# Natural Frequency Question SI Units

 P: 6 i have a basic question about the natural frequency of a system. for a mass (M), spring (k constant) undamped system the natural frequerncy is: w_n=sqrt(k/M) the units of w_n according to a lot of resources i found on the internet & textbooks are [rad/sec], my question is why? if i use the k constant units divided by the mass i get [Hz]: [k]/[M]=[N/m]/[kg]=[kg*m/s^2*m]/[kg]=[1/s^2] [w_n]=sqrt([k]/[M])=[Hz] i'll appreciate a clarification in this subject. thanks.
 P: 5,462 Hello yanaibarr, welcome to Physics Forums Radians are used because the solution to the governing differential equation is in terms of angualar measure y = Asin(x-ct) and radians (not degrees), being the natural numbers you obtain from such an expression.
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P: 12,064
 Quote by yanaibarr if i use the k constant units divided by the mass i get [Hz]: [k]/[M]=[N/m]/[kg]=[kg*m/s^2*m]/[kg]=[1/s^2] [w_n]=sqrt([k]/[M])=[Hz]
Actually you got 1/s, not Hz, for the units. You are assuming that 1/s always means Hz (= cycles/s), but that is not always the case.

Frequency can be measured in rad/s or cycles/s. Both radians and cycles are considered unitless, so both types of frequency can show up as 1/s if you use an equation to figure out the units.

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P: 12,064
 Quote by gsal Actually, Hz is exactly 1/s, and nothing else.
Yes, but 1/s could mean either Hz or rad/s. That was my point. The frequency calculated from the $\sqrt{k/m}$ formula has units of 1/s, but is in rad/s, not Hz.
 P: 6 thank u for the answers. But there is still one thing that bothers me, if i have a first order system the basic transfer function will be: K/(tau*s+1) where K is the Gain, and tau is the system's time constant. tau's units, according to what i've learned, are [sec]. but aren't the s plane units in [rad/sec] (s=jw+sigma)? That means that tau should be given in [sec/rad] to match the "1" in the transfer function. I know that rad can be considered "unitless" but when dealing with actual numbers it matters if the system's time constant is 1 [sec] or 1[sec/rad]= 2*pi [sec]. again, i'll appreciate a clarification. Thanks
 P: 4,663 If you go back to the basic physics equations: 1) F = -kx 2) m d2x/dt2 = -kx 3) d2x/dt2 = -(k/m) x 4) Substitute a solution x(t) = A sin(ωt) + B cos(ωt) 5) Find ω2 = k/m Note that ω has units radians per second. Bob S Added: The SI units for one complete rotation through 4 quadrants is 2pi (2 π) radians. 360 degrees is NOT an SI unit.
 P: 593 In step 4 one could substitute a solution in the form x(t) = A sin(2πft) + B cos(2πft), and then step 5 would have (2πf)^2 = k/m and so on... It just depends what solution form is substituted and whether one like carrying around factors of 2π in the math. (And it often leads to confusion, at least for me!)

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