, we can find its length:

Radians
What are radians? At an AP Physics C level (or even Physics 1 or 2!), you should know that radians make a revolution. A couple of notes:

Angular Velocity
This one's pretty simple. When something is rotating, we measure its average angular velocity instead of its average linear velocity. Denoted by the Greek omega (), it's really what you'd expect it to be:
The most common unit for in Physics is , but it can be degrees per decade, revolutions per minute (RPM), and so on.

Angular Acceleration
Knowing angular velocity, what do you think angular acceleration is?
The units (unsurprisingly) are .

Uniformly Angularly Accelerated Motion (UM)*
Honestly, this is exactly what you'd expect it to be. Our 5 UAM variables are and . Our 5 UM variables are and , which is just (UAM) (UAM Greek-ified). Here are the equations:

Use radians!!

Tangential Velocity
Let's go back to arc length for a moment. Consider an old-school radar (you know, the green ones with a line, and dots?) — the line moves at a constant angular velocity.

sigh. Here's a GIF of what I'm talking about:
But notice how it seems to move a lot faster at the edges of the radar than near the center? While the angular velocity is the same throughout, the distance covered by any point on that line is different. So, we introduce tangential velocity: the angular velocity times the radius (the distance from the center of rotation). Tangential velocity is denoted and has units of meters per second. Make sure that the angular velocity is in radians per second.

A few things to keep in mind:

// skipped: https://www.flippingphysics.com/tangential-acceleration.html
// todo: info boxes?
DO NOT forget that (angular displacement), (angular velocity), and (angular acceleration) refer to the whole object, while (arc length), (tangential velocity), and (tangential acceleration) refer to specific parts of (or points on) an object.

Centripetal Acceleration
If we put an object on a spinning plate (with constant angular velocity), it will have a constant angular velocity, which means zero angular acceleration and zero tangential acceleration. But the tangential velocity isn't constant! Its components change as the spinning plate rotates (remember, tangential velocity stays tangent to the circle). So what's acting on it? A linear acceleration called centripetal acceleration!

Centripetal acceleration is always inwards. If this doesn't make sense (or you like analogies), imagine attaching a soccer ball to a string (magically, so the soccer ball is always held by the string (if you're self studying AP Physics C... you're in dire need of being held)). If you hold the string and start spinning in a circle, you'll find that the soccer ball is in uniform radial motion! Now, it should be easy to make sense of centripetal (sometimes called radial) acceleration: in scenarios without a string, centripetal acceleration is the same as the acceleration caused by the force of tension (in the soccer ball example).

Tangential Acceleration
If the object is speeding up, it must be experiencing an acceleration in the direction it's traveling.. which is along the tangent. The red lines in the following image represent the tangential acceleration (I don't know what the other arrows mean heh).

Total Rotational Acceleration
The total acceleration of something moving rotationally is the sum of the accelerations acting on it: centripetal (radial) and tangential.
Since the two are perpendicular, it follows that the magnitude of the total rotational acceleration is
The radial components of a radial motion acceleration vector at some angle make up the centripetal acceleration:

Ex. A ball is degrees above the lowest point on its circular trajectory. At that moment, its total acceleration is . Find the magnitude of the the radial acceleration.

Sol.

Period ()
The period of an object is the time it takes for one full revolution:

This is easy to derive: .

Let’s start with assuming that the wheel is a “rigid object with shape” — that is, the tangential velocity of any particle in the wheel is (i.e., every particle has the same rotational velocity). So, our expression for the kinetic energy of an object is

Moment of Inertia
The parenthetical, is what we call the moment of inertia, of an object.

So, the kinetic energy of a rotating object, is not too different from what we know about

In APC, we might need to calculate the moment of inertia of a rigid object with shape (and constant density), which we would do with the equation

AP Tip: “thin rod” typically means a rod with negligible radius compared to its length — so, it typically implies a one-dimensional rod.

To take this integral for random shapes, it’s useful to know that, if the density is constant, the mass of some small mass element to the total mass, is the same as its length, area, or volume (whichever is appropriate) to the total length, area, or volume. So, for example, we could find the inertia of a ring:

Note that for a ring, so Also, we can express in terms of angles: We know so we can find the integral:

Arguably, we could’ve solved this very easily considering that is a constant… but this approach allows us to solve more complicated shapes.

What causes rotational kinetic energy? We know that net force changes the translational motion of an object, but what changes the rotational motion of an object?

Torque
Torque is the ability to cause rotational acceleration. The symbol for torque is (tau), and the expression for torque is

where

Newton’s Second Law of Motion (Rotated)
We can use Newton’s Second Law of (translational) Motion and rotation-ify it! The rotational equivalent of net force, or torque, is equal to the rotational equivalent of inertial mass, or moment of inertia, times the rotational equivalent of linear acceleration, or rotational acceleration. So,

In rotational static equilibrium, the object is not moving around any axis of rotation.

Newton’s Universal Law of Gravitation
The force between any two objects is

Where

Near the surface of any plant, we can find the acceleration due to gravity, by setting our two equations for gravity equal to each other and cancelling out

Where is the radius of the planet.

However, we’re faced with a problem: astronauts in space are weightless, which doesn’t make sense given that you were just told that every object in the universe exerts a gravitational pull on every single other object in the entire universe. In fact, we know that (the universal one) will be zero when is zero, which is only zero when is infinite or the mass of either object is zero… but neither of those things is true for the International Space Station (and its contents). The I.S.S experiences an acceleration due to gravity of which means that it, and everything inside of it, is falling. As the astronauts and stuff inside of the international space station fall towards the floor of the I.S.S., the floor falls at the same rate. Finally, while falling, the I.S.S. happens to be moving very quickly in the tangential direction of its orbit — so, it’s exactly like Douglas Adams said: flying is just falling while forgetting to hit the ground. In summary, objects in orbit are falling and experience apparent weightlessness. As a continuation of apparent weightlessness, we can calculate the horizontal and vertical number of ’s* :

Note the difference: it’s possible to have apparent weightlessness (i.e., zero net ’s in all directions) while still experiencing the force of gravity — there just has to be no normal force.

something something gravitational field is the potential of space to have a force of gravity there. gravitational field of an object at a distance from the object is .
* — there shouldn’t be an apostrophe there, but it looks really ugly otherwise…

On the surface of a planet, the gravitational field is basically constant, which means that is constant, the force of gravity is and the gravitational potential energy between an object and the planet (with a given zero line) is However, at larger scales, the gravitational field isn’t constant. So, the force of gravity becomes what we just learned: Newton’s universal gravitation force or whatever. Similarly, the gravitational potential energy becomes