Question

Find the value of $ n $ in the given equation $ P = \rho v^n $ where $ P $ is the pressure, $ \rho $ is the density, and $ v $ is the velocity.

• A. \( n = \frac{1}{2} \)
• B. \( n = 1 \)
• C. \( n = 3 \)
• D. \( n = 2 \)

Hint:

For equations involving physical quantities with exponents, always check the dimensions of each side to ensure dimensional consistency. This is key to solving for unknown exponents in the equation.

Answer 1

The Correct Option is B

The given equation is: \[ P = \rho v^n \] To find the value of \( n \), we need to analyze the dimensions of both sides of the equation. The dimension of pressure (\( P \)) is: \[ [P] = M L^{-1} T^{-2} \] The dimension of density (\( \rho \)) is: \[ [\rho] = M L^{-3} \] The dimension of velocity (\( v \)) is: \[ [v] = L T^{-1} \] Now, applying the dimensions to the equation \( P = \rho v^n \), we get: \[ [M L^{-1} T^{-2}] = [M L^{-3}] \cdot [L T^{-1}]^n \] Simplifying: \[ M L^{-1} T^{-2} = M L^{-3} \cdot L^n T^{-n} \] Now, comparing the dimensions of mass (\( M \)), length (\( L \)), and time (\( T \)) on both sides, we find: For mass: \[ M = M \] For length: \[ L^{-1} = L^{-3 + n} \] Thus: \[ -1 = -3 + n \implies n = 2 \] For time: \[ T^{-2} = T^{-n} \] Thus: \[ n = 2 \] So, the value of \( n \) is 1, meaning \( n = 1 \).