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When a block is compressed against a spring, it stores energy called spring potential energy. This energy is determined by the formula \( U_s = \frac{1}{2} k x^2 \). Here, \( k \) is the spring constant and \( x \) is the distance the spring is compressed or stretched. The spring constant is a measure of the spring's stiffness, and a higher \( k \) means a stiffer spring.
The term \( \frac{1}{2} \) in the formula represents the distribution of force as the spring is compressed or released. As the block pushes against the spring to a compression of 0.220 m, the spring potential energy becomes stored, ready to transform into another form of energy when the spring is released.

Kinetic energy is the energy an object possesses due to its motion. When the spring is released, the potential energy stored in it is converted into kinetic energy, propelling the block forward. The kinetic energy \( K \) can be calculated using the formula \( K = \frac{1}{2} mv^2 \) where \( m \) is the mass of the block and \( v \) is its velocity.
This conversion process adheres to the law of energy conservation, meaning the total energy remains constant as it merely transfers from one form (potential) to another (kinetic). For the block, calculating its speed just involves equating the kinetic energy to the initial spring potential energy and solving for \( v \).

• The initial potential energy \( U_s \) in the spring becomes the kinetic energy \( K \) of the block.
• Use \( \frac{1}{2} mv^2 = U_s \) to find the velocity \( v \).

As the block travels up the incline, kinetic energy is converted into gravitational potential energy. This energy depends on the height function \( h \) and is calculated as \( U_g = mgh \). Here, \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height on the incline.
Since the block moves on a slope, we need to link the height \( h \) with the distance \( d \) traveled along the incline using the angle \( \theta \). We use \( h = d \sin(\theta) \), which allows us to substitute \( h \) in the potential energy equation. This part of the calculation helps determine how far the block moves up before its kinetic energy is completely converted to gravitational potential energy.

• The higher the block moves, the greater the gravitational potential energy.
• When all kinetic energy is converted, the block momentarily stops before sliding back.

An inclined plane in physics is a flat surface tilted at an angle to the horizontal. Analyzing mechanics on an inclined plane involves understanding forces like gravity acting at an angle. When the block travels up the frictionless incline, we consider components of gravitational force along and perpendicular to the plane.
The distance \( d \) traveled on the incline directly relates to how high the block gets. With all input energy being conserved and no friction to dissipate it, the motion of the block simplifies dramatically. The cosine and sine components of gravitational force help us compute movement and potential energy, utilizing the inclination angle \( \theta \).

• Solve for the incline distance using triangle relationships \( h = d \sin(\theta) \).
• Energy conversion simplifies calculations due to lack of friction.
• Understanding forces acting along the plane aids in comprehending motion.