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 August 23rd, 2014, 04:35 AM #1 Senior Member   Joined: Jan 2013 From: Italy Posts: 154 Thanks: 7 Simple exercise on determine some dimensions. Hi, I have this exercise about determine the dimensions in a equation. But I have some problem during the way. This is the exercise: Given that $\displaystyle v = A \cdot \omega \cdot sin(\omega \cdot t )$, where $\displaystyle v$ has dimensions of speed and $\displaystyle t$ is a time, determine the dimensions of $\displaystyle \omega$ and $\displaystyle A$. I have tried this, using SI units, but without success...: $\displaystyle \frac{m}{s} = A \cdot \omega \cdot sin(\omega \cdot sec)$ Please, can you give me a help on how to proceed? Many thanks!
 August 23rd, 2014, 06:53 AM #2 Senior Member     Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics Dimension of $\displaystyle \omega \cdot t$ must be one, because the argument of the sine function must be dimensionless. Thus the unit of $\displaystyle \omega$ is $\displaystyle \frac{1}{\textrm{s}}$. The sine function is also dimensionless, so what is left, is $\displaystyle \frac{\textrm{m}}{\textrm{s}} = \left[ A \right] \cdot \frac{1}{\textrm{s}}$. Hence the unit of $\displaystyle A$ is m. Thanks from beesee
 August 23rd, 2014, 10:20 AM #3 Senior Member   Joined: Jan 2013 From: Italy Posts: 154 Thanks: 7 ok I understand, but, why in the solution of the exercise I have dimensions of $\displaystyle \omega$ as $\displaystyle \frac{rad}{s}$ ?
 August 23rd, 2014, 11:48 AM #4 Senior Member     Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics Radian is an angle unit. If you take a look how it is defined, you can see that it is actually dimensionless. The unit, radian, is put there mostly because of clarity. Or, they may have thought that the argument of the sine function is explicitly in radians (to avoid confusion, maybe(?)), and thus they've put the unit there. Thanks from beesee
August 23rd, 2014, 03:10 PM   #5
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Joined: Jan 2013
From: Italy

Posts: 154
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ok I understand!
Quote:
 Radian is an angle unit. If you take a look how it is defined, you can see that it is actually dimensionless.
yes as here: Radian - Dimensional analysis
Quote:
 they may have thought that the argument of the sine function is explicitly in radians
here in the exercise in can't read anything about the use of radians however.

I could recap all as the following:
Sine must be dimensionless, so giving to omega the rad unit and to t the sec unit (because in the exercise it is stated that t is time, and his unit, in SI, MUST BE sec.), to obtain sine dimentionless we must attribute to omega the unit rad/sec instead of rad.

So we could think like this:
$\frac{m}{s}= m \cdot \frac{rad}{s} \cdot sin(\frac{rad}{s} \cdot s)$
$\frac{m}{s}= m \cdot \frac{\frac{arc \ length }{radius \ length}}{s} \cdot sin(\frac{\frac{arc \ length }{radius \ length}}{s} \cdot s)$
$\frac{m}{s}= m \cdot \frac{1}{s} \cdot sin(\frac{1}{s} \cdot s)$
$\frac{m}{s}= m \cdot \frac{1}{s}$

So $A= m$ and $\omega= \frac{rad}{s}$

or not?

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