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Gravitational potential energy (PE) refers to the energy an object has due to its position in a gravitational field. It is especially relevant when dealing with objects that have the potential to fall from a height. - **Formula**: The formula to calculate gravitational potential energy is given by \[ \mathrm{PE}_G = mgh \] where: - \( m \) represents the mass of the object - \( g \) is the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2\) on Earth) - \( h \) is the height above the reference point. - **Application**: In the exercise, the gravitational potential energy of the falling object is transformed into another form of energy as it stretches a spring. This transformation is key to solving the problem by using the principle of energy conservation. Understanding gravitational potential energy helps us predict how much energy is available to be converted into other forms as an object moves downward.

Elastic potential energy is stored energy in objects that can be stretched or compressed, such as springs. When a spring is stretched or compressed, it can store energy, which can then be released, causing the spring to return to its original shape. - **Formula**: The elastic potential energy (PE) in a spring is given by: \[ \mathrm{PE}_e = \frac{1}{2}kx^2 \] where: - \( k \) is the spring constant, representing how stiff the spring is - \( x \) is the displacement from the spring's equilibrium position (how much you've stretched or compressed it).- **In Context**: In the problem, as the falling object stretches the spring, the kinetic energy (from falling) is converted into elastic potential energy stored in the spring. The spring's energy storage ability relies significantly on its spring constant \( k \) and the extent of displacement \( x \). This concept reveals why springs are capable of storing potential energy through deformation.

The spring constant \( k \) is a vital property of a spring that measures its stiffness. A higher spring constant means a stiffer spring, needing more force to stretch or compress it. - **Understanding \( k \)**: The spring constant is central in calculating how much potential energy a spring can store and how much it stretches under a given force.- **Units**: The spring constant is measured in Newtons per meter (N/m), revealing the amount of force needed for a unit displacement. - **Relevance in Problem Solving**: In the example exercise, the spring constant is given as \( 30 \text{ N/m} \). This value is plugged into the formula for elastic potential energy to determine how much energy the stretched spring can store. The balance between gravity's pull and the spring's resistance is essential, where the spring constant directly influences this equilibrium state.

Mechanical energy conversion is an important principle in physics involving the transformation of energy from one form to another. In many systems, energy is converted between potential, kinetic, and other forms. - **Energy Conservation Principle**: The fundamental rule is that energy within a closed system is conserved, merely changing from one form to another. This concept is employed extensively in the exercise. - **In Practice**: Here, the gravitational potential energy of the falling object is completely converted into elastic potential energy of the spring. Therefore, the equation \[ mgh = \frac{1}{2}kx^2 \] is utilized to ascertain when the object comes to a stop upon stretching the spring. - **Significance**: This concept helps us understand how energy is transitioned seamlessly between systems, ensuring that energy is never lost but reallocated in various forms. It is crucial when one seeks to predict motion outcomes by leveraging the conservation law.