91Ó°ÊÓ

The spring constant, denoted as \( k \), is a measure of how stiff a spring is. The stiffer the spring, the larger the spring constant. In the context of simple harmonic motion, the spring constant influences how quickly the oscillating system returns to its equilibrium position.

In simple harmonic motion of a mass-spring system, the spring constant \( k \) can be determined from the period of oscillation \( T \) using the formula:

• \[ T = 2\pi \sqrt{\frac{m}{k}} \]

This formula shows that the period \( T \) depends on both the mass \( m \) and the spring constant \( k \). By rearranging the formula, you can solve for \( k \):

• \[ k = \frac{4\pi^2 m}{T^2} \]

In our exercise, with a mass \( m = 0.0500 \) kg and a period \( T = 0.500 \) s, we find the spring constant \( k \approx 7.90 \text{ N/m} \).This calculation shows that our spring is moderately stiff, allowing for a brisk return of the stone to equilibrium during vibration.

The frequency of oscillation \( f \) refers to how often the oscillations occur within a given time frame, typically measured in hertz (Hz), which means oscillations per second. In a mass-spring system, the frequency provides insight into how many complete cycles of motion happen each second.

To find the frequency, use the relationship between period \( T \) and frequency \( f \):

• \[ f = \frac{1}{T} \]

Given \( T = 0.500 \) s, the frequency is calculated to be \( 2.00 \text{ Hz} \).

This means the stone completes two full cycles of oscillation every second. The higher the frequency, the faster the oscillations occur, showing that this system has a fairly rapid oscillation, reinforcing the idea of the spring's moderate stiffness impacting the speed of vibrations.

The amplitude of motion in the context of oscillations refers to the maximum distance the oscillating object moves from its equilibrium position. It is a critical parameter as it represents the energy in the motion being directly related to how far the object is set into motion from rest.

In terms of simple harmonic motion, the lack of explicit initial conditions or external forces in the exercise typically means calculating the amplitude directly is not feasible without additional data like energy or force values at maximum displacement. This is often provided or measured in practical applications.

While this exercise doesn't provide specific data for amplitude calculation, it's essential to understand that amplitude is crucial in defining the extent of oscillation, dictating the maximum potential energy the system can possess based on how far the stone is stretched or compressed from the neutral position.