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Potential energy is the energy stored within an object because of its position or configuration in a force field, commonly the gravitational field. When an object is held at a height above the ground, it has gravitational potential energy. For the fish attached to the spring, potential energy is initially stored due to its height before it descends.

The formula to calculate gravitational potential energy is given by:

• \( PE = mgh \)

Where:

• \( m \) is the mass of the object (3.00 kg for the fish).
• \( g \) is the acceleration due to gravity (approx. \( 9.81 \, \mathrm{m/s^2} \)).
• \( h \) is the height or displacement of the object.

As the fish descends, this gravitational potential energy is converted into other forms of energy, specifically kinetic energy and elastic potential energy in the spring.

Kinetic energy is the energy an object possesses due to its motion. Once the fish is released from rest, it starts accelerating downwards due to gravitational pull, gaining kinetic energy. This increase in kinetic energy corresponds to the loss of potential energy as the fish moves down.

The kinetic energy at any point can be calculated using:

• \( KE = \frac{1}{2}mv^2 \)

Where:

• \( m \) is the mass of the object (3.00 kg).
• \( v \) is the velocity of the object.

Initially, the kinetic energy is zero because the fish starts from rest. As it descends, the potential energy diminishes while kinetic energy increases, highlighting the energy transformation in accordance with the law of conservation of energy.

Elastic potential energy refers to the energy stored in elastic materials as they are stretched or compressed. A spring is a classic example where energy is stored within it when stretched or compressed. As the fish descends and the spring stretches, energy is stored as elastic potential energy.

The formula for elastic potential energy is:

• \( EPE = \frac{1}{2}kx^2 \)

Where:

• \( k \) is the force constant of the spring (900 N/m).
• \( x \) is the displacement or change in the spring’s length from its equilibrium position.

This energy plays a key role when assessing the total mechanical energy at any point during the fish's descent, ensuring that the energy changes comply with the principle of energy conservation.

The force constant, also known as the spring constant, is a measure of a spring’s stiffness. It is denoted by \( k \) and defines how much force is required to stretch or compress the spring by a unit of distance.

The greater the force constant, the stiffer the spring. For the exercise, the given spring has a force constant of 900 N/m, meaning for every meter of displacement, the force exerted by the spring is 900 Newtons.

Understanding the force constant is crucial when calculating the elastic potential energy and the forces involved when the spring is stretched or compressed by the fish's weight.

Spring mechanics involves studying the behavior of springs as they are subject to forces, either stretching or compressing. The process is governed by Hooke's Law, which states that the force required to compress or stretch a spring is directly proportional to the displacement from its original (equilibrium) position.

Mathematically, Hooke’s Law is represented as:

• \( F = kx \)

Where:

• \( F \) is the force applied by or on the spring.
• \( k \) is the force constant or spring constant.
• \( x \) is the displacement.

In the example of the fish, as it hangs and stretches the spring, the force exerted by the fish’s weight is balanced by the spring's restoring force, demonstrating fundamental principles of spring mechanics. This understanding also assists in calculating potential and kinetic energies as well as determining maximum and instantaneous speeds as the fish descends.