AcademyGravitation
Academy
Gravitational Potential Energy
Level 1 - Physics topic page in Gravitation.
Principle
Gravitational potential energy tracks how height in a gravitational field stores or releases mechanical energy.
Notation
\(F_{g,y}\)
vertical component of gravitational force
\(\mathrm{N}\)
\(U_g\)
gravitational potential energy
\(\mathrm{J}\)
\(\Delta U_g\)
change in gravitational potential energy
\(\mathrm{J}\)
\(W_g\)
work done by gravity
\(\mathrm{J}\)
\(K\)
kinetic energy
\(\mathrm{J}\)
\(m\)
mass
\(\mathrm{kg}\)
\(g\)
gravitational field strength
\(\mathrm{m\,s^{-2}}\)
\(h\)
vertical height change
\(\mathrm{m}\)
\(y\)
vertical position
\(\mathrm{m}\)
\(\Delta y\)
vertical displacement
\(\mathrm{m}\)
Method
Near Earth, gravity has nearly constant magnitude and points downward. If upward is chosen as positive, a rise increases position while gravity acts in the opposite direction.
Gravity direction
\[F_{g,y}=-mg\]
The sign is negative because the force points downward.
Work for a vertical move
\[W_g=F_{g,y}\Delta y=-mg\Delta y\]
A rise gives negative work by gravity; a fall gives positive work.
Define potential change
\[W_g=-\Delta U_g\]
Near-Earth change
\[\Delta U_g=mg\Delta y\]
Choose a reference level
\[U_g=0\ \text{at a chosen height}\]
Only differences in potential energy are physical, so the zero level is conventional.
Near-Earth form
\[U_g=mgy\]
This form follows when the chosen reference level is at \(y=0\).
Gravity-only motion
\[\Delta K=-\Delta U_g\]
This is the mechanical-energy statement when gravity is the only force doing work.
The graph below shows why only changes in \(U_g\) matter. Changing the reference level shifts the line up or down, but the slope and the energy difference between two heights stay the same.
For the two marked heights, the vertical gap from A to B is the same on both lines, so \(\\Delta U_g\) is unchanged even though the numerical zero differs.
Rules
These are the compact results from the method above.
Near Earth
\[U_g=mgy\]
Energy change
\[\Delta U_g=mg\Delta y\]
Work by gravity
\[W_g=-\Delta U_g\]
Gravity only
\[K_i+U_{g,i}=K_f+U_{g,f}\]
Examples
Question
A mass \(m\) falls through height \(h\) from rest. Find its speed just before impact, ignoring air resistance.
Answer
\[K_i+U_{g,i}=K_f+U_{g,f}\]
\[0+mgh=\frac{1}{2}mv^2\]
\[v=\sqrt{2gh}\]
Checks
- Only changes in gravitational potential energy are physically meaningful.
- Moving upward makes \(\\Delta U_g\) positive near Earth.
- Work by gravity has the opposite sign to \(\\Delta U_g\).
- The formula \(U_g=mgy\) assumes a uniform near-Earth field.