3.6 Non-conservative Forces

Non-conservative forces are the opposite of conservative forces, that is they don’t conserve energy. They can be characterised by failing to meet one or more of the conditions in definition 3.3. I.e. for a non-conservative force, the work done depends on the path taken and there is no potential energy function. Another way of viewing this is that under the action of conservative forces, the work done along a path is equal to the negative of the work done by reversing along the path. So we get back the energy that we put in. But in the case of a non-conservative force like friction, we don’t regain energy we put in by moving an object along a path.

3.6.1 Friction

Let’s look more closely at kinetic friction as an example. Consider a block sliding along a surface which is being slowed down by friction.

The friction force $\boldsymbol{f}_{k}$ always points in the opposite direction to $\operatorname{\mathrm{d}}\!\boldsymbol{r}$ , which points to the right. So, the work done by friction $W_{f}=\int_{\boldsymbol{r}_{1}}^{\boldsymbol{r}_{2}}\boldsymbol{f}_{k}\cdot% \operatorname{\mathrm{d}}\!\boldsymbol{r}$ is always negative. This is true for all paths, even closed ones, so we have

\oint\boldsymbol{f}_{k}\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}<0.

(3.89)

Now let’s add another conservative force and verify that energy is not conserved. Suppose the block is sliding down a slope, let $\boldsymbol{W}_{x}$ denote the component of weight acting parallel to $\operatorname{\mathrm{d}}\!\boldsymbol{r}$ .

By the work-energy theorem, we can write

$\displaystyle\Delta K=\frac{1}{2}mv^{2}-\frac{1}{2}mv_{0}^{2}$	$\displaystyle=\int_{\boldsymbol{r}_{0}}^{\boldsymbol{r}}(\boldsymbol{f}_{k}+% \boldsymbol{W}_{x})\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}$	(3.90)
	$\displaystyle=\int_{\boldsymbol{r_{0}}}^{\boldsymbol{r}}\boldsymbol{W}_{x}% \cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}+\int_{\boldsymbol{r_{0}}}^{% \boldsymbol{r}}\boldsymbol{f}_{k}\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}$	(3.91)
	$\displaystyle=-U(\boldsymbol{r})+\int_{\boldsymbol{r_{0}}}^{\boldsymbol{r}}% \boldsymbol{f}_{k}\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}.$	(3.92)

We know that the weight force is conservative, so we have replaced the integral with the potential energy function. Now let’s rearrange this to get the final total energy on the left:

\frac{1}{2}mv^{2}+U(\boldsymbol{r})=\frac{1}{2}mv_{0}^{2}+\int_{\boldsymbol{r_% {0}}}^{\boldsymbol{r}}\boldsymbol{f}_{k}\cdot\operatorname{\mathrm{d}}\!% \boldsymbol{r}.

(3.93)

Since the integral on the right is negative, the right hand side is always less than $\frac{1}{2}mv_{0}^{2}$ , which is the initial energy. This implies

E_{\text{final}}<E_{\text{initial}},

(3.94)

so energy is lost over time.

3.6.2 Conservative and Non-conservative Forces

In a general system, an object may be under the influence of multiple forces which can be conservative or non-conservative. If we split the resultant force on the system into a conservative part and a non-conservative part: $\boldsymbol{F}=\boldsymbol{F}_{\text{conservative}}+\boldsymbol{F}_{\text{non-% conservative}}$ , then using the work energy theorem again we can write

$\displaystyle\Delta K=W$	$\displaystyle=\int_{\boldsymbol{r}_{1}}^{\boldsymbol{r}_{2}}\boldsymbol{F}% \cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}$	(3.95)
	$\displaystyle=\int_{\boldsymbol{r}_{1}}^{\boldsymbol{r}_{2}}\boldsymbol{F}_{% \text{conservative}}\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}+\int_{% \boldsymbol{r}_{1}}^{\boldsymbol{r}_{2}}\boldsymbol{F}_{\text{non-conservative% }}\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}$	(3.96)
	$\displaystyle=-\Delta U+W_{\text{non-conservative}}.$	(3.97)

If we call the sum of kinetic energy and the potential energy from conservative forces $K+U$ the mechanical energy $E_{\text{mech}}$ , then we get

\Delta E_{\text{mech}}=W_{\text{non-conservative}}.

(3.98)

Consider the case where the non-conservative force is friction, so most of the work done is converted to heat, or thermal energy. So $W_{\text{non-conservative}}=-\Delta E_{\text{thermal}}$ . This implies that we can write energy conservation as

\Delta E_{\text{mech}}+\Delta E_{\text{thermal}}=0.

(3.99)

This implies that $\Delta E_{\text{mech}}\leq 0$ , so the total mechanical energy in a closed system can only stay the same or decrease. We have basically just discovered the first and second laws of thermodynamics, but that is a topic for another time. If our system is not isolated and is acted on by an external force, we can say

\Delta E_{\text{mech}}+\Delta E_{\text{thermal}}=W_{\text{ext}},

(3.100)

where $W_{\text{ext}}$ is the work done by the external force on the system.

Example 3.7.

A 2000kg elevator cable snaps at a height of 20m above a spring with $k=10,000$ Nm^-1. Taking into consideration that the friction of the shaft walls exert a constant force of 15,000N to resist the fall of the elevator, what is the maximum compression of the spring?

Example 3.8.

Consider four possible paths of an object falling that start and end at the same height. Order the paths in terms of the final kinetic energy when there is no friction. What changes if there is friction?

With no friction, the final velocity is the same for all paths because the change in gravitional potential energy $U_{g}$ is the same. With friction, $v_{A}>v_{B}>v_{C}>v_{D}$ .