Question

Wave pulse can travel along a tense string like a violin spring. A series of experiments showed that the wave velocity \( V \) of a pulse depends on the following quantities, the tension \( T \) of the string, the cross-section area \( A \) of the string and density \( \rho \) of the string. Obtain an expression for \( V \) in terms of \( T \), \( A \), and \( \rho \) using dimensional analysis.

• A. \( V = \sqrt{\frac{T}{A \rho}} \)
• B. \( V = \sqrt{\frac{T}{\rho A}} \)
• C. \( V = \sqrt{\frac{A T}{\rho}} \)
• D. None of these

Hint:

Dimensional analysis helps in deriving relationships between physical quantities based on their dimensions.

Answer 1

The Correct Option is B

Step 1: Use dimensional analysis.
The dimensions of velocity \( V \) are \( [M^0 L^1 T^{-1}] \). The dimensions of tension \( T \) are \( [M L^2 T^{-2}] \), area \( A \) is \( [L^2] \), and density \( \rho \) is \( [M L^{-3}] \).
Step 2: Set up the relation.
Let \( V = K \cdot T^a A^b \rho^c \). By equating the dimensions, we solve for \( a \), \( b \), and \( c \) to get the expression: \[ V = \sqrt{\frac{T}{\rho A}} \] Final Answer: \[ \boxed{V = \sqrt{\frac{T}{\rho A}}} \]