if the gravitational force f is given by F=GM1M2/r^2, derive the dimension if the constant G.
If the gravitational force f is given by f= gm, m2/r^2 derive the dimensions of the constant g.
To derive the dimension of the gravitational constant G, we need to substitute the dimensions of the other variables in the equation and solve for the dimensions of G.
Let's break down the dimensions of each variable in the equation:
F: gravitational force
M1: mass of the first object
M2: mass of the second object
r: distance between the objects
The dimension of force (F) is represented as [MLT⁻²] since it is measured in units of mass (M), length (L), and time (T).
The dimensions of the mass of the first object (M1) and the mass of the second object (M2) are both [M].
The dimension of distance (r) is represented as [L].
Substituting these dimensions into the equation:
[MLT⁻²] = G[M][M]/[L]²
Simplifying the equation:
[MLT⁻²] = G[M²]/[L²]
Now, let's equate the dimensions on both sides:
MLT⁻² = GL²M²
Now, we need to solve for the dimension of G.
The dimensions on both sides of the equation should be equal, so we can equate the exponents of each dimension:
1 = L² (Length)
-2 = T (Time)
1 = M² (Mass)
Now, let's substitute these dimensions into the equation:
1 = G × (L²) × (M²) × (T⁻²)
Dividing both sides by (L²) × (M²) × (T⁻²), we get:
G = L⁻²M⁻²T²
Therefore, the dimension of the gravitational constant G is [L⁻²M⁻²T²].