7.1: Principle of work and energy
We start by introducing the most important principle of this chapter.
The principle of work and energy states that the work \(W\) done by a force \(\overrightarrow{\boldsymbol{F}}_{i j}\) on a point mass \(i\), while it moves along a path curve from position \(s_{1}\) to position \(s_{2}\), is equal to the change in its kinetic energy \(T_{i}\).
\[W_{i j, s_{1} \rightarrow s_{2}}=\Delta T_{s_{1} \rightarrow s_{2}} \tag{7.1} \label{7.1}\]
where \(\Delta T_{s_{1} \rightarrow s_{2}}=T_{i}\left(s_{2}\right)-T_{i}\left(s_{1}\right)\) is the change in kinetic energy.
To use the principle in Equation 7.1 we first need to define the work \(W_{i j, s_{1} \rightarrow s_{2}}\) done by a force and the kinetic energy \(T_{i}\) of the point mass.
7.1.1 Work
The work \(W_{i j}\) performed \(\boldsymbol{b y}\) a force \(\overrightarrow{\boldsymbol{F}}_{i j}\) on a point mass \(i\), while it moves from path coordinate \(s_{1}\) to \(s_{2}\) along a path curve \(\overrightarrow{\boldsymbol{r}}_{s}(s)\) is defined as:
\[W_{i j, s_{1} \rightarrow s_{2}} \equiv \int_{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{1}\right)}^{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{2}\right)} \overrightarrow{\boldsymbol{F}}_{i j} \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}} \tag{7.2} \label{7.2}\]
This expression can be simplified by using that \(\mathrm{d} \overrightarrow{\boldsymbol{r}}=\hat{\boldsymbol{s}} \mathrm{d} s\) (see Equation 5.21). We can project the force vector on the path curve, obtaining its tangential component using:
\[F_{i j, t}=\overrightarrow{\boldsymbol{F}}_{i j} \cdot \hat{\boldsymbol{s}} \tag{7.3} \label{7.3}\]
Thus we obtain the following scalar expression for the work:
\[W_{i j, s_{1} \rightarrow s_{2}}=\int_{s_{1}}^{s_{2}} F_{i j, t} \mathrm{~d} s \tag{7.4} \label{7.4}\]
7.1.2 Kinetic energy
The kinetic energy of a point mass \(i\) is defined as one half its mass times its speed squared \(\left(v_{i}^{2}=\left|\overrightarrow{\boldsymbol{v}}_{i}\right|^{2}\right)\), and is indicated by the letter \(T\).
\[T_{i} \equiv \frac{1}{2} m_{i} v_{i}^{2} \tag{7.5} \label{7.5}\]
As an example of the principle of work and energy we determine the change in kinetic energy of a ball that is launched at an initial velocity \(\overrightarrow{\boldsymbol{v}}_{0}=\) \(v_{x, 0} \hat{\imath}+v_{y, 0} \hat{\boldsymbol{\jmath}}\) and experiences constant force \(\overrightarrow{\boldsymbol{F}}_{g}=-m g \hat{\boldsymbol{\jmath}}\) of gravity.
The trajectory of the ball is shown in Figure 7.1. We first determine the work done by the force on the ball using Equation 7.2. By using that \(\mathrm{d} \overrightarrow{\boldsymbol{r}}=\mathrm{d} x \hat{\boldsymbol{\imath}}+\mathrm{d} y \hat{\boldsymbol{\jmath}}\), such that \(\overrightarrow{\boldsymbol{F}}_{g} \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}}=-m g \mathrm{~d} y\) we find:
\[W_{s_{1} \rightarrow s_{2}}=\int_{y_{1}}^{y_{2}}-m g \mathrm{~d} y=-m g\left(y_{2}-y_{1}\right) \tag{7.6} \label{7.6}\]
So, as shown in Figure 7.2, the work done by the gravitational energy is negative and has the same parabolic shape as the trajectory of the ball shown in Figure 7.1. The work is negative because the tangential component of the force initially reduces the speed. After the ball has passed its highest point, the work increases again, because the tangential component of gravity is in the same direction as the velocity vector.
