Check the dimensional correctness of T = 2π√(g/l)
A pendulum with g and l swapped. Swapping l and g turns a time into a frequency.
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.
T = 2 π sqrt(g / l)T[T]=2 pi sqrt(g/l)[T-1]Step by step
1. What each symbol stands for
- T time period [T]
- g acceleration due to gravity [L T-2]
- l length [L]
2. The dimensions of each term
- T [T] [T]
- 2 pi sqrt(g/l) 2 × π × sqrt([L T-2] / [L]) [T-1]
3. Compare
The two sides of an equation must have the same dimensions, but "T" is [T] and "2 pi sqrt(g/l)" is [T-1].
The term 2 pi sqrt(g/l) does not match
A likely corrected form:
T = 2π√(l/g)
The time of one swing of a pendulum of length l.
Check T = 2π√(l/g)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.
When the dimensions of the terms differ the equation cannot be correct, whatever numbers are in it, as long as each symbol means what the table says.
Look up each quantity
T = 2π√(g/l): quick answers
Is T = 2π√(g/l) dimensionally correct?
No. The two sides of an equation must have the same dimensions, but "T" is [T] and "2 pi sqrt(g/l)" is [T⁻¹].
Does a dimension check prove T = 2π√(g/l) 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.