91Ó°ÊÓ

Hooke's Law describes how springs work in terms of force and displacement. It's crucial for understanding simple harmonic motion. In this scenario, the fisherman hangs a fish, which causes the spring to stretch.
- The key relationship in Hooke's Law is:
\[ F = kx \]
Here, \( F \) is the force applied by the fish (due to gravity), \( k \) is the spring's force constant (also known as the spring constant), and \( x \) is the displacement from the spring's original position.
- How to calculate the force constant? You can rearrange the formula to solve for \( k \):
\[ k = \frac{F}{x} \]
For the fish, \( F \) is determined by multiplying the mass of the fish (65.0 kg) by gravitational acceleration (9.81 m/s²), resulting in 637.65 N. With a displacement \( x = 0.120 \) m, the force constant \( k \) comes out to be approximately 5313.75 N/m.
This tells us how stiff the spring is. A larger \( k \) means a stiffer spring, which requires more force to stretch by the same distance.

The period of oscillation, \( T \), is the time it takes to complete one full cycle of motion. For a spring-mass system, you find \( T \) using the following formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
This means that the period depends on both the mass (\( m \)) and the spring constant (\( k \)).
- Heavier objects or softer springs (lower \( k \)) will result in a longer period of oscillation.
- Conversely, lighter objects or stiffer springs (higher \( k \)) will lead to a shorter period.
In our fish scenario, after substituting the mass (65.0 kg) and the calculated spring constant (5313.75 N/m) into the formula, the oscillation period comes out to approximately 0.697 seconds.
This means every 0.697 seconds, the fish completes one full up-and-down motion in the spring. The idea is that this system repeats its motion predictably, governed by the spring's properties and the object's mass.

Angular frequency, \( \omega \), is a measure of how fast the object oscillates in simple harmonic motion. It's defined as:
\[ \omega = \sqrt{\frac{k}{m}} \]
- It offers an alternative way to think about how quickly the system oscillates, compared to the period.
- In rad/s, it describes how many radians the system moves through per second.
For the oscillating fish and spring, with a spring constant \( k \) of 5313.75 N/m and a fish mass \( m \) of 65.0 kg, the angular frequency calculates to approximately 9.022 rad/s.
This value helps further in determining dynamic aspects of the motion, such as the maximum speed, where:
\[ v_{\text{max}} = \omega A \]
Here, \( A \) is the amplitude of the oscillation. For our fish, pulled an additional 0.050 m down, the maximum speed comes out to approximately 0.451 m/s. Angular frequency gives insights into the pace and liveliness of oscillations, linking directly to energy and velocity in the system.