91Ó°ÊÓ

When a spring is pressed or stretched from its equilibrium position, it stores energy due to this deformation. This is what we call "Elastic Potential Energy". It's the energy stored in objects that can be stretched or compressed, much like a compressed spring.
In the case of the exercise at hand, we are dealing with a spring compressed by a certain amount. The formula for elastic potential energy is:

• \( U = \frac{1}{2} k x^2 \)
• where \( U \) is the elastic potential energy, \( k \) is the spring constant, and \( x \) is the compression (or extension) distance.

In this exercise, the spring constant \( k \) is 19.6 N/cm, which is converted to 1960 N/m for calculations in consistent SI units. The spring is compressed by 0.20 meters. Plugging into the formula gives:

• \( U = \frac{1}{2} \times 1960 \, \text{N/m} \times (0.20 \, \text{m})^2 = 39.2 \, \text{J} \)

So, the elastic potential energy stored in the spring is 39.2 Joules.

Gravitational Potential Energy is related to the position of an object in a gravitational field, particularly how high it is from a given reference point.
As the block slides up the incline, driven by the energy from the compressed spring, its gravitational potential energy increases.
The gravitational potential energy can be calculated using the formula:

• \( U_g = mgh \)
• where \( U_g \) is gravitational potential energy, \( m \) is mass, \( g \) is acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)), and \( h \) is the height above the reference point.

For this problem, the spring's total energy was initially stored as elastic potential energy and then transferred into gravitational potential energy as the block moves up the incline.
The change in gravitational potential energy is essentially the same amount as the spring's initial elastic potential energy, which was determined to be 39.2 Joules.

Energy Conservation is the principle stating that the total energy in an isolated system remains constant. In simpler terms, energy cannot be created or destroyed but can only change from one form to another.
For our inclined plane setup, energy from the compressed spring is conserved and transformed into gravitational potential energy as the block ascends.
This conservation can be seen in how both the elastic potential energy and the gravitational potential energy achieve a balance in this system.

• The initial elastic potential energy was 39.2 Joules.
• This entire amount is converted to gravitational potential energy as the block climbs the incline.

To find how far the block travels along the incline, we equate the initial energy with the gravitational potential energy:

• \( mgh = 39.2 \, \text{J} \), leading to \( h = \frac{39.2}{mg} \).

Using the trigonometric relationship \( h = d \sin 30^{\circ} \), where \( d \) is the distance along the incline, we determine that the distance traveled along the incline is 4.00 meters. This perfectly illustrates the conservation of energy in action.