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When we hear the term "gravitational potential energy," it refers to the energy stored in an object due to its position in a gravitational field. In simpler terms, it's the energy an object has because of where it is placed—typically, how high it is above the ground. The formula to calculate this energy is \( \Delta U_g = m g d \) where:

• \( m \) represents the mass of the object (in this scenario, the cabbage),
• \( g \) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\) on Earth), and
• \( d \) denotes the vertical distance the object moves.

When the cabbage is lowered, it loses gravitational potential energy, which we can think of as the energy it could have used to fall. The amount of energy lost depends directly on the distance it falls and its mass.

Spring potential energy is the energy stored in a spring when it is either compressed or stretched. Think of a spring as a sort of energy bank account that stores mechanical energy based on its change from its original shape. This potential energy is given by the formula:\[ \Delta U_s = \frac{1}{2} k x^2 \]where:

• \( k \) is the spring constant that tells us how stiff the spring is, and
• \( x \) is the distance the spring is stretched or compressed from its natural length.

In the problem, as the cabbage stretches the spring, the spring accumulates potential energy. This energy is dependent on the square of the displacement, emphasizing that even small stretches can lead to significant energy storage. At the equilibrium position, the energy gained by the spring is a direct consequence of this stretching.

Equilibrium refers to a state where all forces acting on an object are perfectly balanced, meaning there is no net force causing the object to move. In mechanics, equilibrium can be either static or dynamic, although in this problem, we deal with static equilibrium.When the cabbage is gently lowered, it compresses the spring until the upward force from the spring equals the downward gravitational force. This balance of forces is represented by \[ k x = m g \] indicating that the force exerted by the spring (\( k x \)) matches the weight of the cabbage (\( m g \)).In such situations, understanding equilibrium helps us relate the changes in gravitational and spring potential energies. Since energy is conserved, the energy lost by the cabbage gets transferred to the spring, albeit with some interesting proportional relationships, as shown—the loss in gravitational potential energy is twice the gain in spring potential energy. This clever conservation and distribution of energy are core principles in physics, illustrating how systems naturally balance forces and energies.