Question

If initial conditions for a system are inherently zero, what does it physically mean?

• A. The system is at rest but stores energy
• B. The system is working but does not store energy
• C. The system is at rest or no energy is stored in any of its part
• D. The system is working with zero reference input

Hint:

Think of a pendulum hanging perfectly still at its equilibrium position at \( t = 0 \). Its initial displacement and velocity are zero, and it has no stored potential or kinetic energy relative to its resting state. This represents a system with zero initial conditions.

Answer 1

The Correct Option is C

Step 1: Understand the concept of initial conditions in a system. 
Initial conditions refer to the state of a system at the beginning of an observation or analysis, typically at time \( t = 0 \). These conditions specify the values of the system's state variables, which are the minimum set of variables that fully describe the system's dynamic behavior at any given time. Step 2: Consider the physical meaning of zero initial conditions. 
If the initial conditions for a system are inherently zero, it implies that at the starting point, the system is in a neutral or equilibrium state with respect to its dynamic behavior. This means there is no initial excitation, displacement, velocity, charge, current, or any other form of stored energy within the system that would cause it to evolve or respond even without an external input. 
Step 3: Analyze each option based on the definition of zero initial conditions. 
Option 1 (The system is at rest but stores energy): If the system stores energy, it is not in a state of inherently zero initial conditions. This stored energy would represent non-zero initial values of the state variables (e.g., potential energy in a spring, charge on a capacitor). Option 2 (The system is working but does not store energy): If the system is working, it implies there is some activity or change occurring, which contradicts the idea of being at the initial starting point with no prior excitation. Also, a working system often involves energy storage and transfer. Option 3 (The system is at rest or no energy is stored in any of its part): This option accurately reflects the meaning of zero initial conditions. If the system is at rest, its dynamic variables like velocity are zero. If no energy is stored in any of its parts, it means the initial values of state variables related to energy storage (e.g., displacement in a spring, voltage across a capacitor, current through an inductor) are zero. Therefore, the system is in an unexcited state. Option 4 (The system is working with zero reference input): The reference input is an external signal applied to the system. Zero initial conditions pertain to the internal state of the system at \( t = 0 \), independent of any external input applied subsequently. A system can have zero initial conditions and still respond to a non-zero reference input. 
Step 4: Select the option that best describes the physical meaning of inherently zero initial conditions. 
The most accurate physical interpretation of inherently zero initial conditions is that the system is at rest and contains no stored energy in any of its components at the initial time.