Angular frequency - how is it derived?

In summary, the formula for angular derived is as follows: the angular frequency is equal to the square root of the sum of the squares of the distances from the center of the circle to the positions of the particles.
  • #1
UrbanXrisis
1,196
1
How is the formula for angular derived?

[tex]\omega=\sqrt{\frac{g}{L}}[/tex]

my book has these equations:

[tex]F= -mg sin\theta = m \frac{d^2s}{dt^2}[/tex]

[tex]\frac{d^2 \theta}{dt^2}=-\frac{g}{L}sin\theta[/tex]

[tex]\frac{d^2 \theta}{dt^2}=-\frac{g}{L}\theta[/tex]

[tex]\omega=\sqrt{\frac{g}{L}}[/tex]

what exactly is m [tex]\frac{d^2s}{dt^2}[/tex] and [tex]\frac{d^2 \theta}{dt^2}[/tex]
 
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  • #2
[tex]\mbox{s} [/itex] is arc length...Because the movement is on a circle

[tex]s=L\theta[/tex]

,where [itex] \theta[/itex] is the angle at the center of the circle...

Daniel.
 
  • #3
s is the position of the particle, so the second derivative wtr time denotes the acceleration 'a' of the particle. (Hence F=ma in the first equation)

Solve the D.E. and check what that you get a periodic function. What is its angular frequency?
 
  • #4
Instead of applying Newton's second law for the translation movement [itex] \frac{d\vec{\mbox{p}}}{dt}=\sum_{k} \vec{\mbox{F}}_{k} [/itex],try to apply it for the rotation movement [itex] \frac{d\vec{\mbox{L}}}{dt}=\sum_{k} \vec{\mbox{M}}_{k} [/itex]


Daniel.
 
  • #5
okay, I understand up to here:
[tex]\frac{d^2 \theta}{dt^2}=-\frac{g}{L}\theta[/tex]


I think the next step is this...
[tex]\sqrt{\frac{d^2 \theta}{dt^2}}=\sqrt{-\frac{g}{L}\theta}[/tex]
[tex]\sqrt{\frac{d^2 \theta}{dt^2}}=\frac{d \theta}{dt}= \omega[/tex]
[tex]\omega = \sqrt{\frac{g}{L}[/tex]

but I don't think I'm correct
 
  • #6
Of course not.

It's a definition

[tex] \frac{g}{l}=:\omega_{0}^{2} [/tex]

That's all to it.

Daniel.
 
  • #7
but how do I get [tex]\frac{d^2 \theta}{dt^2}=-\frac{g}{L}\theta[/tex] to go to [tex]\omega=\sqrt{\frac{g}{L}}[/tex]
 
  • #8
How much have you studied about solving differential equations? Here in the USA, very few people study differential equations at the K-12 level.

It turns out that a general solution of the differential equation

[tex]\frac {d^2 \theta} {d t^2} = - \frac {g}{l} \theta[/tex]

is

[tex]\theta = A \sin \left(\sqrt {\frac {g}{l}} t + \theta_0\right)[/tex]

where [itex]A[/itex] and [itex]\theta_0[/itex] are arbitrary constants. You can verify this by working out the second derivative and plugging it back into the differential equation. A general form of a sinusoidal function is

[tex]x = A \sin (\omega t + \theta_0)[/itex]

where [itex]\omega[/itex] is the angular frequency. Matching up the preceding two equations gives you

[tex]\omega = \sqrt {\frac {g}{l}}[/tex]
 

1. What is angular frequency and how is it defined?

Angular frequency is a measurement of the rate of change of an object's angular position over time. It is defined as the number of radians an object rotates through per unit of time.

2. How is angular frequency related to linear frequency?

Angular frequency is related to linear frequency through the equation ω = 2πf, where ω is the angular frequency and f is the linear frequency. This means that the linear frequency is equal to the angular frequency divided by 2π.

3. How is angular frequency derived from circular motion?

Angular frequency is derived from circular motion by considering the relationship between the angular displacement and the linear displacement of an object moving in a circular path. The ratio of these two displacements is equal to the angular frequency.

4. What is the unit of measurement for angular frequency?

The unit of measurement for angular frequency is radians per second (rad/s). This unit is used to describe the change in radians an object rotates through in one second.

5. How is angular frequency used in physics equations?

Angular frequency is used in many physics equations, particularly those involving rotational motion. It is also used in equations related to simple harmonic motion, waves, and oscillations. Examples include the equations for centripetal force, angular velocity, and the period of a pendulum.

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