91Ó°ÊÓ

When a textbook compresses a spring, energy is stored in the spring as potential energy. The potential energy in a spring is given by the formula:

• \[PE = \frac{1}{2}kx^2\]
• where \(k\) is the spring constant and \(x\) is the compression distance.

For our exercise, the textbook compresses the spring by 0.250 meters. With a spring constant \(k\) of 250 N/m, we can calculate the initial spring potential energy that the textbook has stored in the spring.
Plugging in our values, we find the potential energy to be:

• Energy = \(\frac{1}{2} \times 250 \times (0.250)^2\)
  
• This calculates to an energy of 7.8125 Joules.

This energy is crucial, as it will set the textbook in motion when it is released, converting this stored potential energy into other forms of energy as it moves.

As the textbook slides across the table, a force called kinetic friction comes into play. This force opposes the motion of the textbook, working to slow it down. Kinetic friction depends on:

• The normal force \(F_N\), which is the force perpendicular to the surface.
  
• The coefficient of kinetic friction \(\mu_k\), a measure of how much frictional force exists between the surfaces involved.

In our problem, the normal force is caused by gravity acting on the textbook and can be calculated as:
\[ F_N = mg \], where \(m\) is the mass and \(g\) is the acceleration due to gravity, approximated at 9.8 m/s². For the textbook:

• \( F_N = 2.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 \)
  
• Giving a normal force of 24.5 N.

The work done by the kinetic friction is:

• \[ W_f = \mu_k \times F_N \times d \]
• with \(\mu_k = 0.30\), \( F_N = 24.5 \, \text{N} \), and \(d\) the distance traveled.

In this scenario, we witness a classic example of energy conversion. Initially, all energy is stored as spring potential energy when the spring is compressed. As the spring is released, this potential energy converts into kinetic energy, propelling the textbook. This energy conversion process is influenced by the work-energy theorem, which states:

• The work done on an object results in a change in its kinetic energy.

The initial potential energy of the spring (7.8125 Joules) is converted to kinetic energy just as the textbook begins to move. However, as soon as the book slides on the tabletop, kinetic friction starts converting this kinetic energy into thermal energy, slowing down the textbook.
The work done by kinetic friction is calculated by:

• \[ W_f = \mu_k \times F_N \times d \].

Since the kinetic energy becomes zero when the textbook stops, the initial potential energy entirely converts into the work done against friction, which determines the stopping distance \(d\). Understanding these energy transformations and calculations helps solve the problem using the work-energy theorem effectively.