show that following equation is dimensionally correct F=2GMm/(R+h)^2
Answers
Explanation:
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Given:
The given equation is,
F = 2GMm/(R+h)²
Find:
Showing that the equation is dimensionally correct.
Solution:
For an equation to be dimensionally correct, the dimensions on the left-hand side should be equal to the dimensions on the right-hand side.
Given,
F = 2(GMm)/(R+h)²
where F = Force
G = Gravitational constant
M and m are masses.
R and h are distances.
We will now write the dimensions.
Dimensions of F = [M L T⁻²]
∴ Dimensions on the left-hand side,
L.H.S. = [M L T⁻²]
Now, dimensions of G = [M⁻¹ L³ T⁻²]
Dimensions of M = [M L⁰ T⁰]
Dimensions of m = [M L⁰ T⁰]
Dimensions of R = [M⁰ L T⁰]
Dimensions of h = [M⁰ L T⁰]
Dimensions of (R+h) = [M⁰ L T⁰]
Putting the values on the right-hand side, we have
Dimensions on R.H.S. = { [M⁻¹ L³ T⁻²][M L⁰ T⁰][M L⁰ T⁰] } / [M⁰ L T⁰]²
= [M L³ T⁻²]/[M⁰ L² T⁰]
= [M¹⁻⁰ L³⁻² T⁻²⁻⁰]
= [M L T⁻²]
Clearly, Dimensions on L.H.S. = Dimensions on R.H.S.
[M L T⁻²] = [M L T⁻²]
Hence, the given equation is dimensionally correct.
Hence proved.
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