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Gravitational Potential Energy

Level 1 - Physics topic page in Potential Energy and Conservation.

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.

012345-1001020304050yUgreference 1reference 2AB
Changing the zero level shifts the graph vertically without changing \\(\\Delta U_g\\) between two heights.

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.

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Elastic Potential