Wednesday, 9 November 2016

Simple Harmonic Motion - What are the units for omega0?


I'm trying to understand the units in:


mx+kx=0



And the general solution is x(t)=Acos(ω0t)+Bsin(ω0t).


Let ω0=km - the unit for the spring constant k is kgms2 or Nm1, where m is in kg, so that the units of ω0 seem to be "per second" (i.e) 1/s.


But, later we put ω0 in to the cos and sin functions which will return dimensionless ratios. So, The constants A,B must be in m, since x is in m.


What I don't understand is why my book says ω0 has the unit rad/s, I get that the input for cosine is rad or some other angle measure, but where did the radians come from?


My analysis of units only proved 1/s as the actual units..!




I have just been informed that radians are dimensionless. So, that answers part of this question, yet I still don't know why we can't say that dimensionless one in degrees or rotations..? How do I know what kind of cosine and sine table to use with this dimensionless number?



Answer



Ah, good question. The radian is actually a "fake unit." What I mean by that is that the radian is defined as the ratio of distance around a circle (arclength) to the radius of a circle - in other words, it's a ratio of one distance to another distance. For an angle of one radian specifically, the arclength s is equal to the radius r, so you get


1 rad=sr=rr=1



The units of distance (meters or whatever) cancel out, and it turns out that "radian" is just a fancy name for 1!


Incidentally, this also implies that "degree" is just a fancy name for the number π180, and "rotation" is just a fancy name for the number 2π.


This actually addresses the edit to your question. Suppose that you had some object oscillating at ω=π/4rads=0.785rads, and you wanted to evaluate its position after 10 seconds. To get the cosine term, you would plug the numbers in, getting


cos(0.785rads×10s)=cos(7.85 rad)=cos(7.85)


and then you would go to a trig table in radians (or your calculator in radian mode) and look up 7.85.


However, suppose that you were measuring ω0 in degrees per second instead of radians per second. You would instead have


cos(45/s×10s)=cos(450)


If you go look this up in a trig table given in degrees, you will get the same answer as cos(7.85). Why? Well, remember that the unit "degree" is just code for π/180, so this is actually equal to


cos(450×π180)


And 450×π180=7.85, which is just 450 converted to radians. So now you have the same value in the cosine, cos(7.85). Trig tables listed in degrees already have this extra factor of π180 built into them as a convenience for you; basically, if you look up any number θ in a table that uses degrees, what you get is actually the cosine (or sine, or whatever) of θ×π180.



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