Check the dimensional correctness of F = Gm₁m₂/r²
Newton's law of gravitation. The attraction between two masses a distance r apart.
Equation checker
Matching dimensions are necessary, not sufficient. A check cannot see pure numbers such as 1/2 or 2π, so a consistent equation can still be wrong.
F = G m1 m2 / r2F[M L T-2]=G m1 m2 / r2[M L T-2]Step by step
1. What each symbol stands for
- F force [M L T-2]
- G gravitational constant [M-1 L3 T-2]
- m1 mass [M]
- m2 mass [M]
- r length [L]
2. The dimensions of each term
- F [M L T-2] [M L T-2]
- G m1 m2 / r2 [M-1 L3 T-2] × [M] × [M] / [L]2 [M L T-2]
3. Compare
Every term has dimensions [M L T-2], so the equation is dimensionally consistent.
What it proves and what it does not
It can show
- Every term that is added or subtracted has the same dimensions, so the sum makes sense.
- Both sides of the equals sign have the same dimensions.
- Angles inside sin, cos and similar functions are pure numbers.
It cannot show
- A pure number such as 1/2 or 2π has no dimensions, so a check cannot see a missing or wrong one.
- Two quantities with the same dimensions, such as work and torque, cannot be told apart.
- A sign error or a wrong power that keeps the dimensions equal goes unnoticed.
Matching dimensions are necessary but not sufficient. Pure numbers such as 1/2 or 2π have no dimensions, so a check cannot see them, and an equation with matching dimensions can still be wrong.
Look up each quantity
F = Gm₁m₂/r²: quick answers
Is F = Gm₁m₂/r² dimensionally correct?
Yes. Every term has the same dimensions, [M L T⁻²], so F = Gm₁m₂/r² is dimensionally consistent. That does not prove the equation itself is correct.
Does a dimension check prove F = Gm₁m₂/r² is right?
No. A check can show that an equation is wrong, but it cannot show that it is right. Pure numbers such as 1/2 and 2π have no dimensions, so the check cannot see them.