91Ó°ÊÓ

A frictionless incline means that there is no resistance to slow down the movement of the block. On a frictionless surface, all potential energy is converted into other forms of energy as the object moves, because there is no energy lost to friction. This is an idealized scenario where the surface is entirely smooth. Because of this, the total mechanical energy remains constant.

In the given problem, the block slides down a frictionless incline. Thus, the gravitational potential energy (due to its height at the top of the incline) converts entirely into kinetic energy (as it gains speed) and then to spring potential energy (as it compresses the spring). It is important to understand that without friction, calculations become simpler since there's no need to account for thermal energy loss or other forces.

Spring potential energy is stored energy resulting from the compression or extension of a spring. In the problem, the spring is compressed by the block, storing energy in the form of potential energy which can be expressed by the equation:
\[ U_s = \frac{1}{2} k x^2 \]
Where \( U_s \) is the spring potential energy, \( k \) is the spring constant, and \( x \) is the displacement from the spring's equilibrium position.

The spring constant \( k \) is a measure of the stiffness of the spring. A higher spring constant means a stiffer spring that requires more force to compress. When the block compresses the spring by 21 cm, it converts its kinetic energy into spring potential energy, which is then equal to the gravitational potential energy it had initially at the starting point.

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It can be calculated using the formula:
\[ U_g = mgh \]
Here, \( m \) represents the mass of the object, \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) at Earth's surface), and \( h \) is the height above the reference level.

In this scenario, the block initially has gravitational potential energy at the top of the incline. As it slides down the incline, this potential energy is completely transformed into kinetic energy and eventually into spring potential energy as it compresses the spring at the bottom. The amount of gravitational potential energy corresponds directly to the amount of energy needed to compress the spring.

Kinetic energy is the energy that an object possesses due to its motion. It's given by the formula:
\[ K = \frac{1}{2}mv^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.

In this exercise, the block starts from rest, meaning its initial kinetic energy is zero. As it travels down the frictionless incline, gravitational potential energy is converted into kinetic energy, causing the block to speed up. The block's speed is greatest just before it starts compressing the spring. At this point, nearly all the gravitational potential energy has turned into kinetic energy.

Once the block begins to compress the spring, kinetic energy is again transformed into spring potential energy until the block momentarily comes to a complete stop. At this stopping point, all kinetic energy has been transferred into stored energy within the spring.