The Angular Frequency calculator computes the angular frequency of simple harmonic motion. 

INSTRUCTIONS: Choose units and enter the following:

• (κ)  This is the torsion constant for the spring
• (I) This is the the moment of inertia of the oscillating body

Angular Frequency (z): The frequency is returned in hertz.  However, this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

The Angular Frequency based on the torsion constant equation computes the angular frequency of an angular simple harmonic motion, a torsional system such as a coil spring that rotates about some axis, z.

The rotating body in such a system has a moment of inertia  about its axis and the restoring force, analogous to the spring in a linearly oscillating system, is a coil spring like those used in a watch mechanism.

`z = sqrt(κ/i)`

where:

• z - angular frequency
• κ - the torsion constant characterizing the restoring force of the coil spring
• I - the moment of inertia of the oscillating body.

Derivation

The coil spring exerts a restoring torque, `tau_z`,  proportional to the angular displacement `theta` from the equilibrium point:   

     [Eq1] `tau_z = kappa * theta`

The rotational analog of Newton's second law  for a rigid body is:

     [Eq2]  `sum(tau_z) = I*alpha_z = I * (d^2theta) / (dt^2)`

Substitute Eq1 into Eq2 gives us

   [Eq3]  `alpha = (-kappa * theta)/ I = (d^2theta) / (dt^2)`

Note this equation has the same form as:

   [Eq4]  `a_x   = (-k *x) / m = (d^2x) / (dt^2)` ^[1] , where x is replaced by `theta` and k/m is replaced with `kappa/I`   

Angular frequency is:

   [Eq5] `omega = sqrt(k/m)`  ^[2] , and the frequency, f is then:

   [Eq6]  `f = omega/(2*pi)`

Now, again replacing x by `theta` and k/m by `kappa/I`, we get:

   [Eq7] `omega = sqrt(kappa/I)`

   [Eq8]  `f = 1/(2*pi) * sqrt(kappa/I)`

Sources

• Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 434, eq 13.24
• University Physics 12th Edition, Chapter 13, Equation #13.24

1. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 421, eq 13.4 
2. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 423, eq 13.10