Concepts Of Physics MCQ Edition [Volume 1]PhysicsIntroduction to Physics
Let x and a represent distance. Is the equation dx a^2 - x^2 = 1 a ⁻¹ a x dimensionally correct?
Options
- ANo, the left-hand side has dimensions of [L] while the right-hand side is dimensionless.
- BYes, both sides are dimensionless.
- CNo, the left-hand side is dimensionless while the right-hand side has dimensions of [L⁻¹] .
- DYes, both sides have dimensions of [L⁻¹] .
Correct answer
C. No, the left-hand side is dimensionless while the right-hand side has dimensions of [L⁻¹] .
Step-by-step solution
The dimension of dx is [L] . The dimension of a^2 - x^2 is [L^2] = [L] . The dimension of the left-hand side is [L] [L] = [M^0 L^0 T^0] , which is dimensionless. On the right-hand side, the argument of the inverse sine function, a x , is dimensionless. The inverse sine function itself represents an angle, which is also dimensionless. The term 1 a has the dimension [L⁻¹] . Therefore, the dimension of the right-hand side is [L⁻¹] . Since the left-hand side is dimensionless and the right-hand side has dimensions of [L