Check the dimensional correctness of v = √(Y/ρ)
Speed of sound in a solid. Speed of a longitudinal wave in a solid rod from its Young's modulus and density.
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.
v = sqrt(Y / ρ)v[L T-1]=sqrt(Y/rho)[L T-1]Step by step
1. What each symbol stands for
- v velocity [L T-1]
- Y Young's modulus [M L-1 T-2]
- ρ density [M L-3]
2. The dimensions of each term
- v [L T-1] [L T-1]
- sqrt(Y/rho) sqrt([M L-1 T-2] / [M L-3]) [L T-1]
3. Compare
Every term has dimensions [L T-1], 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
v = √(Y/ρ): quick answers
Is v = √(Y/ρ) dimensionally correct?
Yes. Every term has the same dimensions, [L T⁻¹], so v = √(Y/ρ) is dimensionally consistent. That does not prove the equation itself is correct.
Does a dimension check prove v = √(Y/ρ) 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.