91Ó°ÊÓ

Kinetic Energy (KE) is the energy that an object possesses due to its motion. It's a key form of energy, especially in dynamic systems. When an object is moved or in motion, it exhibits kinetic energy, which can be calculated using the formula: - \[ K = \frac{1}{2}mv^2 \] - Here, \( m \) is the mass of the object and \( v \) is its velocity. Throughout the problem, as the block moves vertically, its velocity will change, resulting in a change in kinetic energy. At various points along the vertical path (like at \(-5.0\, \text{cm}, -10\, \text{cm}\), etc.), the block will exhibit a specific kinetic energy depending on its speed at those moments.

Energy conservation is crucial because it tells us how kinetic energy relates to potential energies. For example, at each point, the sum of kinetic and potential energies must remain constant, assuming no external forces are doing work.

To find the kinetic energy at each point in this exercise, knowing how gravitational and elastic potential energies change is vital. The conservation principle \( K + \Delta U_g + \Delta U_e = \text{constant} \) is used to compute the kinetic energy at various displacements of the block.

Gravitational Potential Energy (GPE) is the energy stored in an object due to its position in a gravitational field. It's particularly relevant when dealing with vertical motions, as seen in this problem. The GPE can be calculated with:- \[ U_g = mgy \] - Where \( m \) is the mass, \( g \) is the gravitational acceleration (approximately \( 9.8 \, \text{m/s}^2 \)), and \( y \) is the height above a reference point. In our exercise, since the block moves downward, its height \( y \) is negative, leading to a decrease in GPE.

In the calculations provided, GPE reduction matches the displacement of the block, like going from \(-5\, \text{cm}\) to \(-20\, \text{cm}\). The changes in GPE were specifically quantified for the new positions: - At \(-5\, \text{cm}\): \( \Delta U_g = -1.0 \, \text{J} \) - At \(-10\, \text{cm}\): \( \Delta U_g = -2.0 \, \text{J} \) - And so on.

Understanding how GPE changes help us appreciate how energy shifts within the system as the spring dictates the path of motion, and ultimately how the potential energy converts to kinetic.

Elastic Potential Energy (EPE) is the energy stored in elastic materials, like springs, when they are compressed or stretched. For springs, the EPE can be expressed with Hooke's Law as:- \[ U_e = \frac{1}{2}ky^2 \] - Where \( k \) is the spring constant (given as \( 200\, \text{N/m} \) in this problem), and \( y \) is the displacement from the spring's natural length. When the block is attached and displaces \(-5.0\, \text{cm}\) and further, the spring stretches, storing elastic potential energy.

As we analyze this problem, the spring's displacement from equilibrium plays a critical role in how energy is distributed within the system. Each marked distance represents how much potential energy is retained in the spring: - At \(-5\, \text{cm}\): \( \Delta U_e = 0.25 \, \text{J} \) - At \(-10\, \text{cm}\): \( \Delta U_e = 1.0 \, \text{J} \)

This substantial storage of elastic energy is vital in determining how the block's movement conserves total mechanical energy. As the spring's tension changes, it's crucial to maintain that the total energy of the system, combining kinetic, gravitational, and elastic potential energies, remains conserved.