# Why do the units in the period of a mass-spring SHM not work out? [closed]

I am a high school physics teacher having students use the period of a mass-spring system with a known mass to determine the spring constant. We are practicing linearizing functions, so rather than plotting period vs mass, students are rearranging the usual period equation to allow $$k$$ to come out as a slope.

The units in $$T_s = 2 \pi \sqrt{m/k}$$ simply don't seem to work out to be seconds (there appears to be a missing unit of distance).

$$\sqrt\frac{kg}{kg m s^2} = \sqrt{s^2/m}$$

What am I missing? This seems terribly obvious, but the units of the RHS just don't work out to be seconds. This seems independent of the 'unitless radians' conversations that I've seen as the core of any similar questions on this topic.

## closed as off-topic by sammy gerbil, GiorgioP, Kyle Kanos, Dvij Mankad, ZeroTheHeroMar 13 at 4:41

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• Steinkamp, it may be more convenient if you have your students plot $T^2 vs. m$, in which case the slope of the resulting line will be $4 \pi^2/k$. – David White Mar 8 at 17:10
• In addition to what @DavidWhite pointed out, you should notice that your units for $k$ are mistaken. – Dvij Mankad Mar 12 at 0:02

## 1 Answer

$$\omega=\sqrt{\frac{k}{M}}$$

$$F=k\,x\quad \Rightarrow\quad [k]=\frac{\mathrm{N}}{\mathrm{m}}=\frac{\mathrm{kg\, m/s^2}}{\mathrm{m}}$$

$$[\omega]=\sqrt{\frac{\mathrm{kg\, m}}{\mathrm{m\,s^2\,kg}}}=\frac{1}{\mathrm{s}}$$

$$[T]=\frac{2\,\pi}{[\omega]}=\mathrm{s}\quad \surd$$

• Wow I feel silly. Maybe keeping track of my units in k would have helped! I simply did not have the 'per meter' unit in my original k. – Steinkamp Mar 8 at 16:38
• Eli, I've fixed a bit the notation: brackets should not be used around unit symbols, but around quantities, because the brackets are a kind of operator meaning "unit of <quantity>". – Massimo Ortolano Mar 8 at 18:24
• Actually brackets should indicate physical dimensions and shouldn't be used for units. – GiorgioP Mar 9 at 6:56