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First, I am not a native English-speaking student so I am not good at physics definitions in English. I participated in the MIT e-learning course on classical physics. The 1st lesson is about 3 fundamental physical quantities (time, length and mass).

It mentions dimensional units/quantity and dimensional state. I couldn't find the meaning of those in a dictionary. Can someone give me an specific explanation?

Also, in the lesson, why are the quantities always associated with some power of unknown value like $\alpha,\beta$, etc when we predict the equation? And how do we know what quantities should be added into the equation of a model?

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  • $\begingroup$ Your second question makes no sense to me...perhaps give an example or elaborate? $\endgroup$
    – evil999man
    Mar 6, 2014 at 14:21
  • $\begingroup$ ocw.mit.edu/courses/physics/… In this clip, in the Galieo section he mention about how to make an equation that express the time when a thing drop at a certain length h. He associate Mass, time, length to create the equation but I don't know how how he know what quantity should be add into an equation. $\endgroup$
    – aukxn
    Mar 6, 2014 at 14:34
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    $\begingroup$ I've never heard the term "dimensional state" so I'll pass on that question. Powers $\alpha$, $\beta$, etc: I'll try this one. I think the idea here is that any mechanical quantity (thus excluding electrical quantities) can be expressed as $\mathrm{kg}^\alpha\mathrm{m}^\beta\mathrm{s}^\gamma$ The problem is finding the numerical values of the exponents. It's sometimes a matter of guess-and-try to find the exponents, depending on what question is being asked. $\endgroup$
    – garyp
    Mar 6, 2014 at 14:34
  • $\begingroup$ DO you think the unit Kelvin could appear in speed? And what makes you say speed is in m/s given that distance is in m and time in s $\endgroup$
    – evil999man
    Mar 6, 2014 at 14:37
  • $\begingroup$ Galileo, and everyone else, sometimes have to make educated guesses. Sometimes we find the correct quantities simply by requiring the resulting expressions be internally consistent. $\endgroup$
    – garyp
    Mar 6, 2014 at 14:38

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Perhaps you are confused between dimension and unit.

Note that $cm$ and $m$ are different units but have same dimension of length. See? It's simple. They have only different magnitudes.

You have to understand that you cannot subtract or add 1 kg from 1 metre. Makes no sense, right?

Suppose you want to know about speed. You know that it is $\frac {distance}{time}$ Hence its units are $\frac {m}{s}$ and its dimensions are clear by formula.

You see that if a formula says that $1 kg = 1 s$ , It makes no sense, right?

So you check what the thing you wanna find about depends on and let analyse how to multiply and divide them to get the same dimension of thing you are looking for.

Note that this still will not give you perfect formula as $\frac {distance}{time}$ and $2*\frac {distance}{time}$ have same dimensions, you will be short of a constant.

Also it cannot predict equations like $v=u+at$

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I'm not sure exactly what you are asking, so I will simply write some relevant facts that might answer your question.

Dimensional analysis is a powerful tool for solving problems in physics. If we want the formula for a quantity $Q$, we can guess the formula for $Q$ by first writing a product of all relevant dimensional quantities raised to unknown powers $\alpha,\beta,\gamma\ldots$ and second making sure that our formula has the same dimension on the left and right-hand sides and solve for $\alpha,\beta,\gamma\ldots$

It might be the case that there exists a dimensionless combination of the relevant dimensional quantities. In that case, our method is not as useful.

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  • $\begingroup$ Thank for the answer, but can you explain the definition of "dimensional" quantities. I don't know why people say dimension of time, dimension of mass, etc. Does dimensional quantities have any relation to those. Also, how do we know there are dimensionless value (not unit/quantities). For example, some constants in physical equation. $\endgroup$
    – aukxn
    Mar 6, 2014 at 15:23
  • $\begingroup$ It means the units. We measure mass in kg, time in seconds, etc. A dimensionless quantity is a number that doesn't have units. Examples include numbers that arise from mathematical equations such as $\pi$ or $e$ or $3$, as well as some other quantities where the units end up "cancelling out", for example the decibel, the correlation coefficient in statistics, coefficients of friction, the Atwood number, ... $\endgroup$
    – Stella Biderman
    Mar 6, 2014 at 21:00

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