Frequency and Period
Frequency and Period
You’ve encountered frequency and period when dealing with springs and simple harmonic motion, and you will encounter them again in the chapter on waves. These terms are also relevant to rotational motion, and SAT II Physics has been known to test the relation between angular velocity and angular frequency and period.
Angular Frequency
Angular frequency, f, is defined as the number of circular revolutions in a given time interval. It is commonly measured in units of Hertz (Hz), where 1 Hz = 1 s–1. For example, the second hand on a clock completes one revolution every 60 seconds and therefore has an angular frequency of 1 /60 Hz.
The relationship between frequency and angular velocity is:
For example, the second hand of a clock has an angular velocity of s. Plugging that value into the equation above, we get
which we already determined to be the frequency of the second hand of a clock.
Angular Period
Angular period, T, is defined as the time required to complete one revolution and is related to frequency by the equation:
Since we know that the frequency of the second hand is 1/60 Hz, we can quickly see that the period of the second hand is 60 s. It takes 60 seconds for the second hand to complete a revolution, so the period of the second hand is 60 seconds. Period and angular velocity are related by the equation
Example
The Earth makes a complete rotation around the sun once every 365.25 days. What is the Earth’s angular velocity?
The question tells us that the Earth has a period of T = 365.25 days. If we plug this value into the equation relating period and angular velocity, we find:
Note, however, that this equation only gives us the Earth’s angular velocity in terms of radians per day. In terms of radians per second, the correct answer is:
Relation of Angular Variables to Linear Variables
At any given moment, a rotating particle has an instantaneous linear velocity and an instantaneous linear acceleration. For instance, a particle P that is rotating counterclockwise will have an instantaneous velocity in the positive y direction at the moment it is at the positive x-axis. In general, a rotating particle has an instantaneous velocity that is tangent to the circle described by its rotation and an instantaneous acceleration that points toward the center of the circle.
On SAT II Physics, you may be called upon to determine a particle’s linear velocity or acceleration given its angular velocity or acceleration, or vice versa. Let’s take a look at how this is done.
Distance
We saw earlier that the angular position, , of a rotating particle is related to the length of the arc, l, between the particle’s present position and the positive x-axis by the equation = l/r, or l = r. Similarly, for any angular displacement, , we can say that the length, l, of the arc made by a particle undergoing that displacement is
Note that the length of the arc gives us a particle’s distance traveled rather than its displacement, since displacement is a vector quantity measuring only the straight-line distance between two points, and not the length of the route traveled between those two points.
Velocity and Acceleration
Given the relationship we have determined between arc distance traveled, l, and angular displacement, , we can now find expressions to relate linear and angular velocity and acceleration.
We can express the instantaneous linear velocity of a rotating particle as v = l/t, where l is the distance traveled along the arc. From this formula, we can derive a formula relating linear and angular velocity:
In turn, we can express linear acceleration as a = v/t, giving us this formula relating linear and angular acceleration:
Example
The radius of the Earth is approximately m. What is the instantaneous velocity of a point on the surface of the Earth at the equator?
We know that the period of the Earth’s rotation is 24 hours, or seconds. From the equation relating period, T, to angular velocity, , we can find the angular velocity of the Earth:
Now that we know the Earth’s angular velocity, we simply plug that value into the equation for linear velocity:
They may not notice it, but people living at the equator are moving faster than the speed of sound.
Equations of Rotational Kinematics
In Chapter 2 we defined the kinematic equations for bodies moving at constant acceleration. As we have seen, there are very clear rotational counterparts for linear displacement, velocity, and acceleration, so we are able to develop an analogous set of five equations for solving problems in rotational kinematics:
In these equations, is the object’s initial angular velocity at its initial position, .
Any questions on SAT II Physics that call upon your knowledge of the kinematic equations will almost certainly be of the translational variety. However, it’s worth noting just how deep the parallels between translational and rotational kinematics run.
Vector Notation of Rotational Variables
Angular velocity and angular acceleration are vector quantities; the equations above define their magnitudes but not their directions. Given that objects with angular velocity or acceleration are moving in a circle, how do we determine the direction of the vector? It may seem strange, but the direction of the vector for angular velocity or acceleration is actually perpendicular to the plane in which the object is rotating.
We determine the direction of the angular velocity vector using the right-hand rule. Take your right hand and curl your fingers along the path of the rotating particle or body. Your thumb then points in the direction of the angular velocity of the body. Note that the angular velocity is along the body’s axis of rotation.
The figure below illustrates a top spinning counterclockwise on a table. The right-hand rule shows that its angular velocity is in the upward direction. Note that if the top were rotating clockwise, then its angular velocity would be in the downward direction.
To find the direction of a rigid body’s angular acceleration, you must first find the direction of the body’s angular velocity. Then, if the magnitude of the angular velocity is increasing, the angular acceleration is in the same direction as the angular velocity vector. On the other hand, if the magnitude of the angular velocity is decreasing, then the angular acceleration points in the direction opposite the angular velocity vector.
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