if the gravitational force F is given by F=GM1M2/r^2 derive the dimension of the constant G.
G is drived 4rm newtons law of gravitatn,d dimensn 4 force is=MLT-2 & d dimensn 4 G=M-1L3T-2
M2l-2t-2
To derive the dimensions of the gravitational constant G in the equation F = GM1M2/r^2, we need to express both sides of the equation in terms of their fundamental dimensions.
Let's start by breaking down the variables and constants:
- F represents force, which has dimensions of mass × acceleration (M × L × T^-2).
- M1 and M2 represent masses, which have dimensions of mass (M).
- r represents distance, which has dimensions of length (L).
- G is the gravitational constant, which we need to derive the dimensions for.
Now, let's rewrite the equation by substituting the dimensions of each variable and constant:
[M × L × T^-2] = [G] × [M] × [M] / [L^2]
Next, let's assign dimensions to each term on both sides:
[M1 × L1 × T^-2] = [G] × [M2] × [M2] / [L2^2]
Now, we can simplify the equation by canceling out the common dimensions:
L1T^-2 = G × M2^2 / L2^2
To compare both sides of the equation, we need to express L1 and L2, the dimensions of length, in terms of the fundamental dimensions. Since they represent the same physical quantity, they should have the same fundamental dimensions, denoted by L.
Now, we can rewrite the equation as follows:
LT^-2 = G × M^2 / L^2
To equalize the dimensions on both sides, we can multiply the left side by T^2:
L = GT^2 × M^2 / L^2
Simplifying further, we can cancel out one power of L from the numerator and denominator:
L^3 = GT^2 × M^2
At this point, we have L cubed on one side and the product of G, T squared, and M squared on the other side. To maintain dimensional equality, G must have dimensions of L^3 × T^-2 × M^-2.
Therefore, we can conclude that the dimensions of the gravitational constant G are [L^3][T^-2][M^-2].