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Understanding the spring constant is essential in analyzing harmonic oscillation. The spring constant, denoted by \( k \), measures the stiffness of a spring. A larger spring constant indicates a stiffer spring, which requires more force to stretch or compress it. To calculate the spring constant, you need the angular frequency \( \omega \) and the mass \( m \) attached to the spring. The angular frequency in oscillatory motion is given by \( \omega = 2\pi f \), where \( f \) is the frequency of oscillation. Previously, we found the frequency to be \( 2.00 \text{ Hz} \). Thus, the angular frequency is \{4\pi \text{ rad/s}\}. We use the formula \( \omega = \sqrt{\frac{k}{m}} \) to connect \( k \) with the observed motion. By substituting \( \omega = 4\pi \) and \( m = 0.0500 \text{ kg} \) into the formula, we solve for \( k \):

• \(4\pi = \sqrt{\frac{k}{0.0500}} \)
• Square both sides to eliminate the square root: \((4\pi)^2 = \frac{k}{0.0500} \)
• Multiply through by \(0.0500\): \( k = (4\pi)^2 \times 0.0500 \)
• Resulting in \( k = 7.90 \text{ N/m} \)

This calculation shows how we derive the spring constant from more easily measurable quantities, like the frequency and mass.

The frequency of oscillation in a mechanical system reflects how often the system completes a full cycle of motion per second, measured in hertz (Hz). It is crucial for understanding the behavior of oscillating systems like springs, pendulums, and other similar devices. The original exercise provides the period \( T \), which is the time for one complete cycle, as \(0.500 \text{ s}\). Frequency is the reciprocal of the period, calculated using the simple formula: \( f = \frac{1}{T} \). For this problem:

• Period \( T = 0.500 \text{ s} \)
• Frequency \( f = \frac{1}{0.500} = 2.00 \text{ Hz} \)

High frequency implies that the oscillations occur more quickly, while low frequency means slower oscillations. Understanding frequency helps determine how various factors, like the mass and spring stiffness, influence the motion characteristics.

The amplitude in harmonic motion represents the maximum extent of the oscillation from its equilibrium position. It's a measure of the energy contained in the motion and directly affects how far the oscillating object travels. The maximum speed of the object, given as \( v_{max} = 15.0 \text{ cm/s} \), relates to the amplitude \( A \) through the angular frequency \( \omega \) by the equation \( v_{max} = A \omega \). To find the amplitude:

• Convert maximum speed to meters per second: \( 0.150 \text{ m/s} \)
• Use the previously calculated \( \omega = 4\pi \text{ rad/s} \)
• Rearrange and solve the equation for \( A \): \( A = \frac{v_{max}}{\omega} \)
• Substitute known values: \( A = \frac{0.150}{4\pi} \)
• Result in \( A \approx 0.01194 \text{ m} = 1.19 \text{ cm} \)

The amplitude gives insight into the size of the oscillation, affecting both the potential and kinetic energy in the system.