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In SHM, $a=- x\omega^2$ and $max(a) = ±A\omega^2$, but what does $\omega$ mean, because i cannot see anything related to circular motion in SHM like to and from motion of a particle on $y$ axis about a mean position? Is it just a symbol given to $\sqrt{\frac{k}{m}}$, if yes, then why we are using omega to calculate frequency by $\omega = 2\pi*f$?

meaning of the symbols:

$x =$ displacement of particle,

$A =$ amplitude,

$a =$ acceleration of particle

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  • $\begingroup$ Please define all symbols used - I can pretty well guess what $k$ and $m$ are, but I actually don't know what $a$ and $A$ are. Not all textbooks/teachers use the same symbols. $\endgroup$
    – AXensen
    May 22, 2023 at 10:01
  • $\begingroup$ done, now you can see it again sir. $\endgroup$
    – user367859
    May 22, 2023 at 10:09
  • $\begingroup$ In essence, it's a trick to keep the equations simple. Using $\omega$, you don't have factors of $2\pi$ all over the place. $\endgroup$
    – John Doty
    May 22, 2023 at 12:52

3 Answers 3

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$\omega$ is the "angular frequency." If you plot the particle's velocity and position as a function of time, the particle moves around a circle (see the comments for a discussion of the controversy on whether this is a circle or an ellipse)

enter image description here

The frequency $f=2\pi\omega$ is 1/the time it takes to go around the entire circle. But the angular frequency is 1/the time it takes to go around 1 radian of the circle.

Physicists like to use angular frequency because a lot of mathematical expressions take a simpler form when we do that - it reduces the number of $2\pi$'s we need to explicitly write in formulas. For example the motion of the particle is $\cos(\omega t)$. And as you pointed out, the angular frequency has a nice expression in terms of $k$ and $m$; $\omega=\sqrt{k/m}$. Another example is that when a ball attached to a string is swung around your head, the centripetal acceleration is $a=\omega^2r$ (where $\omega$ is the angular frequency of that circular motion and $r$ is the length of the string; the acceleration $a$ is inward).

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  • $\begingroup$ In its present form, this explanation starts with a wrong statement. The phase trajectory (the curve in the v-x plane) for a generic initial condition is not a circle but an ellipse. Moreover, even though it is true that the projection of a uniform circular motion on one axis is an SHM, the connection between the two problems may not be evident without a good background. $\endgroup$ May 22, 2023 at 10:22
  • $\begingroup$ @GiorgioP-DoomsdayClockIsAt-90 Could be an ellipse, could be a circle. The two quantities have different units. I'm free to scale them so that the plot looks like a circle for any harmonic oscillator - and only then does "angular frequency" make sense. It's not clear to me that an "ellipse" with different units in the axes is clearly an ellipse and not a circle. I'll define the quantities in the circular motion example (although it's kind of just a quick aside at the end, not a critical part of the discussion). I'd agree the answer may be advanced for the asker, but it is the answer. $\endgroup$
    – AXensen
    May 22, 2023 at 10:30
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Simple harmonic motion and circular motion are mathematically closely related.

On a unit circle (circle with radius 1), every point has the coordinates $(\cos\theta,\sin\theta)$ for some angle $\theta$ ($t$ in the image below) between the x-axis and the radius vector that ends at that point.

enter image description here
(image source)

Now, if you take the angle $\theta$ and let it increase at a steady rate (constant angular velocity $\omega$) starting from zero, the point $(x, y)$ will spin around the circle in a counter-clockwise direction.

If you then separately plot each of the $(x, y)$ coordinates as a function of $\theta$, you'll get exactly the plots of the cosine and sine functions (in that order).

enter image description here
(image source)

Simple harmonic motion is the same motion as that of the projection (shadow) of a particle going uniformly around a circle (when you project / shine light from the side). On the other hand, you can also think about circular motion as being a superposition of two orthogonal simple harmonic motions that are out of phase by $\pi/2$ ($\cos\theta$ is just $\sin\theta$ shifted by $\pi/2$). That just means that if you have 2 vectors oscillating around the origin orthogonally to each other in that particular way, when you add them up, the resultant vector will go around in a circle (check again the first two images in this answer to see how that would work).

enter image description here
(image source)

Now, you don't have to think about $\omega$ in terms of going around a circle. Remember that the period of both the sine and the cosine functions is $2\pi$ - the pattern repeats after that. The angular frequency $\omega$ just tells you how fast you go through that cycle. In other words, it's a number of full cycles (periods) that fit onto the $0$-to-$2\pi$ span on the input axis (when I say "fit", I also allow for a non-whole number of cycles - e.g. 8.5 would mean 8 full cycles + a half way through the 9th one). The $2\pi$ span/interval is of interest just because sine and cosine have a period of $2\pi$; it's is a quirk of how these two functions are defined.

enter image description here

The angular frequency is different from the usual frequency (whether temporal or spatial, otherwise know as the "wavenumber") in that the usual frequency tells you how many full cycles fit on the $0$-to-$1$ span on the input axis. E.g. if the input variable is time, and the frequency is $f = 12 \text{ Hz}$, that means that 12 full cycles fit in the $0$-to-$1$ timespan (by the time $1 \text{ s}$ has passed, 12 oscillations have happened).

enter image description here

When the frequency is 1, the number of cycles that fits on any span is the same as the length of that span, so $f = 1$ corresponds to $\omega = 2\pi$ ($2\pi$ cycles fit on the $2\pi$ span), which is where $\omega = 2\pi f$ comes from ($2\pi$ is just the conversion factor between the two).

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It is just an analogy with the circular motion. If we define the frequency $f$ of a harmonic oscillator as the number of oscillations ("back and forth") in one second, then we can define the quantity $\omega=2\pi f$ to simplify the mathematical forms of all the formula involved. Since in circular motion the angular velocity is $2\pi f$, we use the same symbol and call it "angular frequency" (or, also, "pulsation").

If you want, you can regard the SHM as the projection of a uniform circular motion on one of the diameters; in this way, you can regard $\omega$ as the angular velocity of the particle undergoing the circular motion and generating the harmonic motion.

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