Find the units and dimensions of 'G' the gravitational constant in the following equation, F=(GM2M1/r squared where F is the force of attraction between the two bodies of mass respectively.
And r is the separation distance before the two bodies
The units and dimensions of the gravitational constant, G, can be found by analyzing the given equation.
In the equation, F = (GM1M2/r^2), F represents force, which has the unit of Newtons (N).
The masses, M1 and M2, have the unit of kilograms (kg).
The separation distance, r, is given in meters (m).
Therefore, we can express the equation as:
[N] = [G] * [kg] * [kg] / [m^2]
To find the units of G, we can rearrange the equation:
[G] = [N] * [m^2] / ([kg] * [kg])
Simplifying further:
[G] = [N] / [kg^2] * [m^2]
Therefore, the units of G, the gravitational constant, are:
[N] / [kg^2] * [m^2]
The dimensions of G are derived from the units:
Dimension of G = (dimension of N) / (dimension of kg^2) * (dimension of m^2)
The dimension of N is [M][L][T]^-2 (mass times length divided by time squared).
The dimension of kg is [M] (mass).
The dimension of m^2 is [L]^2 (length squared).
Substituting in the dimensions, we get:
Dimension of G = ([M][L][T]^-2) / ([M]^2) * ([L]^2)
Simplifying further:
Dimension of G = [M]^-1 * [T]^-2 * [L]^0
So, the dimensions of G are:
[M]^-1 * [T]^-2 * [L]^0
To find the units and dimensions of the gravitational constant 'G' in the equation F = (GM2M1/r^2), we can analyze the equation using dimensional analysis.
The equation can be rearranged as:
F = G(M2M1/r^2)
Now let's analyze each term separately:
F is the force of attraction, which has the dimensions of force (mass × acceleration) and is measured in units of Newtons (N).
M1 and M2 are the masses of the two bodies, which have the dimensions of mass and are measured in units of kilograms (kg).
r is the separation distance between the two bodies and has the dimensions of length and is measured in units of meters (m).
Now, let's break down the dimensions of the equation by substituting the dimensions of each quantity:
[N] = [G] × [kg] × [kg] / [m]^2
We can simplify this equation by canceling out the kg and m terms:
[N] = [G] × kg^2 / m^2
To isolate the dimensions of G, we rearrange the equation as:
[G] = [N] × m^2 / kg^2
Now, substituting their base units:
[N] = kg × m / s^2
[m] = m
[kg] = kg
So, [G] = (kg × m / s^2) × m^2 / kg^2
By canceling out the like terms, we get:
[G] = m^3 / (kg × s^2)
Therefore, the units of 'G' are cubic meters per kilogram per second squared (m^3 / (kg × s^2)) and the dimensions are [L^3 M^(-1) T^(-2)].