Check the dimensional correctness of W = Fs cos θ
Work done by a force. Work when the force makes an angle θ with the displacement. The angle has no dimension.
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.
W = F s cos(θ)W[M L2 T-2]=F s cos(theta)[M L2 T-2]Step by step
1. What each symbol stands for
- W work [M L2 T-2]
- F force [M L T-2]
- s displacement [L]
- θ angle [M0 L0 T0]
2. The dimensions of each term
- W [M L2 T-2] [M L2 T-2]
- F s cos(theta) [M L T-2] × [L] × cos([M0 L0 T0]) [M L2 T-2]
3. Compare
Every term has dimensions [M L2 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
W = Fs cos θ: quick answers
Is W = Fs cos θ dimensionally correct?
Yes. Every term has the same dimensions, [M L² T⁻²], so W = Fs cos θ is dimensionally consistent. That does not prove the equation itself is correct.
Does a dimension check prove W = Fs cos θ 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.