91Ó°ÊÓ

Hooke's Law is a fundamental concept in physics that provides insight into how springs behave. It describes the relationship between the force applied to a spring and the resulting displacement or stretch from its natural length. Simply put, when you pull or push a spring, it resists with a force that is directly proportional to the amount you stretch or compress it. This relationship is described by the formula:
\[ F = kx \]

• \( F \) is the force applied on the spring (in newtons, N).
• \( k \) is the spring constant, a measure of the spring's stiffness (in newtons per meter, N/m).
• \( x \) is the displacement of the spring from its equilibrium position (in meters, m).

Changes in the force or displacement will affect the spring's behavior. This is why Hooke's Law is crucial for determining how much force is necessary to achieve a specific extension or compression. Knowing Hooke's Law helps us solve problems like the one in our exercise effectively.

The spring constant, denoted by \( k \), is a crucial parameter in understanding a spring's characteristics. It tells us the amount of force needed to stretch or compress a spring by one unit of length. It also indicates how stiff or flexible a spring is.
For example, in the exercise provided, the spring has a spring constant of 150 N/m. This means if you were to stretch or compress the spring by 1 meter from its natural length, it would require a force of 150 newtons.
Here are some essential points about the spring constant:

• A larger spring constant signifies a stiffer spring that requires more force to stretch.
• A smaller spring constant means a more flexible spring that requires less force for the same amount of stretch.
• The spring constant is unique to each spring and depends on its material and structure.

Understanding the spring constant helps in predicting how a spring will react under varying forces and displacements, making it possible to calculate precisely activities like the work done on a spring.

The work done in stretching or compressing a spring is a critical measure of energy transfer. This is quantified by the work done formula in the context of springs which is derived from integrating Hooke's Law. The formula is:
\[ W = \frac{1}{2}k(x_f^2 - x_i^2) \]
In this formula:

• \( W \) is the work done on the spring, calculated in joules (J).
• \( x_i \) and \( x_f \) are the initial and final displacements of the spring from its rest position, respectively.
• \( k \) remains the spring constant, which indicates the inherent stiffness.

To solve the exercise:

• The initial displacement \( x_i = 0.10 \, \text{m} \) and final displacement \( x_f = 0.30 \, \text{m} \)
• Substituting these into the formula: \[ W = \frac{1}{2} \times 150 \times (0.30^2 - 0.10^2) \] results in 6 joules.

This measure of work represents the energy required to transition the spring from its initial to its final position, under the forces applied described by the spring constant.