Non-Conservative Forces Continued
Non-Conservative Forces and Work-Energy Theorem
We know that the net work done will be the sum of the work done by all of the individual forces:
\begin{align*} F_{\text{net}} & = F_{\text{conservative}}+F_{\text{non-conservative}} \\ & (\text{the work done by these forces yields}) \\ W_{\text{net}} & = W_{\text{conservative}}+W_{\text{non-conservative}} \\ & (\text{using the work-energy theorem}) \\ \Delta E_k & = W_{\text{conservative}}+W_{\text{non-conservative}}\\ &(\text{using}~W_{\text{conservative}}=-\Delta PE) \\ \Delta E_k & = -\Delta E_p +W_{\text{non-conservative}} \\ W_{\text{non-conservative}} & = \Delta E_k + \Delta E_p \end{align*}
When the non-conservative forces oppose the motion, the work done by the non-conservative forces is negative, causing a decrease in the mechanical energy of the system. When the non-conservative forces do positive work, energy is added to the system. If the sum of the non-conservative forces is zero then mechanical energy is conserved.
Example: Sliding Footballer
Question
Consider the situation shown where a football player slides to a stop on level ground. Using energy considerations, calculate the distance the \(\text{65.0}\) \(\text{kg}\) football player slides, given that his initial speed is \(\text{6.00}\) \(\text{m·s$^{-1}$}\) and the force of friction against him is a constant \(\text{450}\) \(\text{N}\).
Solution
Step 1: Analyse the problem and determine what is given
Friction stops the player by converting his kinetic energy into other forms, includingthermal energy. In terms of the work-energy theorem, the work done by friction, which isnegative, is added to the initial kinetic energy to reduce it to zero. The work done byfriction is negative,because \(F_f\) is in the opposite direction of the motion (that is,\(θ=\text{180}\text{ º}\), and so\(\cos\theta=-1\)). Thus \(W_{\text{non-conservative}} = -F_f \Delta x\).
There is no change in potential energy.
Step 2: Next we calculate the distance using the conservation of energy
We begin with conservation of energy:
\[W_{\text{non-conservative}} = \Delta EK + \Delta PE\]The equation expands to:
\begin{align*} W_{\text{non-conservative}} & = \Delta E_k + \Delta E_p \\ & = E_{k,f} - E_{k,i} + (0) \\ -F_f \Delta x & = E_{k,f} - E_{k,i} \\ -F_f \Delta x & = (0) - E_{k,i} \\ F_f \Delta x & = \frac{1}{2}mv_i^2 \\ \Delta x &= \frac{mv_i^2}{2F_f} \\ & = \frac{(\text{65.0})(\text{6.00})^2}{2(\text{450})} \\ & = \text{2.60}\text{ m} \end{align*}Step 3: Quote the final answer
The footballer comes to a stop after sliding for \(\text{2.60}\) \(\text{m}\).
Discussion
The most important point of this example is that the amount ofnon-conservative work equals the change in mechanical energy. For example, you must work harderto stop a truck, with its large mechanical energy, than to stop a mosquito.
Example: Sliding Up a Slope
Question
The same \(\text{65.0}\) \(\text{kg}\) footballer running at the same speed of \(\text{6.00}\) \(\text{m·s$^{-1}$}\) dives up the inclined embankment at the side of the field. The force of friction is still \(\text{450}\) \(\text{N}\) as it is the same surface, but the surface is inclined at \(\text{5}\) \(\text{º}\). How far does he slide now?
Step 1: Analyse the question
Friction stops the player by converting his kinetic energy into other forms, including thermal energy, just in the previous worked example. The difference in this case is that the height of the player will change which means a non-zero change to gravitational potential energy.
The work done by friction is negative, because \(F_f\) is in the opposite direction of the motion (that is, \(θ=\text{180}\text{ º}\)).
We sketch the situation showing that the footballer slides a distance \(d\) up the slope.
In this case, the work done by the non-conservative friction force on the player reduces the mechanical energy he has from his kinetic energy at zero height, to the final mechanical energy he has by moving through distance \(d\) to reach height \(h\) along the incline. This is expressed by the equation:
\begin{align*} W_{\text{non-conservative}} & = \Delta E_k + \Delta E_p \\ & = E_{k,f} - E_{k,i} + E_{p,f} - E_{p,i} \\ W_{\text{non-conservative}} + E_{k,i} + E_{p,i} & = E_{k,f} + E_{p,f} \end{align*}We know that:
- the work done by friction is \(W_{\text{non-conservative}} = - F_f d\),
- the initial potential energy is \(E_{p,i} = mg(0)=0\),
- the initial kinetic energy is \(E_{k,i}=\frac{1}{2}mv_i^2\),
- the final kinetic energy is \(E_{k,f}=0\), and
- the final potential energy is \(E_{p,f}=mgh=mgd \sin \theta\).
Step 2: Solve for the distance
\begin{align*} W_{\text{non-conservative}} + E_{k,i} + E_{p,i} & = E_{k,f} + E_{p,f} \\ (-F_fd) + \frac{1}{2}mv_i^2 + 0 & = 0 +mgd\sin \theta \\ (-F_fd) - mgd\sin \theta &=-\frac{1}{2}mv_i^2 \\ d (F_f + mg\sin\theta) & = \frac{1}{2}mv_i^2 \\ d & = \frac{\frac{1}{2}mv_i^2}{F_f + mg\sin\theta} \\ & = \frac{\frac{1}{2}(\text{65.0})(\text{6.00})^2}{\text{450}+(\text{65.0})(\text{9.8})\sin(\text{5.00})}\\ & = \text{2.31}\text{ m} \end{align*}Step 3: Quote the final answer
The player slides for \(\text{2.31}\) \(\text{m}\) before stopping.
As might have been expected, the footballer slides a shorter distance by sliding uphill. Note that the problem could also have been solved in terms of the forces directly and the work energy theorem, instead of using the potential energy.This method would have required combining the normal force and force of gravity vectors,which no longer cancel each other because they point in different directions, and friction,to find the net force.
You could then use the net force and the net work to find the distance \(d\) that reduces the kinetic energy to zero. By applying conservation of energy and using the potential energy instead, we need only consider the gravitational potential energy, without combining and resolving force vectors.This simplifies the solution considerably.
This lesson is part of:
Work, Energy and Power