From the principle of work and energy Equation 7.1 we see that the change in kinetic energy \(\Delta T_{12}\) equals \(W_{s_{1} \rightarrow s_{2}}\), and follows the same curve in Figure 7.2. Note that the work could also have been determined by projecting the force on the path curve using angle \(\alpha\) and using the scalar equation Equation 7.4 but this would have been more difficult.
7.1.3 Derivation of the principle
Here we present the derivation of the principle of work and energy. For a force \(\overrightarrow{\boldsymbol{F}}_{i j}\) acting on a point mass \(i\) we have from Newton’s second law that:
\[\overrightarrow{\boldsymbol{F}}_{i j}=m_{i} \overrightarrow{\boldsymbol{a}}_{i} \tag{7.7} \label{7.7}\]
\[\int_{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{1}\right)}^{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{2}\right)}\left[\overrightarrow{\boldsymbol{F}}_{i j}\right] \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}}=\int_{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{1}\right)}^{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{2}\right)}\left[m_{i} \overrightarrow{\boldsymbol{a}}_{i}\right] \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}} \tag{7.8} \label{7.8}\]
We see directly that the left side of this equation is equal to Equation 7.2 and can be replaced by the work \(W_{i j, s_{1} \rightarrow s_{2}}\). To prove the principle in Equation 7.1 we still have to show that the right side is equal to the change in kinetic energy.
Derivation. Kinetic energy
To simplify the right side of Equation 7.8, we evaluate the integral over the dot product \(\overrightarrow{\boldsymbol{a}}_{i} \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}}\).
\[m_{i} \int_{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{1}\right)}^{\overrightarrow{\boldsymbol{r}}_{s}\left(s_{2}\right)} \overrightarrow{\boldsymbol{a}}_{i} \cdot \mathrm{d} \overrightarrow{\boldsymbol{r}}=m_{i} \int_{s_{1}}^{s_{2}} \overrightarrow{\boldsymbol{a}}_{i} \cdot \hat{\boldsymbol{s}} \mathrm{d} s=m_{i} \int_{s_{1}}^{s_{2}} a_{i, t}(s) \mathrm{d} s \tag{7.9} \label{7.9}\]
Since \(a_{i, t}=\ddot{s}_{i}\) the rightmost integral over \(a_{i, t}(s) \mathrm{d} s\) is almost identical to Equation 5.45, and can be integrated with \(a_{s} \mathrm{~d} s=v_{s} \mathrm{~d} v\) yielding:
\[m_{i} \int_{s_{1}}^{s_{2}} a_{i, t} \mathrm{~d} s=m_{i} \int_{v_{1}}^{v_{2}} v_{s, i} \mathrm{~d} v_{s, i}=\left[\frac{1}{2} m_{i} v_{s, i}^{2}\right]_{v_{1}}^{v_{2}}=\frac{1}{2} m_{i} v_{2}^{2}-\frac{1}{2} m_{i} v_{1}^{2} \tag{7.10} \label{7.10}\]
The right side of this equation shows that the integral is simply equivalent to the change in the kinetic energy \(T_{i}=\frac{1}{2} m_{i} v_{i}^{2}\), and proves that the integral over Newton’s second law Equation 7.8 leads to the principle of work and energy Eq. \((7.1)\).
7.1.4 Work and energy for a system
To analyse a system of point masses, on which multiple forces \(F_{j}\) are working, the principle of work and energy can be extended by summing the work Equation 7.1 over all point masses \(i\) and forces \(j\) to obtain:
\[W_{\mathrm{tot}}=\sum_{i} \sum_{j} W_{i j}=\sum_{i} \Delta T_{i}=\Delta T_{\mathrm{tot}} \tag{7.11} \label{7.11}\]
Thus the sum of the work of all forces on all point masses equals the total increase in kinetic energy.