91Ó°ÊÓ

The oscillation frequency in Simple Harmonic Motion (SHM) describes how fast an object moves back and forth through its cycle. For a spring-mass system, it's vital for determining how quickly the oscillations occur.
The frequency is largely dependent on two factors:

• The spring constant, denoted as \( k \)
• The mass of the system, denoted as \( m \)

The mathematical expression linking frequency \( f \), spring constant \( k \), and mass \( m \) is:\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]In this exercise, when the sack of gravel leaves the beam, the mass changes from 400 kg to 225 kg. This affects the frequency, as shown in the calculation:\[ f' = f \cdot \sqrt{\frac{400}{225}} \approx 0.800 \text{ cycles/s} \]The frequency increases since the system mass decreases. This highlights the inverse relationship between frequency and mass.

The spring constant, \( k \), provides a measure of a spring's ability to resist deformation. It is a fixed value for a given spring and plays a crucial role in SHM.
This constant strongly influences both the frequency and the potential energy stored in the spring. The spring constant is a reflection of the spring's stiffness and is critical in determining how the system behaves:

• A higher \( k \) value means a stiffer spring.
• A lower \( k \) value means a more flexible spring.

In the exercise, as the gravel leaves the beam, the spring constant remains constant, but its effect on the system changes due to the reduced mass. As seen in the frequency and amplitude equations, the spring constant helps determine the new motion properties when mass changes, while ensuring the total energy conservation in a different mass scenario.

In SHM, energy conservation is a fundamental principle that predicts how the system evolves over time. Here, the total mechanical energy is a mix of kinetic and potential energy that stays constant if no external force acts upon it.
Consider:

• At maximum displacement (either upward or downward), all the energy is potential.
• At equilibrium, the energy is entirely kinetic.

The exercise illustrates the two cases:- **Case (a):** When the sack falls at maximum upward displacement, potential energy is at peak.The system conserves energy, thus keeping the amplitude constant at 40 cm. The kinetic component holds no value at this point.- **Case (b):** When the sack falls off at maximum speed, energy is again fully conserved. This time, energy is primarily kinetic as we are at the equilibrium point. The amplitude changes due to the new balance of kinetic energy causing dynamics in displacement capability, calculated by:\[ A' = A \cdot \sqrt{\frac{m}{m'}} \approx 0.533 \text{ m} \]So ultimately, whether potential or kinetic, energy rules the outcomes of SHM.

Amplitude in this context refers to how far the object moves from its equilibrium position during oscillation. It shows the maximum extent of the oscillation, and is directly linked to the energy in the system.
If no energy is lost, the amplitude reflects the total energy's potential axis:

• In part (a) of the problem, since the system changes as the sack of gravel departs at maximum displacement, the amplitude stays constant as energy mainly transitions in potential energy form.
• For part (b), a shift due to loss of mass at equilibrium results in a new amplitude.

This new amplitude accounts for changes in the potential for maximum displacement capability, calculated as approximately 53.3 cm. The derivation of this change relied on the conservation principles and the altered mass. The amplitude change factors significantly into determining the properties and dynamics of SHM post any mass alteration during the motion cycle.