Gravitational potential energy is a crucial concept in understanding how energy is stored and transformed in physical systems. It is defined as the energy stored in an object due to its height above a reference point, typically the ground. The formula for gravitational potential energy, denoted as \( U_g \), is given by:
\( U_g = mgh \)
or equivalently, \( U_g = mg y \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)), and \( h \) or \( y \) is the height of the object above the reference point.
When an object falls, gravity does work on it, which results in a change in its kinetic energy. The work done by gravity can be calculated using the equation:
\( W_g = -mg \Delta y \)
where \( \Delta y \) is the change in height. If an object falls from a height of 10 meters to 4 meters, the displacement \( \Delta y \) is negative, indicating a decrease in height. For a 5.1 kg box falling from 10 m to 4 m, the work done by gravity can be calculated as follows:
\( W_g = -5.1 \times 9.8 \times (-6) = 300 \, \text{J} \)
This positive work indicates that gravity is aiding the box's motion, increasing its kinetic energy from 0 J to 300 J as it falls.
To understand the relationship between gravitational potential energy and kinetic energy, consider the initial and final gravitational potential energies. The initial potential energy when the box is at 10 m is:
\( U_{g, \text{initial}} = mgh_{\text{initial}} = 5.1 \times 9.8 \times 10 = 500 \, \text{J} \)
At a height of 4 m, the final potential energy is:
\( U_{g, \text{final}} = mgh_{\text{final}} = 5.1 \times 9.8 \times 4 = 200 \, \text{J} \)
The change in gravitational potential energy is calculated as:
\( \Delta U_g = U_{g, \text{final}} - U_{g, \text{initial}} = 200 - 500 = -300 \, \text{J} \)
This negative change indicates a loss of stored energy as the box falls. Notably, this change in gravitational potential energy can also be expressed as:
\( \Delta U_g = mg \Delta y \)
Thus, the work done by gravity is equal to the negative change in gravitational potential energy:
\( W_g = -\Delta U_g \)
This relationship highlights the conservation of energy principle, where the energy lost in potential energy is converted into kinetic energy as the object falls. Conversely, when an object rises, the work done by gravity is negative, leading to a decrease in kinetic energy and an increase in potential energy.
In summary, gravitational potential energy and kinetic energy are interconnected through the work done by gravity, illustrating the fundamental principles of energy conservation in mechanics.
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