91Ó°ÊÓ

The spring constant, symbolized as \( k \), is a measure of a spring's stiffness. It's defined as the force required to stretch or compress the spring by a unit distance. In our exercise, the spring constant is given as \( k = 800 \text{ N/m} \). This means it takes 800 Newtons of force to compress or extend the spring by a single meter. The higher the spring constant, the stiffer the spring is, and the more force is required for displacement. Understanding the spring constant is crucial for solving problems related to spring potential energy as it directly influences how much energy the spring can store. In formulas, it plays a key role in determining both potential energy stored and the compression distance of the spring.

Energy conservation is a fundamental principle stating that energy cannot be created or destroyed, only transformed from one form to another. In mechanical systems, this often involves converting between kinetic, potential, and elastic (spring) energy. In the given exercise, energy conservation helps us understand how the initial gravitational potential energy (when a book is held above the spring) is transformed into spring potential energy as the spring compresses. This conservation allows us to equate the potential energy due to gravity with the elastic potential energy of the spring at maximum compression, simplifying calculations and ensuring accuracy.

Compression distance, denoted by \( x \), is an essential measurement in determining how much a spring is compressed or stretched. For a given amount of potential energy, the spring's compression distance can be calculated using the formula \( PE = \frac{1}{2} k x^2 \). By rearranging this formula, we solved for \( x \) to find how far the spring must be compressed to store 1.20 Joules of energy:\[ x = \sqrt{\frac{2 \times PE}{k}} \]Using the spring constant \( k = 800 \text{ N/m} \) and \( PE = 1.20 \text{ J} \), we get \( x \approx 0.0546 \text{ meters} \). This calculation shows that a small compression distance can correspond to a significant amount of stored energy, especially in springs with high spring constants like ours.

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It is given by the formula \( PE_{gravity} = mgh \), where \( m \) is the mass, \( g \) is the gravitational acceleration, and \( h \) is the height above a reference point. In the context of our exercise, when a book is placed on a vertically oriented spring, its GPE is converted to spring potential energy at maximum compression. The equation \( mgh = \frac{1}{2} k x^2 \) signifies this energy transformation. Given the book's mass of 1.60 kg, \( g = 9.8 \text{ m/s}^2 \), and equating this with the spring's energy equation, we determined the compression distance at maximum spring compression is approximately \( 0.0392 \text{ meters} \). Understanding GPE's role in this process helps clarify how potential energy is managed in mechanical systems and offers insight into designing practical applications.