91Ó°ÊÓ

Kinetic Energy is the energy a body possesses due to its motion. In our scenario with the block and spring, kinetic energy helps us understand how much energy the block has as it moves after being released from the depressed spring.
Kinetic energy (\(K\)) can be determined using the equation:\[ K = \frac{1}{2} m v^2\]For this problem, however, we use the conservation of energy principle to directly calculate kinetic energy without needing to evaluate velocity.

• Initially, the spring is compressed, meaning the block's kinetic energy at \(y = 0\) is zero, as it's just being released.
• As the block moves upwards, it gains kinetic energy when the spring's potential energy and gravitational potential energy decrease.

The conservation of energy equation in the exercise shows how kinetic energy is calculated by balancing the spring's potential energy decrease with the increase in gravitational potential energy. This approach simplifies the calculation immensely, as it directly relates to changes in potential energies.

Gravitational Potential Energy (GPE) refers to the energy stored by an object's position relative to a height in a gravitational field.
In our problem, gravitational potential energy (\(U_g\)) varies as the block rises from its initial release point. It is calculated using:\[ U_g = m g y \]where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity (often \(9.81 \text{ m/s}^2\) for calculations, though in this exercise weight provides a direct relation), and \(y\) is the height above a defined reference point.

• When \(y = 0\), GPE is zero since this is our reference point.
• As the block ascends to different heights like 0.050 m or 0.10 m, its GPE increases, showing the block stores more energy due to its elevated position.
• This incremental addition of potential energy is crucial, as the energy from the spring transforms partly into this gravitational form.

Through each step of the height change, GPE is calculated to track the energy's transformation within the system, illustrating how energy conserves altogether.

Spring Potential Energy is the energy stored in a compressed or stretched spring. This energy is crucial in our problem as the primary source of energy that allows the block to move upwards after being released.
The potential energy stored in a spring (\(U_s\)) is calculated by:\[ U_s = \frac{1}{2} k x^2 \]where \(k\) is the spring constant, and \(x\) is the distance from the spring's natural length (zero force position).
If the spring is initially compressed by 25 cm:

• At the starting point (\(y = 0\)), the potential energy is 19.375 J, since all energy is stored in the compressed spring.
• As the block moves upwards and the spring decompresses, this stored energy decreases, becoming kinetic and gravitational potential energy. For \(0.25 - y\) deformations, \(U_s\) constantly changes.

Understanding these energy conversions helps in applying the conservation of energy principles effectively, and explains why the block has kinetic energy as the system evolves.