91Ó°ÊÓ

The spring constant is a measure of a spring's stiffness. It tells us how much force is needed to compress or stretch the spring by a certain amount. In the given problem, the spring constant is provided as \(250 \, \text{N/m}\). This means that for every meter the spring is compressed or stretched, 250 Newtons of force are required. The spring constant, denoted by \(k\), is crucial in determining how a spring will behave under compression or tension. It is a material and design property unique to each spring. A higher spring constant signifies a stiffer spring, while a lower value indicates a more flexible spring.

Compression distance refers to how much a spring is compressed from its natural length. In this problem, the spring in the pen is initially compressed by \(5.0 \, \text{mm}\) and then further compressed by an additional \(6.0 \, \text{mm}\) for a total of \(11.0 \, \text{mm}\). It is important to convert these measurements into meters when using them in calculations, making it \(0.005 \, \text{m}\) and \(0.011 \, \text{m}\), respectively. Understanding compression distance is essential because it directly affects the force exerted by the spring and the work done, as the force is proportional to the change in compression distance.

The spring force is the force exerted by a spring when it is compressed or stretched. According to Hooke's Law, this force is directly proportional to the displacement of the spring from its rest position and is calculated as \(F = -kx\). Here, \(k\) is the spring constant, and \(x\) is the displacement (compression or stretch distance). The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement. In our context, the spring force resists the compression as the pen is made ready for writing. This resistance is precisely what allows the pen's mechanisms to function effectively.

The work-energy principle states that work done by forces on an object will result in a change in energy of that object. For springs, the work done is related to the potential energy stored in the spring. The work done by a spring force can be calculated with the formula: \[ W = \frac{1}{2}k(x_1^2 - x_2^2) \] This formula considers the squares of the initial \(x_1\) and final \(x_2\) compression distances. The work is negative in this problem, meaning the spring's force is opposing the compression. This concept helps us analyze mechanical systems where energy is stored and released, such as in pen mechanisms or shock absorbers. Understanding this principle allows us to comprehend how energy conservation and transformation occur in physical systems.