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Mathematics for Physics

103 Gravitational Potential Energy Formula Derivation

A derivation of the gravitational potential energy formula that is used to obtain \(U_{G} = mgy\) from the work done by gravity.

definition [d] (Gravitational Potential Energy Formula Derivation) From Knight: the work done by the constant gravitational force \(F_{G} = -mg\) through a vertical displacement is

  • \(W_{G} = mg(y_{i} - y_{f}) = mgy_{i} - mgy_{f}\) .

With \(\Delta U_{G} = -W_{G}\),

  • \(U_{f} - U_{i} = -(mgy_{i} - mgy_{f}) = mgy_{f} - mgy_{i}\) ,

so

  • \(U_{G} = mgy\)

when \(U_{G} = 0\) is chosen at \(y = 0\).

where

  • \(F_{G}\) is the gravitational force.
  • \(m\) is the mass.
  • \(g\) is the gravitational acceleration.
  • \(y\) is the height.
  • \(y_{i}\) is the initial height.
  • \(y_{f}\) is the final height.
  • \(W_{G}\) is the work done by gravity.
  • \(\Delta U_{G}\) is the change in gravitational potential energy.
  • \(U_{i}\) is the initial gravitational potential energy.
  • \(U_{f}\) is the final gravitational potential energy.
  • \(U_{G}\) is the gravitational potential energy.

103.1 Elementary Example

103.1.1 Simple

For \(m = 2\,\mathrm{kg}\) raised from \(y_{i} = 0\) to \(y_{f} = 3\,\mathrm{m}\) with \(g = 9.8\,\mathrm{m/s}^{2}\),

\[ U_{f} - U_{i} = mgy_{f} - mgy_{i} = 58.8\,\mathrm{J} \]

where

  • \(m\) is the mass.
  • \(g\) is the gravitational acceleration.
  • \(y_{i}\) is the initial height.
  • \(y_{f}\) is the final height.
  • \(U_{i}\) is the initial gravitational potential energy.
  • \(U_{f}\) is the final gravitational potential energy.
  • \(U_{G} = mgy\) is the gravitational potential energy formula.
  • \(W_{G}\) is the work done by gravity.
  • \(\Delta U_{G} = -W_{G}\) relates potential change to work.

103.1.2 General

Raising the same mass from \(y_{i} = 1\,\mathrm{m}\) to \(y_{f} = 4\,\mathrm{m}\) gives

\[ \Delta U_{G} = mg(y_{f} - y_{i}) = (2)(9.8)(3) = 58.8\,\mathrm{J} \]

where

  • \(m\) is the mass.
  • \(g\) is the gravitational acceleration.
  • \(y_{i}\) is the initial height.
  • \(y_{f}\) is the final height.
  • \(\Delta U_{G}\) is the change in gravitational potential energy.