91Ó°ÊÓ

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. In the context of Hooke's Law, it is pivotal to determining how easily a spring (or elastic cord) stretches or compresses in response to an applied force. The stiffer the spring, the higher the spring constant.

• The spring constant \( k \) is expressed in Newtons per meter (\( \ ext{N/m} \)).
• A large \( k \) value indicates a stiff spring that does not stretch much under force.
• A smaller \( k \) value implies a more elastic spring that extends easily under force.

Hooke's Law, formulated as \( F = k \times \Delta x \), highlights the linear relationship between the force \( F \) exerted on a spring and the extension \( \Delta x \). When calculating \( k \), one must rearrange this equation to \( k = \frac{F}{\Delta x} \), allowing one to find how resistant the material is to deformation when an external force is applied.

Force and extension are fundamental to understanding how springs operate according to Hooke's Law. Here, force refers to any external push or pull applied to the spring, measured in Newtons (\( \mathrm{N} \)). Extension or compression, on the other hand, represents the change in length a spring undergoes.
In our exercise, we had two forces applied to the elastic cord, 75 N and 180 N:

• Under a force of 75 N, the cord length remained 65 cm, meaning no extension occurred.
• Under a force of 180 N, the cord extended to 85 cm.

The change in length, denoted as \( \Delta x \), can be found by subtracting the initial length from the new length after the force is applied. Here, \( \Delta x = 85 \text{ cm} - 65 \text{ cm} = 20 \text{ cm} = 0.20 \text{ m} \).
Accurately calculating the extension is essential as it determines the proportional response or deformation of the spring under the applied force.

Elasticity is the tendency of a material to return to its original form after being deformed by an external force. In the context of springs, elasticity plays a critical role in explaining how springs behave under tension or compression.

• Materials with high elasticity can stretch significantly and then regain their original shape.
• Low elasticity materials might permanently deform under force.
• The elastic limit is the maximum extent to which the material can be stretched without suffering permanent deformation.

For the elastic cord in the problem, its ability to return to the original length upon removal of the weight demonstrates its elasticity qualities. By applying Hooke's Law and calculating the spring constant \( k \), we can quantify how elastic a material is. A constant 900 \( \mathrm{N/m} \) indicates a relatively stiff cord that requires more force for a significant extension, showing good elasticity but with a high resistance.