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Mathematics for Physics

121 Potential Energy

An energy associated with position that is used to compute the work done by a conservative force between two points.

Note: Also called \(U\). Also called \(V\).

definition [d] (Potential Energy) From Hall: the potential energy of the system is the function

  • \(V(x) = -\displaystyle\int F(x)\, dx\) ,

so that

  • \(F(x) = -\dfrac{dV}{dx}\) .

where

  • \(V(x)\) is the potential energy.
  • \(F(x)\) is the force.
  • \(x\) is the position.

definition [d] (Potential Energy) From Shankar: introducing \(U(x)\) with \(F(x) = -dU/dx\), the change in potential energy is

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

where

  • \(U(x)\) is the potential energy.
  • \(F(x)\) is the force.
  • \(x_{1}\) and \(x_{2}\) are the initial and final positions.

definition [d] (Gravitational Potential Energy) From Knight: near Earth’s surface,

  • \(U_{G} = mgy\) ,

and for two point masses,

  • \(U_{G} = -\dfrac{GMm}{r}\) .

where

  • \(m\) and \(M\) are masses.
  • \(g\) is the gravitational acceleration near Earth.
  • \(y\) is the height.
  • \(r\) is the separation.
  • \(G\) is Newton’s gravitational constant.

definition [d] (Elastic Potential Energy) From Knight: the elastic potential energy is

  • \(U_{\mathrm{Sp}} = \dfrac{1}{2}k(\Delta s)^{2}\) ,

where

  • \(k\) is the spring constant.
  • \(\Delta s\) is the displacement of the spring from its equilibrium length.

121.1 Elementary Example

121.1.1 Simple

A mass \(m = 2\,\mathrm{kg}\) raised by \(y = 3\,\mathrm{m}\) near Earth has

\[ U_{G} = mgy = (2)(9.8)(3) = 58.8\,\mathrm{J} \]

where

  • \(U_{G}\) is the gravitational potential energy.

121.1.2 General

A spring with \(k = 200\,\mathrm{N/m}\) stretched by \(\Delta s = 0.10\,\mathrm{m}\) stores

\[ U_{\mathrm{Sp}} = \dfrac{1}{2}(200)(0.10)^{2} = 1\,\mathrm{J} \]

where

  • \(U_{\mathrm{Sp}}\) is the elastic potential energy.

121.2 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(V(x)=-\int F(x)\,dx\) and \(F=-dV/dx\).
  2. Shankar, R. Fundamentals of Physics. Yale University Press. — \(U(x_{2})-U(x_{1})=-\int_{x_{1}}^{x_{2}} F(x)\,dx\).
  3. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(U_{G}=mgy\), \(U_{G}=-\dfrac{GMm}{r}\), \(U_{\mathrm{Sp}}=\dfrac{1}{2}k(\Delta s)^{2}\).
  4. Logan, J. D. A First Course in Differential Equations. — spring potential \(V(x)=\dfrac{1}{2}kx^{2}\).