91Ó°ÊÓ

When two masses collide and stick together, they create an oscillating system, especially if one mass is attached to a spring, as seen in this exercise. To better understand the oscillation, we calculate its frequency. Oscillation frequency is how often the system goes back and forth in one second. It's connected to the angular frequency, denoted as \(\omega\), which is given by the formula:

• \(\omega = \sqrt{\frac{k}{m}}\)

Here, \(k\) represents the spring constant (80 N/m), and \(m\) is the combined mass (20 kg). After doing the math: \(\omega = 2.00\) rad/s.

The frequency \(f\), measured in hertz, can be found using \(f = \frac{\omega}{2\pi}\). This gives us a value that tells us how often the spring goes through a complete cycle. Understanding this helps anticipate how quickly oscillations occur after the collision. For instance, with \(\omega = 2.00\) rad/s in this scenario, the cycle repeats approximately every 3.14 seconds, equating to a frequency of around 0.318 Hz.

Kinetic energy, often symbolized as KE, is the energy an object possesses due to its motion. In this context, the system's kinetic energy right after the collision is crucial for determining the oscillation's amplitude. Before any oscillation occurs, the kinetic energy of the combined masses right after collision is calculated using:

• \( KE = \frac{1}{2}mv_f^2 \)

Given that \(m\) stands for the system's total mass (20 kg) and \(v_f\) is the velocity (1 m/s), the kinetic energy is \(10\) Joules.

This energy is transferred to the spring and converted into potential energy at the oscillation's maximum displacement (amplitude). By using energy conservation:

• \(\frac{1}{2}mv_f^2 = \frac{1}{2}kA^2\)

we find the amplitude, \(A = 0.50\) m. This ensures that we understand how far the spring stretches or compresses at its peak energy state, giving a tangible measure of motion range.

The spring constant, usually denoted by \(k\), is a measure of a spring's stiffness. It plays a major role in determining how the system behaves after the collision. In our example, the spring constant is specified as 80 N/m, indicating a relatively stiff spring.

The spring constant affects both the oscillation frequency and the system's energy dynamics. It is present in both the angular frequency formula \(\omega = \sqrt{\frac{k}{m}}\) and the amplitude calculation from the kinetic and potential energy relation. A larger \(k\) means the spring is harder to compress or stretch, leading to faster oscillations. Conversely, a smaller \(k\) would mean more flexible oscillations, with cycles taking longer.

In practical terms, understanding \(k\) helps predict how the oscillating masses will behave once they've gained velocity and attached to a spring. It allows us to anticipate physical behaviors such as the period of oscillation (here, \(\pi\) seconds for a full cycle), which is critical when designing systems like vehicle suspension or measuring equipment where oscillations serve a functional purpose.