91Ó°ÊÓ

The spring constant, denoted as \( k \), is a fundamental concept in Hooke's Law. It represents the stiffness of the spring or elastic object. The higher the spring constant, the stiffer the spring is. Hooke’s Law is mathematically expressed as \( F = kx \), where \( F \) is the force applied on the spring, \( k \) is the spring constant, and \( x \) is the displacement from its equilibrium position.
Understanding the spring constant is crucial for predicting how a spring will behave under various forces. A spring with a large \( k \) will require a significant force to stretch or compress it, compared to a spring with a smaller \( k \).
For instance, in the context of the balloon launching problem, the spring constant is given as \( 100 \, \text{N/m} \). This means for every meter the hose is stretched, a force of \( 100 \, \text{Newtons} \) is applied. This information helps us determine the behavior of the catapult and how effectively it can launch a balloon.

When a spring or elastic object returns to its equilibrium position after being stretched or compressed, it performs work. The formula to calculate the work done by a spring is \( W = \frac{1}{2} k x^2 \). Here, \( W \) represents the work done, \( k \) is the spring constant, and \( x \) is the maximum displacement.
This equation highlights that the work done depends on both the square of the displacement and the spring constant. The further the spring is compressed or stretched, and the stiffer the spring (higher \( k \)), more work is done.
In the example given, the hose is stretched at \( 5.00 \, \text{m} \), so the work done by the hose when it releases is calculated as \( 1250 \, \text{Joules} \). Understanding this concept allows us to calculate the energy transfer involved when a spring-like component is used in devices such as catapults or other mechanisms employing elastic potential.

Elastic potential energy is the energy stored in a spring or elastic material when it is stretched or compressed. This stored energy is released as kinetic energy when the spring returns to its original shape. The amount of elastic potential energy is also given by the formula \( U = \frac{1}{2} k x^2 \), where \( U \) is the elastic potential energy, \( k \) is the spring constant, and \( x \) is the displacement.
The concept of elastic potential energy helps us understand how springs function as energy storage devices. When the balloon in the pouch is stretched, the hose stores energy that will be converted to kinetic energy to propel the balloon when released.
In our context, elastic potential energy calculations mirror those of the work done by the spring. So when the hose is stretched to \( 5.00 \, \text{m} \), it holds \( 1250 \, \text{Joules} \) of elastic potential energy. This energy is crucial for tasks where rapid release and conversion of energy are needed, like launching projectiles in this example.