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The spring stiffness constant, often represented by the symbol \( k \), is a measure of the rigidity of the spring. In other words, it tells us how much force is needed to stretch or compress a spring by a unit distance. According to Hooke's Law, the force \( F \) required to change the length of a spring by a distance \( x \) is given by:\[ F = kx \]In the case of the fisherman's scale, when a 2.4 kg fish is hung, the spring stretches 3.6 cm or 0.036 m. The weight of the fish, which acts as the force here, is calculated by \( F = mg \), where \( m \) is mass and \( g \) is acceleration due to gravity (9.8 m/s²).

• Force, \( F = 2.4 \times 9.8 = 23.52 \text{ N} \).

Now, by substituting into Hooke's Law:

• \( 23.52 = k \times 0.036 \), solve for \( k \) to find \( k = 653.33 \text{ N/m} \).

This constant implies that for every meter the spring stretches, it requires a force of 653.33 N, characterizing its stiffness.

The amplitude of oscillation refers to the maximum distance from the equilibrium position that the object moves during oscillation. In the context of a mass-spring system like the fisherman's scale, it shows how far the spring stretches or compresses beyond its natural resting point.
When the fisherman's fish is further pulled down by 2.1 cm (or 0.021 m) from its equilibrium position, this distance becomes the amplitude. Fundamentally, amplitude reflects how energetic the oscillation is, signifying the extreme points of the system's motion cycle.

• Amplitude, \( A = 0.021 \text{ m} \).

Knowing the amplitude is crucial for understanding the full extent of the oscillatory motion and predicting the forces involved at maximum displacement. It also impacts the potential energy stored in the spring at the points of maximum compression or stretch.

The frequency of oscillation is a measure of how many oscillations occur in a unit of time, typically expressed in hertz (Hz), where 1 Hz equals one cycle per second. This concept is integral to understanding the dynamics of any periodic motion, including the oscillation of a spring-mass system.
For a mass attached to a spring, the frequency \( f \) is calculated using the formula:\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]where \( k \) is the spring stiffness constant and \( m \) is the mass. By substituting the previously calculated \( k = 653.33 \text{ N/m} \) and \( m = 2.4 \text{ kg} \):

• \( f = \frac{1}{2\pi} \sqrt{\frac{653.33}{2.4}} \approx 2.62 \text{ Hz} \).

This frequency indicates that the system completes roughly 2.62 cycles per second. It helps in understanding the behavior of the oscillating system and how quickly the fish moves back and forth. Higher frequencies imply more rapid oscillations, while lower frequencies indicate slower movements.