91Ó°ÊÓ

Gravitational potential energy (GPE) is an essential concept in physics and is the energy stored in an object positioned at a height. When an object is elevated in a gravitational field, it possesses potential energy due to its height above the Earth's surface. This energy depends on three factors: the object's mass, the height it is lifted to, and the gravitational pull.

GPE can be calculated using the formula: \[GPE = mgh\]where:

• \(m\) is the mass of the object in kilograms (kg).
• \(g\) is the acceleration due to gravity, typically \(9.81\, ext{m/s}^2\) on Earth.
• \(h\) is the height in meters (m) above the reference point.

In the context of our exercise, as the block of mass \(2\, ext{kg}\) is released, the initial gravitational potential energy of the block gets entirely transformed into other forms of energy following the conservation of energy principle. This conversion allows us to calculate how much the spring will compress when the block falls on it.

Spring potential energy is the energy stored in a spring when it is compressed or stretched from its equilibrium position. This type of potential energy obeys the conservation of energy principle, meaning the energy put into the spring will be stored until it's released.

The potential energy stored in a spring can be calculated with the formula:\[rac{1}{2} k x^2\]where:

• \(k\) is the spring constant in Newtons per meter (N/m), which measures the spring's stiffness.
• \(x\) is the displacement from the spring's equilibrium position in meters (m).

In our problem, as the falling block compresses the spring, its gravitational potential energy converts to the spring's potential energy. At maximum compression, the energy balance helps us find the compression distance \(x\), reflecting the amount of energy stored in the spring.

Hooke's Law is a fundamental principle for describing the behavior of springs. It defines how forces affect a spring, specifically stating how much a spring will deform under a given force. Hooke's Law shows that the force needed to compress or stretch a spring by some distance is proportional to that distance.

Mathematically, Hooke's Law is expressed as:\[F = -kx\]where:

• \(F\) represents the force applied on the spring in Newtons (N).
• \(k\) is the spring constant (N/m), a measure of the spring's stiffness.
• \(x\) is the compression or elongation distance from its natural length.

In the context of our problem, Hooke's Law supports the calculation of energy transformations in the spring. It enables us to understand how much a spring compresses when subjected to the force exerted by the weight of the block. The spring's stiffness (represented by \(k = 200\, ext{N/m}\)) and the force from the block determine the spring's compression in this scenario.