91Ó°ÊÓ

Potential energy is a type of energy that is stored within an object due to its position or state. In mechanics, it's most commonly associated with gravitational potential energy, which is energy that an object possesses because of its height above a reference point. In our exercise, the gravitational potential energy is initially present due to the block being positioned at a certain height on a frictionless incline.
The formula to calculate gravitational potential energy is: \[ U_g = mgh \]where:

• \( U_g \) is the gravitational potential energy,
• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity, and
• \( h \) is the height of the object above the reference point.

In our problem, the block moves down the incline covering a distance \( d + x \). The height \( h \) is calculated using the incline's angle \( \theta \) by the relation \( h = (d + x) \sin \theta \). Consequently, the potential energy as the block moves is related to both the distance it travels and the angle of the incline.

Kinetic energy is the energy an object possesses due to its motion. When the block slides down the ramp, it transforms potential energy into kinetic energy as it gains speed. Initially, the block has no kinetic energy because it starts at rest. As it slides down the incline, gravitational potential energy is converted to kinetic energy, allowing the block to move faster.The formula for kinetic energy is:\[ KE = \frac{1}{2}mv^2 \]where:

• \( KE \) is the kinetic energy,
• \( m \) is the mass of the object, and
• \( v \) is the velocity of the object.

In our problem, as the block reaches the spring, all of the gravitational potential energy has been converted to kinetic energy. This kinetic energy will eventually be used to compress the spring, illustrating how energy transforms from one form to another.

Spring force is a particular type of force that acts upon a compressed or stretched spring. It follows Hooke’s Law, which states that the force needed to extend or compress a spring by some distance is proportional to that distance. The spring force is responsible for stopping the block by converting kinetic energy into spring potential energy.The formula according to Hooke's Law is:\[ F_s = -kx \]where:

• \( F_s \) is the force exerted by the spring,
• \( k \) is the spring constant, a measure of the stiffness of the spring, and
• \( x \) is the displacement of the spring from its equilibrium position.

When the block compresses the spring, all the moving (kinetic) energy of the block is converted into potential energy of the spring: \[ U_s = \frac{1}{2} k x^2 \]This potential energy is stored in the spring as it compresses, and at the maximum compression, all the energy from the block's descent has transitioned into this form. Thus, the block is momentarily at rest before the process could potentially reverse and move the block back if there were no other forces acting on it.