91Ó°ÊÓ

In a spring-mass system, potential energy plays a key role when an external force compresses a spring. The spring stores this energy as 'potential energy'. The potential energy in an elastic spring is determined using the formula:

• \( E_{s} = \frac{1}{2} k x^2 \)

Here, \( k \) is the spring constant, which indicates the stiffness of the spring, and \( x \) is the compression distance. Substituting the values from the problem, we have \( k = 400 \mathrm{~N/m} \) and \( x = 0.10 \mathrm{~m} \) (since 10 cm = 0.10 m). Thus, your calculation would look like this:

• \( E_{s} = \frac{1}{2} \times 400 \times (0.10)^2 = 2 \mathrm{~J} \)

This means that the spring stores 2 joules of potential energy when it is fully compressed. Each element within the equation reflects how much the spring is able to store based on its physical properties.
Understanding potential energy in springs allows us to predict how energy stored can later contribute to the motion of the mass once it is released.

Kinetic energy arises when the potential energy stored in the spring is converted into motion energy once the object is released. The kinetic energy of an object depends on its mass and velocity, and it is described by the equation:

• \( KE = \frac{1}{2} mv^2 \)

For the given problem, where the block has a mass \( m \) of \( 1 \mathrm{~kg} \) and the potential energy of the spring is converted to kinetic energy at 2 joules, we can use the formula to solve for velocity \( v \):

• Plug \( KE = 2 \mathrm{~J} \) and \( m = 1 \mathrm{~kg} \) into the equation: \( 2 = \frac{1}{2} \times 1 \times v^2 \)
• Rearrange the equation to solve for \( v^2 \): \( v^2 = 4 \)
• Take the square root: \( v = \sqrt{4} = 2 \mathrm{~m/s} \)

This tells us that when all the potential energy is converted to kinetic energy, the velocity of the block reaches \( 2 \mathrm{~m/s} \). It illustrates how energy conversions within the system determine the object's motion.

The principle of energy conservation is fundamental when analyzing spring-mass systems. It states that energy cannot be created or destroyed, only transformed from one form to another. In the context of our problem:

• The potential energy stored in the compressed spring is converted entirely into kinetic energy of the block.
• Throughout this process, the total mechanical energy within the system remains constant.

By calculating the potential energy, we knew that 2 joules were stored in the spring. Upon release, this 2 joules became kinetic energy, allowing us to determine the velocity of the block using energy conservation. The transformation looks like:

• Potential Energy (Spring) \( \rightarrow \) Kinetic Energy (Block)

If there were no external forces (like friction), the energy would retain its magnitude throughout the motion. This demonstrates a basic understanding of energy conservation, highlighting how shifts in energy states govern the dynamics of motion in spring-mass systems.