91Ó°ÊÓ

Gravitational potential energy is a type of energy that depends on an object's position relative to the ground. When an object is lifted to a height, it gains gravitational potential energy. This energy is calculated using the formula:

• \( ext{Gravitational Potential Energy} = mgh \)

where:

• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity \( (9.8 \, \text{m/s}^2) \),
• \( h \) is the height above the reference point.

When the object in our exercise, with a mass \( M \), descends by a distance \( l \), it loses gravitational potential energy. This loss is expressed as:

• \( Mgl \)

This indicates the energy lost to the surroundings as the object moves downward due to gravity.

Hooke's Law relates the force necessary to either compress or extend a spring to the distance it is compressed or stretched. It's a fundamental principle that explains how springs work. The formula can be expressed as:

• \( F = kx \)

where:

• \( F \) is the force applied to the spring,
• \( k \) is the spring constant, a measure of the spring's stiffness,
• \( x \) is the displacement of the spring from its original length.

In our scenario with the wire, applying Hooke's Law involves the mass pulling on the wire causing elongation. The force exerted by the mass, \( F = Mg \), results in the elongation \( l \). By rearranging Hooke’s Law, we find the spring constant \( k \) for the wire:

• \( k = \frac{Mg}{l} \)

This equation helps us understand how the wire behaves similarly to a spring, stretching due to the force applied by the weight.

The spring constant, represented by \( k \), is a crucial factor that describes how responsive a spring is to an applied force. A higher spring constant means the spring is stiffer and requires more force to stretch or compress. Conversely, a lower spring constant indicates a less stiff spring, which is easier to deform.
In the context of a wire behaving like a spring, the spring constant can be determined using Hooke's Law:

• \( k = \frac{F}{l} \)

In our specific example:

• \( k = \frac{Mg}{l} \)

This shows the wire's resistance to deformation. Understanding the spring constant helps predict how much the wire will stretch given a particular force, providing insight into the material properties of the wire.

Energy conservation is a fundamental principle in physics stating that energy cannot be created or destroyed, only transformed from one form to another. In the scenario with the suspended mass and the wire, energy conservation plays a crucial role. As the mass descends and stretches the wire, gravitational potential energy is converted into elastic potential energy stored in the wire.
The initial gravitational potential energy lost by the mass as it falls a distance \( l \) is:

• \( Mgl \)

Part of this energy is stored as elastic potential energy in the wire described by the formula:

• \( \frac{1}{2} kl^2 \)

After substituting the spring constant \( k \) we derived earlier \( \left( k = \frac{Mg}{l} \right) \), the elastic potential energy simplifies to:

• \( \frac{Mgl}{2} \)

This demonstrates the conservation of energy where the total initial gravitational energy is shared between lost energy and stored elastic energy, maintaining the overall energy balance.