91Ó°ÊÓ

Potential Energy is the energy stored in an object due to its position or configuration. In the context of the exercise, we are looking at the potential energy stored in a spring. When a spring is stretched or compressed, it has potential energy because it can perform work if allowed to return to its relaxed, or equilibrium, position.

The spring constant, denoted as \(k\), indicates how stiff the spring is. A larger \(k\) means a stiffer spring, which requires more force to stretch or compress by a given length. The potential energy \(U\) in a spring is given by:

• \( U = \frac{1}{2} k x^2 \)

where \(x\) is the displacement from the spring's relaxed length. In this exercise, the spring is stretched by 10 cm initially, which equates to 20 Joules of potential energy.

As the block moves, this energy changes as the spring shifts back to its relaxed position. Calculating potential energy at each stage helps determine the changes in kinetic energy and the effects of work done by other forces.

Friction is a force that opposes motion, converting kinetic energy into thermal energy. The work done by friction is a vital factor in determining the net energy available to convert into kinetic energy.

In this exercise, the block experiences a constant frictional force of 80 N as it slides. The work done by this frictional force \(W_f\) over a distance \(d\) is calculated using:

• \( W_f = f \cdot d \)

where \(f\) is the frictional force, and \(d\) is the distance moved by the block.

As the block moves 2 cm from its initial position, the work done by friction is 1.6 Joules. When the block returns to the relaxed position of the spring, the total work done by friction amounts to 8 Joules. This reduction in mechanical energy due to friction affects the amount of kinetic energy the block possesses at various points in its motion.

The spring constant \(k\) is a measure of a spring’s stiffness, indicating the force needed to stretch or compress the spring by one unit of length. It reflects the spring's ability to store potential energy. The units of the spring constant are typically Newtons per meter (N/m).

In the given exercise, the spring constant \(k\) is 4000 N/m. This indicates a rather stiff spring, requiring significant force to displace it. The stiffness of the spring directly influences the potential energy stored or released during compression or extension.

• A higher spring constant means the spring stores more energy for a given displacement.
• Determining \(k\) allows us to calculate potential energy and relate it to kinetic energy and work done by other forces.

Understanding the spring constant is essential to solving problems involving forces and energy transformations in spring-related exercises. It helps clarify how energy is transferred between potential and kinetic forms in these systems.