The Angular Frequency calculator computes the angular frequency of simple harmonic motion.
INSTRUCTIONS: Choose units and enter the following:
Angular Frequency (z): The frequency is returned in hertz. However, this can be automatically converted to compatible units via the pull-down menu.
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:
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)`