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The spring constant, denoted as \( k \), is a critical parameter in Hooke's Law, representing the stiffness of a spring. A higher spring constant means a stiffer spring that requires more force to achieve the same stretch. The formula for Hooke's Law is \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the distance stretched.
To find the spring constant, you rearrange the equation to solve for \( k \) as \( k = \frac{F}{x} \). In the given exercise, a force of 2 N stretches the rubber band by 0.02 m. By substituting these values into the equation, we get \( k = \frac{2}{0.02} = 100 \, \mathrm{N/m} \).
This tells us that for every meter of stretch, 100 Newtons of force are needed. Understanding this concept allows us to predict how much force is required for any given stretch for the same spring or rubber band.

Calculating the work done on a spring involves understanding energy transfer. When you stretch or compress a spring, you are transferring energy into it, which is stored as potential energy. The work done, \( W \), when stretching or compressing a spring can be calculated using the formula:

• \[ W = \frac{1}{2} k x^2 \]

Here, \( k \) is the spring constant, and \( x \) is the distance stretched or compressed.
In the given problem, after calculating the spring constant \( k = 100 \, \mathrm{N/m} \) and the new extension distance \( x = 0.04 \, \mathrm{m} \), we substitute back into the work formula to calculate the work done: \( W = \frac{1}{2} \times 100 \times (0.04)^2 = 0.08 \, \mathrm{J} \).
This value, 0.08 joules, represents the energy required to stretch the rubber band from its original length to 4 cm. Understanding how to calculate work done helps in recognizing how energy is stored and transferred in mechanical systems.

Physics problem solving requires a step-by-step approach to ensure that all concepts are correctly understood and applied. When dealing with problems involving Hooke's Law and similar concepts:

• Identify what is given and what needs to be found.
• Use the relevant physics laws and formulas, like Hooke's Law and the work done formula in this case.
• Start by calculating needed constants, such as the spring constant, based on given data.
• Apply these constants to find unknown values, like the new stretch distance or the work done.

For instance, in the rubber band problem, we first determined the spring constant using the initial conditions. With this spring constant, we could then find the new extension when a different force was applied, finally allowing us to calculate the work done. Breaking down each step ensures clarity and reduces the possibility of errors, providing a better grasp of physics concepts.