Angular Frequency and Angular Velocity

In summary, both angular frequency in Simple Harmonic Motion and angular velocity in Rotational Motion share the same symbol, "omega" (w). Although there is a mathematical relationship between the two, they are not physically equivalent. The angular frequency is constant while the angular velocity changes throughout the motion. Additionally, while angular frequency is a scalar, angular velocity is a vector. However, both play a role in describing the motion of uniform circular motion, with simple harmonic motion being the x component of this motion.
  • #1
mnandlall
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Both angular frequency in Simple Harmonic Motion and angular velocity in Rotational Motion are given the same symbol, "omega" (w). I was wondering if these two values are equivalent. For instance, if there was a cylinder attached to spring and rolling down a ramp, would the angular frequency of the harmonic motion equal the angular speed of the rotating cylinder. I am thinking that this is not true, because the angular frequency is constant while the angular velocity will be changing throughout the motion. Does any relationship exist between the two values?
 
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  • #2
There is a relationship between the two, but it is mathematical rather than physical. If you have something in uniform circular motion about the origin then you can write the location as:
r cos(wt+c) x + r sin(wt+c) y
where r is the radius of the circle, w is the angular velocity, c is the phase (determined by the position at t=0), and x and y are unit vectors along the x and y-axis respectively.

You can see then that simple harmonic motion is mathematically just the x component of uniform circular motion.

More generally, any time there is a an expression of the form cos(at+b) we call the "a" term angular frequency and the "b" term phase.
 
  • #3
There is one important difference, angular frequency is a scalar, while angular velocity is a vector. Otherwise, they are equivalent.

Claude.
 

1. What is the difference between angular frequency and angular velocity?

Angular frequency is the rate of change of angular displacement and is measured in radians per second. Angular velocity is the rate of change of angular displacement with respect to time and is measured in radians per second squared. In other words, angular frequency represents the frequency of rotation, while angular velocity represents the speed of rotation.

2. How are angular frequency and angular velocity related?

Angular frequency and angular velocity are mathematically related by the formula ω = 2πf, where ω is angular frequency, f is frequency, and 2π is a conversion factor between radians and cycles. This means that as angular frequency increases, so does angular velocity.

3. How are angular frequency and linear frequency related?

Angular frequency and linear frequency are related by the formula f = ω/2π, where f is linear frequency. This means that as angular frequency increases, linear frequency also increases. However, the units of measurement are different, with angular frequency measured in radians per second and linear frequency measured in cycles per second or hertz.

4. How does angular frequency affect the period of rotation?

The period of rotation is the time it takes for one complete rotation. It is inversely proportional to angular frequency, meaning that as angular frequency increases, the period decreases. This is because a higher angular frequency means that the object is rotating faster, completing one full rotation in a shorter amount of time.

5. How can angular frequency and angular velocity be measured?

Angular frequency and angular velocity can be measured using specialized sensors and instruments, such as gyroscopes and accelerometers. These devices can detect and measure the rate of rotation and provide accurate readings of angular frequency and angular velocity. They are commonly used in fields such as aerospace engineering, robotics, and navigation systems.

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