Gravitational Potential Energy

• The formula for GPE in a uniform gravitational field is derived from the definition of Work Done ($W = F \times d$). To lift an object of mass $m$ at a constant speed, an upward force equal to its weight ($W = mg$) must be applied.

• When this force moves the object through a vertical displacement $\Delta h$, the work done on the object is $W = mg \Delta h$. This work is stored as the object's potential energy.

Key Formula: $\Delta E_p = m g \Delta h$

• This equation assumes a uniform gravitational field, where the value of $g$ remains constant (approximately $9.81 \, N/kg$ near the Earth's surface).

• GPE is always measured relative to a reference level (or 'zero point'). The choice of this level is arbitrary; it could be the ground, a tabletop, or the bottom of a pit.

• Because the choice of $h=0$ is flexible, the absolute value of GPE is less important than the change in GPE ($\Delta E_p$) during a process.

• If an object moves below the chosen reference level, its height $h$ becomes negative, resulting in a negative GPE value. This simply indicates the object has less energy than it would at the reference point.

• In a closed system without resistive forces (like friction or air resistance), the sum of kinetic energy ($E_k$) and potential energy ($E_p$) remains constant. This is known as the Principle of Conservation of Mechanical Energy.

• As an object falls, its GPE decreases while its kinetic energy increases. The loss in GPE is exactly equal to the gain in KE: $\Delta E_p = \Delta E_k \implies mgh = \frac{1}{2}mv^2$

• This relationship allows for the calculation of an object's final velocity based solely on its initial height, independent of the path taken, provided friction is negligible.

• GPE vs. Kinetic Energy: GPE is energy of position (stored), whereas Kinetic Energy is energy of motion. GPE depends on height, while KE depends on velocity squared ($v^2$).

• Check Units: Always ensure mass is in kg and height is in meters. If a problem gives mass in grams or height in centimeters, convert them first to avoid magnitude errors.

• Identify the Zero Point: Explicitly state where your $h=0$ level is. Consistency is key; once you choose a reference level for a problem, do not change it midway through your calculations.

• Energy Transfers: In multi-step problems, look for phrases like 'starts from rest' (initial $E_k = 0$) or 'reaches maximum height' (final $E_k = 0$). These are clues for setting up conservation equations.

• Sanity Check: If an object is falling, its GPE must decrease. If your calculation shows an increase in GPE for a falling object, check the sign of your height change or your reference level.