10 Chapter 10: Rotational Motion
Section 10.1: Uniform Circular Motion
Textbook Section 4.4: Uniform Circular Motion
Textbook Section 10.1: Rotational Variables
Uniform Circular Motion refers to motion in a circular path at a constant speed. Examples of this include carousels, racetracks, and stellar orbits. Rotational motion refers to objects moving in a circle around a central point, and translational motion refers to motion in a straight line. You can combine these types of motion; for example, putting a spin on a football as you throw it.
The circumference of a circle is the distance around the edge, and is calculated as where is the radius of the circle. The angle through which a clock hand might move is given at , where is some arc length (distance swept out by the tip of the clock hand in some given time interval).
A radian is defined at 360 of a circle divided by , or 57.3. radians are equivalent to one revolution (one full trip around the circle and back to your starting place).
The angular velocity of an object moving in a circular path tells you how many you move through in a certain time period, and is calculated as:
This is the average angular velocity of an object spinning in the plane (the direction of the angular velocity is perpendicular to the plane, or in the direction of ). Instantaneous angular velocity is found as following:
The units of angular velocity are given in radians per second (rads/s). The angular displacement of an object moving in a circular path is given by .
Note that can be a number larger than 2! Typically when moving in a circle, we start over when we reach our starting point again, but not when calculating angular displacements.
Clockwise motion (cw) around a circle gives us a negative angular velocity, and counterclockwise motion (ccw) gives us a positive angular velocity. We assume for now that our rotating objects are rigid bodies (do not deform, bend, or stretch while moving in a circle) and that every point on the object has the same angular velocity.
Angular distance can be given at , and velocity is equivalent to . We can re-write the former equation as , and plug into the latter to receive . Now we can relate translational (also called linear) velocity and our angular velocity if we know the radius of the circle traced out by our object.
Section 10.2: Centripetal Acceleration and Force
As an object moves around a circular path, the magnitude of its velocity can remain constant, but the direction is constantly changing, so it is undergoing an angular acceleration. This acceleration due to an object moving in a circular path at a constant speed is called centripetal acceleration, and is given by:
Textbook Section 6.3: Centripetal Force
The force causing this acceleration is the centripetal force. Note that this is NOT a new force! It’s not magically arising out of the fact that something travels in a circle. The centripetal force was already on your diagram in some form — it may be the friction force,