91Ó°ÊÓ

Hooke's Law describes the behavior of springs. It states that the force needed to compress or extend a spring by some distance is proportional to that distance. Mathematically, it is represented as:
\( F = kx \)
Here, \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement. In this exercise, the weight of the stone compressed the spring, and we used Hooke's Law to find the spring constant. By rearranging the formula to \( k = \frac{F}{x} \), and substituting the known values, we calculated the spring constant as 784 N/m. Understanding Hooke's Law is crucial for solving problems involving springs. It shows that the force required to compress a spring increases linearly with the distance compressed.

Elastic potential energy is the energy stored in an elastic object, like a spring, when it is compressed or stretched. The formula for elastic potential energy is:
\( U = \frac{1}{2}kx^2 \)
Here, \( U \) is the elastic potential energy, \( k \) is the spring constant, and \( x \) is the compression or extension distance. In our exercise, the spring was further compressed by 0.30 m, making the total compression 0.40 m. We used the spring constant calculated earlier and plugged in the values to find the elastic potential energy. By substituting \( k = 784 \) N/m and \( x = 0.40 \) m in the formula, we got the elastic potential energy as 62.72 J. This concept helps us understand how energy is stored and transferred within a spring system.

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is given by the formula:
\( U_g = mgh \)
In this formula, \( U_g \) represents gravitational potential energy, \( m \) is mass, \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the height. In our problem, we calculated the change in gravitational potential energy by determining the height the stone would rise after the spring is released. We equated the elastic potential energy (62.72 J) to the change in gravitational potential energy and solved for \( h \). This gave us a height of 0.80 m. This conversion between energy forms shows the principle of conservation of energy.

Spring compression is the act of pressing a spring together, reducing its length. The amount of compression affects the force stored and the potential energy within the spring. When the stone was placed on the spring, it compressed initially by 0.10 m. This displacement was crucial to calculating the spring constant using Hooke’s Law. Further, the stone was compressed an additional 0.30 m, making the total compression 0.40 m. This total compression was used to determine the elastic potential energy before the stone was released. Understanding the concepts of compression and displacement is essential for solving problems involving springs.

Physics problem-solving often involves multiple steps and the application of different principles. Here's a structured approach we used for this exercise:

• Identify given values and what needs to be found.
• Use fundamental laws and formulas, like Hooke's Law for spring constant.
• Calculate the required quantities step-by-step, such as spring constants and potential energies.
• Apply energy conservation principles to find relationships between different forms of energy.
• Recheck units and calculations to ensure accuracy.

This structured method helps break down complex problems into manageable steps, leading to accurate solutions. Practice and familiarity with key concepts enhance problem-solving skills in physics.