Phui Yeng
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#1
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#1
Hi, I have learnt that in any equation where each term has the same base units, the equation is said to be homogenous or balanced. However, are ALL homogenous equations correct equations? What is the explanation to that?
Thank you very much!
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Kallisto
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In mathematics an equation is called homogenous, if its a conditional equation which contains a zero. May I ask you in which part of physics do you use a homogenous equation at the moment? perhaps I'm able to give a better answer by getting more informations.
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Stonebridge
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#3
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(Original post by Phui Yeng)
Hi, I have learnt that in any equation where each term has the same base units, the equation is said to be homogenous or balanced. However, are ALL homogenous equations correct equations? What is the explanation to that?
Thank you very much!
Homogeneous means that the units (dimensions of mass, length, time etc) are the same on both sides of the equation.
So in
p = hdg (pressure at a depth h in a liquid of density d)
the units on the left are pressure, which is Force / area or N/m2

On the right we have h (metres) d (kg/m3) and g (m/s2)

So on the left we have
Force which is defined in F=ma so the units of force are kg.m/s2
and area which is m2
Check for yourself that F/A reduces to kg.m-1.s-2
On the right you have hdg
h = m (metre)
d is kg.m-3 (mass/volume)
g is m.s-2 (acceleration)
Check for yourself that this also reduces to
kg.m-1.s-2

So the equation is homogeneous.
This does not mean it's correct though.
The correct equation could be, for example, p=2hdg
There is no way of knowing, purely from consideration of the units (homogeneity) that the equation is correct.
What you can say is that if the equation is not homogeneous it is not correct.
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Kallisto
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#4
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(Original post by Stonebridge)
Homogeneous means that the units (dimensions of mass, length, time etc) are the same on both sides of the equation. (...)
Does apply this meaning too, if there are different units on both sides of the equation which mean the same?

Example:
1 J = 1 kg m² / s²; the units are different, but the meaning is the same. So is this an homologous equation?
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Stonebridge
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#5
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#5
(Original post by Kallisto)
Does apply this meaning too, if there are different units on both sides of the equation which mean the same?

Example:
1 J = 1 kg m² / s²; the units are different, but the meaning is the same. So is this an homologous equation?
It means when you break down the derived units (such as J) to the fundamental (base) units (kg, m, s etc) they are the same on both sides.

Joule is work done and is force x distance
Force is mass x acceleration and is (kg.ms-2)
So joule is (kg.ms-2)m
This becomes kgm2s-2
So any equation with energy on the left has units kg.m2.s-2 on both sides if it's homogeneous.
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Kallisto
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(Original post by Stonebridge)
x
I see. Equations can be made homologous by derivating the unit on one of the sides to get the same units in equation. Right?
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Stonebridge
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#7
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(Original post by Kallisto)
I see. Equations can be made homologous by derivating the unit on one of the sides to get the same units in equation. Right?
All true equations MUST have the same units (= combination of base units) on both sides.
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Kallisto
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#8
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(Original post by Stonebridge)
All true equations MUST have the same units (= combination of base units) on both sides.
To come to an end: if the comnbination of base units, as you have done in comment #5, result in the same unit, so its an homologous equation. So equations in terms of homologous can be investigated by thinking about and combinating the base units. Do I have this right now?
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Stonebridge
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Yes. It's possible even to derive an equation by consideration of the base units. There are examples in most physics text books.
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Phui Yeng
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#10
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#10
(Original post by Kallisto)
In mathematics an equation is called homogenous, if its a conditional equation which contains a zero. May I ask you in which part of physics do you use a homogenous equation at the moment? perhaps I'm able to give a better answer by getting more informations.
The first topic in AS level Physics.
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Phui Yeng
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#11
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#11
(Original post by Stonebridge)
Homogeneous means that the units (dimensions of mass, length, time etc) are the same on both sides of the equation.
So in
p = hdg (pressure at a depth h in a liquid of density d)
the units on the left are pressure, which is Force / area or N/m2

On the right we have h (metres) d (kg/m3) and g (m/s2)

So on the left we have
Force which is defined in F=ma so the units of force are kg.m/s2
and area which is m2
Check for yourself that F/A reduces to kg.m-1.s-2
On the right you have hdg
h = m (metre)
d is kg.m-3 (mass/volume)
g is m.s-2 (acceleration)
Check for yourself that this also reduces to
kg.m-1.s-2

So the equation is homogeneous.
This does not mean it's correct though.
The correct equation could be, for example, p=2hdg
There is no way of knowing, purely from consideration of the units (homogeneity) that the equation is correct.
What you can say is that if the equation is not homogeneous it is not correct.

Thank you very much, this has helped me a lot!
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Sourish sharma
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#12
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Please tell about homogeneity of dimension
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