Using dimensional analyisis to verify formulae

[BogMonkey]

Heres a question I never figured out.

"Using dimensional analysis, determine which formula (a) or (b) is correct for the description of the force due to gravity between a body of mass m (kg) and another of mass M (kg) separated by a distance of r (m). The gravitational constant G has units (m^3s^-2kg^-1)
(a) FG = G m M / r^2 (b) FG = G m M ./ r"

I've been looking up dimensional analysis but can't find any information relating to validating formulae.

 

[Doc Al]

For an equation to be physically meaningful, the units on each side must match. Check the units in these two and see what you find.

 

[sir-pinski]

Dimensional analysis is a powerful tool that can be used to verify and validate formulae in physics. It involves checking the units of the variables involved in a formula to ensure that they are consistent and make sense in relation to the physical quantities being described. In this case, we can use dimensional analysis to determine which formula, (a) or (b), is correct for the force due to gravity between two bodies.

First, let's look at the units of each variable in both formulas. In formula (a), we have G (m^3s^-2kg^-1), m (kg), M (kg), and r (m). In formula (b), we have G (m^3s^-2kg^-1), m (kg), M (kg), and r (m). So both formulas have the same units for all variables, which is a good start.

Next, we need to look at the units of the overall force, FG. In formula (a), the units for FG would be (m^3s^-2kg^-1) * kg * kg / m^2 = m^3s^-2kg^-1, which is consistent with the units of G. In formula (b), the units for FG would be (m^3s^-2kg^-1) * kg * kg / m = m^2s^-2kg^-1, which is not consistent with the units of G.

Therefore, based on dimensional analysis, we can conclude that formula (a) is correct for the force due to gravity between two bodies. This is because it results in the correct units for the overall force, FG, and all the variables have consistent units.

In summary, dimensional analysis is a useful tool for verifying and validating formulae in physics. By checking the units of the variables and the overall result, we can determine which formula is correct and ensure that it accurately describes the physical quantity being studied.