Why are Some Constants Dimensionless?

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In summary, pure numbers like 1, 2, 3... are dimensionless because they do not have units. However, Avogadro's Number, Plank's Constant, and the Gravitational Constant have dimensions because they are associated with physical quantities that have units. This can be seen in equations such as F = -GmM/r^2, where the units on both sides must be compatible. Similarly, 1/2 in 1/2 mv^2 is dimensionless because it is a constant that can be factored out of the equation. The speed of light, c, is also dimensionless in Special and General Relativity, as it is equal to 1 and can be thought of as a
  • #1
physics kiddy
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Why are pure numbers like 1 , 2 , 3 ... dimensionless and Avogadro's Number, Plank's Constant, Gravitational Constant dimensional ?
 
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  • #2
Because the latter have dimensions! The Gravitational constant, G, appears in F= -GmM/r^2. In the metric system (mks), r has units of meters, both m and M have units of kg so that "mM/r^2" have units of [itex]kg^2/m^2[/itex]. F, a force, has units of "[itex]kg m^2/sec^2[/itex]". In order to make the units on both sides of the equation the same, G must have units of [itex]1/(kg sec^2)[/itex].

Avogadro's number is the number of molecules per mole. The number of molecules does not depend on any units so Avogadro's number has units of [itex]mol^{-1}[/itex].

Plank's constant is the "h" in [itex]h\nu[/itex] where E is energy, and so has units of [itex]kg m^2/sec^2[/itex] while [itex]\nu[/itex], a frequency, is "number of cycles per second". "Number of cycles", like "number of molecules" is just a number without units. Since we need to have left "[itex]kg m^2/sec^2[/itex]" we need . That means that h must have units of [itex]kg m^2[/itex] in the numerator and one "sec" in the denominator: [itex]kg m^2/sec[/itex].
 
  • #3
Why is 1/2 in 1/2 mv^2 dimensionless ?
 
  • #4
Because mv^2 has the unit same as that of Energy.
 
  • #5
aati2sh said:
Because mv^2 has the unit same as that of Energy.

Please elaborate it mathematically.
 
  • #6
Let this constant (which turns out to be 1/2) be C. Let k and k' be different dimensionless numbers. It's quite easy to see that we can set up

[tex]C\cdot k\cdot \left(1\ \mathrm{J}\right)=k'\cdot\left(1\ \mathrm{J}\right)[/tex]

And so

[tex]C=\dfrac{k'}{k}[/tex]

And so C's dimensionless.

I wanted to point out that c, the speed of light, is dimensionless, equal to 1, in Special (and General, I'd imagine) Relativity. Basically, [itex]299792458\ \mathrm{m}=1\ \mathrm{s}[/itex].

I've always found it useful to think of units as constants that are, in some cases, incompatible with one another, so the simplest form is just their product.
 

1. What are dimensionless quantities?

Dimensionless quantities are numerical values that do not have any physical dimensions. They are used to represent ratios, proportions, and other mathematical relationships between physical quantities.

2. Why are dimensionless quantities important in science?

Dimensionless quantities are important in science because they allow us to compare and analyze different physical phenomena without the influence of units. They also help in simplifying complex equations and making them more generalizable.

3. How are dimensionless quantities calculated?

Dimensionless quantities are calculated by dividing two physical quantities with the same dimensions. The resulting value will be dimensionless, representing the relationship between the two quantities.

4. What is the significance of dimensionless quantities in dimensional analysis?

Dimensionless quantities play a crucial role in dimensional analysis as they allow us to identify and eliminate redundant terms in equations. This helps in reducing the complexity of equations and making them more applicable to different situations.

5. Can dimensionless quantities have units?

No, dimensionless quantities do not have units as they represent pure numbers. They are also known as unitless quantities or quantities of dimension one.

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