Question

Which of the following expressions can be deduced on the basis of dimensional analysis? (All symbols have their usual meanings)

• A. \( F = \epsilon r \, v \)
• B. \( s = ut + \frac{1}{2} a t^2 \)
• C. \( x = A \cos \omega t \)
• D. \( N = N_0 2^t \)

Hint:

Dimensional analysis helps in checking the consistency of equations, especially when deriving relationships between physical quantities. Always ensure that the units on both sides of the equation match.

Answer 1

The Correct Option is B

Dimensional analysis helps in deriving expressions that are dimensionally consistent. We analyze each of the options: - Option (1) \( F = \epsilon r v \) cannot be derived directly using dimensional analysis as it involves unknown constants and factors not defined by basic physical dimensions. - Option (2) \( s = ut + \frac{1}{2} a t^2 \) is the standard kinematic equation for displacement under constant acceleration, which can indeed be derived using dimensional analysis, as the units of each term are consistent with displacement. - Option (3) \( x = A \cos \omega t \) is a solution to simple harmonic motion, and while it is physically correct, its dimensional consistency needs more information for derivation. - Option (4) \( N = N_0 2^t \) involves exponential growth, and dimensional analysis does not provide a simple way to deduce this relationship. Thus, the correct answer is the kinematic equation \( s = ut + \frac{1}{2} a t^2 \), which is a standard result in mechanics.