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Consider, a particle is moving in a harmonic oscillator potential : $V=\frac{1}{2}m\omega^2x^2$. The force on the particle will be : $F=-m\omega^2x$.

What is the unit of $\omega$ here ? Is it $Hz$ or $rad\;s^{-1}$ ? From the force equation, it appears that if the unit of $\omega$ is taken as $s^{-1}$ or $Hz$, the unit of force comes as $Newton$, but if it is taken as $rad\;s^{-1}$ it doesn't seem to be coming in $Newton$.

Qmechanic
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bubucodex
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2 Answers2

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Even though the units for $\omega$ are radians/sec, radians have no physical dimension. In both cases the dimensions will be the same. That is, $\text Hz$ and $\text{rad}\ s^{-1}$ have the same physical dimensions since radians are dimensionless.

In the equation $F=-m\omega^2x$, the dimensional units on the right hand are $\text kg\ m\ s^{-2}$ or physical dimensions $[M][L][T]^{-2}$ which is the dimensions for force, consistent with the units of force, or $\text {Newton}$.

joseph h
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  • Suppose, in a certain problem I take the numerical value of $\omega$ as 40. Does this mean that, $\omega=40;Hz=40;rad;s^{-1}$ ? – bubucodex Feb 05 '23 at 06:48
  • It means that $\omega=40\ \text{rad}s^{-1}$ as units but that it has dimension equal to that of Hz, which is inverse time. Remember that radians have no dimension. It's incorrect to say Hz=rad/sec, but the dimensions of $s^{-1}$ are the same. Cheers. – joseph h Feb 05 '23 at 06:51
  • Thanks. But I have seen in some cases that the frequency in $Hz$ is multiplied by $2\pi$ to obtain angular frequency $\omega$ in $rad;s^{-1}$, for example, https://physics.stackexchange.com/questions/146112/simple-harmonic-motion-frequency – bubucodex Feb 05 '23 at 07:01
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    Yes, and that's completely fine. But still remember that rad/sec and Hz have the same dimensions. Don't confuse the terms "units" and "dimensions". That's an important distinction here. Cheers. – joseph h Feb 05 '23 at 07:06
  • Useful answer (+1). However, BIPM brochure (9-th ed., sect. 2.3.3) says, "The dimensions of the derived quantities are written as products of powers of the dimensions of the base quantities using the equations that relate the derived quantities to the base quantities. In general the dimension of any quantity Q is written in the form of a dimensional product, $dim(Q)=T^{\alpha}L^{\beta}M^{\gamma}I^{\delta}\Theta^{'epsilon}N^{\zeta}J^{\eta}$", without square brackets. – GiorgioP-DoomsdayClockIsAt-90 Feb 05 '23 at 08:55
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The definition of Hertz is 1/s. Radians or degree's are essentially unitless by convention, therefor your units for the angular frequency would still be in 1/s or Hz.

Look at this for the unit conventions of radians and degrees: https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless

Shane Gervais
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