A ceiling fan is rotating in clockwise direction (as viewed from below) but it is slowing down. What are the directions of \(\omega\) and \(\alpha ?\)

Angular velocity, denoted by the Greek letter 'omega': \(\omega\), is a vector quantity representing how fast an object rotates around a fixed point or axis. It's similar to how we describe speed in linear motion, but instead of distance per unit time, angular velocity tells us how many radians (a measure of angle in a circle) an object covers per unit time.

In essence, angular velocity describes both the rotational speed and the direction of the rotating body. With clockwise or counterclockwise motion, the direction can be up or down when using the right-hand rule—which we'll discuss in a later section. For our ceiling fan slowing down while spinning clockwise, the direction of its angular velocity is vertically downward. This is important because it influences the direction of angular acceleration when changes to the rotational speed occur.

Angular acceleration, symbolized by the Greek letter 'alpha': \(\alpha\), is the rate at which an object's angular velocity changes over time. When an object speeds up or slows down its rotation, angular acceleration is present. Just like angular velocity, it has both magnitude and direction.

To calculate the angular acceleration, you'll need to consider changes in the angular velocity over a given time interval. It's crucial to remember that if an object is slowing down, its angular acceleration is directed opposite to its angular velocity. In our textbook example, the ceiling fan has an angular acceleration pointing upwards because it's decelerating while its angular velocity points downwards.

The right-hand rule is a handy mnemonic useful in vector fields, specifically in physics, to determine the direction of certain quantities in three-dimensional space, like angular velocity or magnetic field vectors.

To apply this rule to rotational motion, you simply point your thumb in the direction the object is moving towards, or away from, depending on the context. Then, with your fingers curled, the curling direction represents the rotation direction. When looking at the ceiling fan that's slowing down, if you point your thumb upward, representing the direction of decreasing speed (or angular acceleration), your fingers would curl in the direction of rotation, which is clockwise when viewed from below.

Rotational motion is motion around an axis. It is a complex form of motion because different parts of the rotating object move at different speeds; however, the angular velocity is the same for all parts. This is why we can use a single scalar value to describe the rotational speed of any point on the object.

In the context of our ceiling fan example, each blade of the fan rotates through the same angle in the same time interval, meaning all points on the blades have the same angular velocity. This uniform motion is central to simplifying complex rotational systems and understanding the kinematics and dynamics involved in objects like spinning wheels, planets orbiting a star, or, in our case, ceiling fans.