ANS:C - second

A mercury manometer (U-tube type) exemplifies a first-order system. In a first-order system, the response of the system to a change in input is proportional to the first derivative of the output. For a mercury manometer, the height of the mercury column responds to pressure changes in a manner consistent with a first-order system, where the rate of change of the mercury height is proportional to the difference between the current height and the equilibrium height. This is analogous to the behavior of other first-order systems such as RC (resistor-capacitor) circuits and thermal systems. A mercury manometer (U-tube type) exemplifies a first-order system due to the way it responds to changes in pressure. Here’s a detailed explanation:

1. System Description:
   • A U-tube manometer consists of a U-shaped tube filled with mercury (or another liquid).
   • One end of the U-tube is open to the atmosphere, and the other end is connected to the pressure source to be measured.
   • When there is a pressure difference between the two ends, the mercury levels in the two arms of the U-tube change until they reach a new equilibrium.
2. First-Order System Characteristics:
   • A first-order system is characterized by a single energy storage element, which, for the manometer, is the potential energy of the mercury column.
   • The response of a first-order system to a step input (sudden change in pressure) is an exponential function, which describes how the system gradually approaches its new equilibrium position.
3. Dynamic Response:
   • When there is a sudden change in pressure, the mercury in the U-tube moves until the pressure difference is balanced by the height difference of the mercury columns.
   • The rate at which the mercury moves is initially fast and then slows down as it approaches the new equilibrium. This behavior is typical of a first-order response.
   • The time constant (τ\tauτ) for the mercury manometer depends on factors like the density of mercury, the viscosity of mercury, and the geometry of the U-tube. The time constant describes how quickly the mercury column reaches 63.2% of the total height change after a sudden pressure change.
4. Mathematical Representation:
   • If h(t)h(t)h(t) represents the height difference between the two columns at time ttt, the first-order differential equation describing the system might look like: τdh(t)dt+h(t)=KΔP\tau \frac{dh(t)}{dt} + h(t) = K \Delta Pτdtdh(t)​+h(t)=KΔP where ΔP\Delta PΔP is the pressure difference, KKK is a constant related to the system's sensitivity, and τ\tauτ is the time constant.
   • This equation shows that the rate of change of the height difference (dh(t)/dtdh(t)/dtdh(t)/dt) is proportional to the difference between the current height and the equilibrium height, which is a hallmark of first-order systems.
5. Practical Implications:
   • Understanding that the mercury manometer behaves as a first-order system helps in predicting its behavior in response to pressure changes.
   • For example, if you know the time constant, you can estimate how quickly the manometer will settle to its new reading after a change in pressure.

In summary, a mercury manometer exemplifies a first-order system due to its exponential response to changes in pressure, characterized by a time constant that governs the speed of its response. This behavior is similar to other first-order systems where a single energy storage element determines the system's dynamic response.