A question on Dimensional Analysis

[Yashbhatt]

Dimensional Analysis is really simple and I have read that it is only used to verify the derived equations. But I don't understand how we work with constants in Dimensional Analysis. For example, if we are given KE depends on mass and velocity, we can easily derive KE = 1/2 mv².

In the above case we assume the proportionality to be dimensionless but what do we do if we want to derive the Newtonian relation between gravity and mass/distance. Do we still assume the constant G to dimensionless and later give it dimensions to fit the other units?

 

[jk86]

For example, if we are given KE depends on mass and velocity, we can easily derive KE = 1/2 mv².

From dimensional analysis you can derive that it depends on [mass][velocity]², but the factor of $\frac{1}{2}$ comes from the work-energy theorem:

$$W=\int_1^2 Fvdt = \int_1^2 \vec{F}\cdot d\vec{x} = \int_1^2\left(m\frac{dv}{dt}\right)vdt = \frac{1}{2}m(v_2^2-v_1^2)$$

In the above case we assume the proportionality to be dimensionless but what do we do if we want to derive the Newtonian relation between gravity and mass/distance. Do we still assume the constant G to dimensionless and later give it dimensions to fit the other units?

The dimensions of [force]=[mass][distance]/[time]² and the dimensions of [m₁m₂/r²]=[mass]²/[distance]², so you know you need a dimensionful constant that compensates for the difference. The way this was studied originally was not in terms of a direct relation like this but rather in terms of proportionality. In other words, you first derive/measure that the force of gravity is proportional to the product of the masses and also to the inverse square of the distance, then you worry about measuring the proportionality constant.

 

[Adjoint]

We can use dimensional analysis in two ways:

1. To verify if an equation is incorrect:
Suppose you have an equation and you want to make sure it is dimensionally correct. For this, you must know the dimension of all physical quantities in the equation. Plus you should also know the dimension of any constant which is present in the equation. By the way, a constant can have no dimension or it might have dimension.

2. To find dimension of a constant or any physical quantity:
Suppose you know an equation is correct. Now you want to find dimension of a quantity or constant present in the equation. In this case you must know the dimensions of all other quantities expect the one you want to find the dimension of.

Example of 1:
Suppose you are given a equation F = GMm/r² and you are not sure if this equation is dimensionally correct or not.
To check if this equation is dimensionally correct you must know the dimensions of all quantities. You must already know the dimension of the constant G too.

Example of 2:
Suppose you know F = GMm/r² is dimensionally correct. You know dimensions of F, m (or M) and r. Then by using dimensional analysis you can find the dimension of G.

By the way, you certainly know, if a equation is dimensionally incorrect it must be an wrong equation; but if an equation is dimensionally correct, it still might not be a physically correct equation.

 

[Yashbhatt]

So, we decide experimentally if the constant is dimensionless or not?

 

[UltrafastPED]

So, we decide experimentally if the constant is dimensionless or not?

No, it depends upon how the constant was created.

In Newtonian gravity the force is proportional to M*m/r^2, which has units of [mass]^2 divided by [distance]^2. But we want units of force, so the constant G must have the correct units to yield a force, but must also have the correct magnitude to convert from the units that are being used for mass and distance; the standard units in SI will be kg and meters, and force is in Newtons, so G must have units equivalent to Newtons*[meters]^2/[kg]^2.

In other cases the constant appears for numerical/geometric reasons: 4 pi comes from an integration over the surface of a sphere. These are dimensionless.

 

[Yashbhatt]

But in this case Force was already a known quantity. So, we could adjust the dimensions of G but we couldn't have done so if it were a completely new quantity.

 

[UltrafastPED]

The magnitude of G is experimentally determined, nut the dimensions are not.

You can always determine the dimensions for a constant by writing the known dimensions for both sides, then cancel the common factors.