7.3: Potential energy of specific force fields
The concept of potential energy is mainly useful for conservative forces. These forces can ’store’ energy and return it later. As examples we determine the potential energy of a gravitational force field and of a spring, which are both conservative forces.
7.3.1 Gravitational energy
The gravitational force \(\overrightarrow{\boldsymbol{F}}_{i g}=m_{i} \overrightarrow{\boldsymbol{g}}\) is given by Equation 6.27. If the gravitational force acts along the \(y\)-axis, with \(\overrightarrow{\boldsymbol{g}}=-g \hat{\boldsymbol{\jmath}}\), it follows from Equation 7.22 that:
\[\overrightarrow{\boldsymbol{F}}_{i g}=-m_{i} g \hat{\boldsymbol{\jmath}}=-\frac{\partial V_{i g}}{\partial y} \hat{\boldsymbol{\jmath}} \tag{7.35} \label{7.35}\]
By integrating this equation over \(y\) we obtain the potential energy function for a mass in a constant gravitational field:
\[V_{i g}\left(y_{i}\right)=\int_{y_{0}}^{y_{i}} m_{i} g \mathrm{~d} y=\left[m_{i} g y\right]_{y_{0}}^{y_{i}}=m_{i} g y_{i}+C \tag{7.36} \label{7.36}\]
The integration constant \(C=-m_{i} g y_{0}\) can be arbitrarily chosen by selecting a height \(y_{0}\) at which the potential energy is zero. Its choice does not affect the dynamics.
7.3.2 Centre of mass and gravity
To determine the gravitational energy of objects that consist of many point masses \(m_{i}\) with position vectors \(\overrightarrow{\boldsymbol{r}}_{j}\), we define the centre of mass (CoM) and its position vector \(\overrightarrow{\boldsymbol{r}}_{G}\) for such a system.
The centre-of-mass position vector \(\overrightarrow{\boldsymbol{r}}_{G}\) of a system of point masses \(m_{i}\), with a total mass \(m_{\text {tot }}\), is the mass weighted average of all position vectors \(\overrightarrow{\boldsymbol{r}}_{i}\), as defined by the following equations.
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{G} & \equiv \frac{1}{m_{\mathrm{tot}}} \sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i} \tag{7.37} \label{7.37}\\[4pt] m_{\mathrm{tot}} & =\sum_{i} m_{i} \tag{7.38} \label{7.38}\\[4pt] m_{\mathrm{tot}} \overrightarrow{\boldsymbol{r}}_{G} & =\sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i} \tag{7.39} \label{7.39}\\[4pt] m_{\mathrm{tot}} \overrightarrow{\boldsymbol{r}}_{G} & =\int_{V} \overrightarrow{\boldsymbol{r}} \rho \mathrm{d} V \tag{7.40} \label{7.40}\end{align}\]
If the object has a mass density \(\rho\) (in \(\left.\mathrm{kg} / \mathrm{m}^{3}\right)\) that is distributed over a certain volume \(V\), then the last equation gives the volume integral that determines the centre of mass. By taking the time derivatives of Equation 7.39 we obtain useful expressions for the velocity \(\overrightarrow{\boldsymbol{v}}_{G}\) and the acceleration vector \(\overrightarrow{\boldsymbol{a}}_{G}\) of the CoM:
\[\begin{align} m_{\mathrm{tot}} \overrightarrow{\boldsymbol{v}}_{G} & =\sum_{i} m_{i} \overrightarrow{\boldsymbol{v}}_{i} \tag{7.41} \label{7.41}\\[4pt] m_{\mathrm{tot}} \overrightarrow{\boldsymbol{a}}_{G} & =\sum_{i} m_{i} \overrightarrow{\boldsymbol{a}}_{i} \tag{7.42} \label{7.42}\end{align}\]
When the gravitational force \(\overrightarrow{\boldsymbol{F}}_{i g}=-m_{i} g \hat{\boldsymbol{\jmath}}\) acts on the system, the \(y\) coordinate of every point mass is given by \(y_{i}=\overrightarrow{\boldsymbol{r}}_{i} \cdot \hat{\boldsymbol{\jmath}}\), such that the potential energy of the whole system of point masses is given by:
\[V_{g, \text { tot }}=\sum_{i} V_{i g}=\sum_{i} m_{i} g y_{i}=g \sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i} \cdot \hat{\boldsymbol{\jmath}}=m_{\mathrm{tot}} \overrightarrow{\boldsymbol{r}}_{G} \cdot \hat{\boldsymbol{\jmath}}=m_{\mathrm{tot}} y_{G} \tag{7.43} \label{7.43}\]
where we used Equation 7.39. This equation shows that the gravitational potential energy of a system of point masses is identical to the potential energy that the system would have if all point masses would be located at its centre of mass at a height \(y_{G}\). The centre of mass is therefore also sometimes called the centre of gravity. We use the subscript \(G\) to indicate the centre of mass.
7.3.3 Spring energy
The properties of a spring that is aligned along the positive \(x\)-axis and connected at the origin can be analysed in a similar way. When the spring is displaced by a distance \(x_{k e}\) from its relaxed length, the spring force of the spring on a mass \(i\) (Eq. 6.33) is related to the potential energy:
\[\overrightarrow{\boldsymbol{F}}_{i k}\left(x_{k e}\right)=-k x_{k e} \hat{\boldsymbol{\imath}}=-\frac{\partial V_{i k}}{\partial x} \hat{\boldsymbol{\imath}} \tag{7.44} \label{7.44}\]
By integrating this equation over \(x\) we obtain:
\[V_{i k}\left(x_{k e}\right)=\int_{0}^{x_{k e}} k x \mathrm{~d} x=\left[\frac{1}{2} k x^{2}\right]_{0}^{x_{k e}}=\frac{1}{2} k x_{k e}^{2} \tag{7.45} \label{7.45}\]
If the length of the spring equals its rest length, \(x_{k e}=\left(x_{i}-L_{0}\right)=0\), its potential energy is chosen to be zero by starting the integral from \(x=0\).
7.3.4 Other conservative forces
Another important example of conservative forces are contact forces. Very often these forces act on objects that have zero velocity, or have a velocity perpendicular to the contact force, such that the power from the contace force is zero \((P=\overrightarrow{\boldsymbol{F}} \cdot \overrightarrow{\boldsymbol{v}}=0)\). In these cases the work done, and therefore the potential energy of these forces is zero. Thus, contact forces are conservative forces, since they do not change the total energy in the system.