Question

Time constant for U-tube manometer is given by

• A. \(\sqrt{L/2g}\)
• B. \(\sqrt{2Lg}\)
• C. \(2g\sqrt{L}\)
• D. \(L\sqrt{2g}\)

Hint:

The natural frequency of oscillation of the manometer liquid is $\omega_n = \sqrt{2g/L}$.
Since the time constant $\tau$ is the reciprocal of natural frequency ($\tau = 1/\omega_n$), we can write $\tau = \sqrt{L/2g}$ immediately.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the mathematical expression for the time constant ($\tau$) of a U-tube manometer.
A U-tube manometer is a standard device used to measure pressure differences, and its dynamic response to pressure changes is modeled as a second-order system.

Step 2: Key Formula or Approach:
The dynamic behavior of the liquid column in a U-tube manometer of length $L$ can be described by a second-order differential equation derived from Newton's second law:
\[ \frac{d^2 y}{dt^2} + \frac{32 \mu}{\rho d^2} \frac{dy}{dt} + \frac{2g}{L} y = \frac{2g}{L} x(t) \] where $y(t)$ is the displacement of the liquid level, and $x(t)$ is the applied pressure head.
The standard form of a second-order system is: \[ \tau^2 \frac{d^2 y}{dt^2} + 2\zeta\tau \frac{dy}{dt} + y = x(t) \] where $\tau$ is the system time constant (or period of natural oscillation divided by $2\pi$).

Step 3: Detailed Explanation:
Let us divide the manometer differential equation by the coefficient of the $y$ term, which is $\frac{2g}{L}$, to bring it to the standard form:
Multiplying the entire equation by $\frac{L}{2g}$ gives: \[ \left(\frac{L}{2g}\right) \frac{d^2 y}{dt^2} + \left(\frac{16 \mu L}{\rho g d^2}\right) \frac{dy}{dt} + y = x(t) \] By comparing this equation directly to the standard second-order form: \[ \tau^2 = \frac{L}{2g} \] Taking the square root of both sides to solve for the time constant $\tau$: \[ \tau = \sqrt{\frac{L}{2g}} \] This represents the characteristic time constant of the manometer, which depends solely on the length of the liquid column $L$ and the acceleration due to gravity $g$.

Step 4: Final Answer
Thus, the time constant for a U-tube manometer is $\sqrt{L/2g}$, corresponding to option (A).