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The equation of a simple harmonic motion can be $x=A \cos(\omega t)$.

$\omega$ therefore has units of $radians/sec$.

I was solving some problems when I found a statement on my notes

$x=\left(1+\omega_{0} t\right)(e)^{-\omega_{0} t}$

with the variables having the usual meanings. I believe the statement is absurd because the left hand side has dimensions of distance and the right hand side has of radians (as $\omega t $ is in radians ) not to mention that the exponential has units of radians as well.

Could anyone please point out my mistake. I'd be glad even for a hint. Thank you.

Kashmiri
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  • I believe you copied the note incorrectly. – Bill N Dec 26 '20 at 14:52
  • I’m voting to close this question because it's primarily about units and algebra, not a broad physics concept outside of the mathematical mistake. It is also most likely related to a copying error. – Bill N Dec 26 '20 at 14:54
  • Thank you dear Bill, I downloaded the note from the University and found the same thing. I checked it over and again for an hour and only then posted here. For me it was more probable that I'm at error then the teacher that's why I posted because I thought maybe there is something that I'm failing to understand. :) – Kashmiri Dec 26 '20 at 15:06
  • Ok. The note is incomplete. – Bill N Dec 26 '20 at 15:54
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    Does this answer your question? Are units of angle really dimensionless? – John Rennie Dec 26 '20 at 16:11
  • In the last equation, $x$ is dimensionless. The units are consistent - You're wrong about your assertion that anything in that equation has units of length. Some context would probably make this clear. – Brick Dec 28 '20 at 21:20

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"Radians" aren't really units. E.g. $\frac{\pi}{2}$ (or $90^{\circ}$) is just a dimensionless number. That's why in physics/math we prefer "radians" to degrees.

$\mathrm{radians/second}$ is thus really $\mathrm{s^{-1}}$, in $\text{SI}$ units.

Your expression:

$$x=\left(1+\omega_{0} t\right)e^{-\omega_{0} t}$$

cannot refer to a distance, unless its RHS is multiplied by some factor $A$ with units $\mathrm{m}$.

Gert
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  • Thank you dear Gert, but Wikipedia says"The radian, denoted by the symbol ${\displaystyle {\text{rad}}}$ is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics". – Kashmiri Dec 26 '20 at 15:11
  • radian = distance / distance. Its dimensionless – R. Emery Dec 26 '20 at 15:55
  • Well, Wiki is wrong on this. That's not the first time either! It's like claiming the number $5$ has SI units: not true. – Gert Dec 26 '20 at 20:28
  • NIST also calls radian an SI unit, but it's a dimensionless unit. So you're both right. https://physics.nist.gov/cuu/Units/SIdiagram.html – Brick Dec 29 '20 at 18:04