Rotational Kinematics
Rotational Kinematics
You are now going to fall in love with the word angular. You’ll find that for every term in kinematics that you’re familiar with, there’s an “angular” counterpart: angular displacement, angular velocity, angular acceleration, etc. And you’ll find that, “angular” aside, very little changes when dealing with rotational kinematics.
Angular Position, Displacement, Velocity, and Acceleration
SAT II Physics is unlikely to have any questions that simply ask you to calculate the angular position, displacement, velocity, or acceleration of a rotating body. However, these concepts form the basis of rotational mechanics, and the questions you will encounter on SAT II Physics will certainly be easier if you’re familiar with these fundamentals.
Angular Position
By convention, we measure angles in a circle in a counterclockwise direction from the positive x-axis. The angular position of a particle is the angle, , made between the line connecting that particle to the origin, O, and the positive x-axis, measured counterclockwise. Let’s take the example of a point P on a rotating wheel:
In this figure, point P has an angular position of . Note that every point on the line has the same angular position: the angular position of a point does not depend on how far that point is from the origin, O.
We can relate the angular position of P to the length of the arc of the circle between P and the x-axis by means of an easy equation:
In this equation, l is the length of the arc, and r is the radius of the circle.
Angular Displacement
Now imagine that the wheel is rotated so that every point on line moves from an initial angular position of to a final angular position of . The angular displacement, , of line is:
For example, if you rotate a wheel counterclockwise such that the angular position of line changes from = 45º = π/4 to = 135º = 3π/4, as illustrated below, then the angular displacement of line is 90º or π/2 radians.
For line to move in the way described above, every point along the line must rotate 90º counterclockwise. By definition, the particles that make up a rigid body must stay in the same relative position to one another. As a result, the angular displacement is the same for every point in a rotating rigid body.
Also note that the angular distance a point has rotated may or may not equal that point’s angular displacement. For example, if you rotate a record 45º clockwise and then 20º counterclockwise, the angular displacement of the record is 25º, although the particles have traveled a total angular distance of 65º. Hopefully, you’ve already had it hammered into your head that distance and displacement are not the same thing: well, the same distinction applies with angular distance and angular displacement.
Angular Velocity
Angular velocity, , is defined as the change in the angular displacement over time. Average angular velocity, , is defined by:
Angular velocity is typically given in units of rad/s. As with angular displacement, the angular velocity of every point on a rotating object is identical.
Angular Acceleration
Angular acceleration, , is defined as the rate of change of angular velocity over time. Average angular acceleration, , is defined by:
Angular acceleration is typically given in units of rad/s2.
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