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Mathematics for Physics

122 Potential Energy Formula Derivation

A derivation of the potential energy formula that is used to obtain \(U(x_{2})-U(x_{1})=-\displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx\) from the work-energy theorem.

Related definitions:

definition [d] (Potential Energy Formula Derivation) From Shankar: start from the rate of change of kinetic energy

  • \(\dfrac{dK}{dt} = \dfrac{d}{dt}\left(\dfrac{1}{2}mv^{2}\right) = m\dfrac{dv}{dt}\, v\) .

With \(F(x) = m\,\dfrac{dv}{dt}\) and \(v = \dfrac{dx}{dt}\),

  • \(\dfrac{dK}{dt} = F(x)\dfrac{dx}{dt}\) .

Integrating from \(t_{1}\) to \(t_{2}\) gives the work-energy theorem

  • \(K(t_{2}) - K(t_{1}) = \displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx\) .

Writing the integral via an antiderivative \(G\) with \(\dfrac{dG}{dx} = F(x)\),

  • \(K_{2} - K_{1} = G(x_{2}) - G(x_{1})\) ,

hence

  • \(K_{2} - G(x_{2}) = K_{1} - G(x_{1})\) .

Setting \(U(x) = -G(x)\) so that \(F(x) = -\dfrac{dU}{dx}\) yields conservation

  • \(K_{2} + U_{2} = K_{1} + U_{1}\)

and the potential-energy change

  • \(U(x_{2}) - U(x_{1}) = -\displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx\) .

where

  • \(K\) is the kinetic energy.
  • \(\dfrac{dK}{dt}\) is the rate of change of kinetic energy.
  • \(t\) is time.
  • \(m\) is the mass.
  • \(v\) is the velocity.
  • \(\dfrac{dv}{dt}\) is the acceleration.
  • \(F(x)\) is a position-dependent force.
  • \(x\) is the position.
  • \(dx\) is the position differential.
  • \(\dfrac{dx}{dt}\) is the velocity written as a derivative of position.
  • \(t_{1}\) and \(t_{2}\) are the initial and final times.
  • \(x_{1}\) and \(x_{2}\) are the initial and final positions.
  • \(K(t_{1})\) and \(K(t_{2})\) are the kinetic energies at those times.
  • \(K_{1}\) and \(K_{2}\) are the same kinetic energies at \(x_{1}\) and \(x_{2}\).
  • \(G(x)\) is an antiderivative of \(F\).
  • \(\dfrac{dG}{dx}\) is the derivative of \(G\) with respect to \(x\).
  • \(U(x)\) is the potential energy.
  • \(U_{1}\) and \(U_{2}\) are the potential energies at \(x_{1}\) and \(x_{2}\).
  • \(U(x_{1})\) and \(U(x_{2})\) are the same potential energies written as functions of position.
  • \(\dfrac{dU}{dx}\) is the derivative of \(U\) with respect to \(x\).
  • \(\displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx\) is the work done by \(F\) from \(x_{1}\) to \(x_{2}\).
  • \(W = \displaystyle\int_{x_{1}}^{x_{2}} F(x)\, dx\) is that work.
  • \(\Delta U = -W\) is the change in potential energy.

122.1 Elementary Example

122.1.1 Simple

For a constant force \(F = 4\,\mathrm{N}\) from \(x_{1} = 0\) to \(x_{2} = 3\,\mathrm{m}\),

\[ U(x_{2}) - U(x_{1}) = -\int_{0}^{3} 4\, dx = -12\,\mathrm{J} \]

where

  • \(F\) is the constant force.
  • \(x_{1}\) is the initial position.
  • \(x_{2}\) is the final position.
  • \(dx\) is the position differential.
  • \(U(x_{1})\) is the initial potential energy.
  • \(U(x_{2})\) is the final potential energy.
  • \(W = \displaystyle\int_{x_{1}}^{x_{2}} F\, dx\) is the work done by \(F\).
  • \(\Delta U = -W\) is the change in potential energy.

122.1.2 General

For \(F(x) = -6x\) from \(x_{1} = 1\) to \(x_{2} = 2\),

\[ U(2) - U(1) = -\int_{1}^{2} (-6x)\, dx = 9\,\mathrm{J} \]

where

  • \(F(x)\) is the position-dependent force.
  • \(x\) is the position.
  • \(x_{1}\) is the initial position.
  • \(x_{2}\) is the final position.
  • \(dx\) is the position differential.
  • \(U(1)\) is the potential energy at \(x = 1\).
  • \(U(2)\) is the potential energy at \(x = 2\).
  • \(W = \displaystyle\int_{1}^{2} F(x)\, dx\) is the work done by \(F\).
  • \(\Delta U = -W\) is the change in potential energy.