Final answer:
The equilibrium height of a ball suspended from a spring is typically where the spring's potential energy is zero, meaning where it is neither stretched nor compressed. Without the specific height-time function, a numerical answer cannot be provided, but the equilibrium concept is reliant on energy conservation principles.
Step-by-step explanation:
The height of the ball at its equilibrium can often be determined by analyzing the forces in action and their respective energy conversions. In this case, we look at the spring's potential energy and the ball's gravitational potential energy. Usually, the spring's potential energy is zero at its equilibrium. Without the complete function provided for the ball's height as a function of time, we have to rely on the fundamental principles of energy conservation in physics. If we consider the point where the kinetic energy is transferred to potential energy, this is where the system would be in equilibrium, assuming no external forces act on the system.
Using the equation mgh, the gravitational potential energy, where m is mass, g is gravitational acceleration, and h is height, we could find the ball's height at equilibrium by setting the kinetic energy equal to the potential energy. However, without a specific equation provided for the variation of height with time or the initial conditions, it is not possible to give a numerical answer here. In problems like these, where the motion of the ball is typically sinusoidal in nature due to the spring, the equilibrium position is generally at the position where the spring is neither stretched nor compressed.