Dissipative Forces
![[url=https://pixabay.com/en/sailboat-boat-ship-sail-sailing-2423484/]"Sailboat"[/url] by Denis_Azarenko is in the [url=http://creativecommons.org/publicdomain/zero/1.0/]Public Domain, CC0[/url]
A sailboat is a wonderful example of a non-conservative force (the wind) doing positive work on the boat.](https://www.geogebra.org/resource/ZEy535as/6nZUsYrwmzOq6Zxg/material-ZEy535as.png)
Before discussing sailboats, let's continue the discussion of our simple ball/earth system. In order for the ball-earth system to gain or lose both kinetic and potential energy, something outside the system would have to provide energy or take energy. That could be me. I could do work and lift the ball against the will of gravity by exchanging chemical energy from my breakfast for the ability to do work on the ball and lift it away from earth. The work I do in that process would need to be part of the sum. Work done by friction or air drag or a motor would also be encompassed by that sum.
Another name given to such non-conservative forces is dissipative forces. Dissipate in English just means to scatter around or disperse something. These non-conservative or dissipative forces do just that with energy. Little bits of energy go to many different places. Consider, for instance air drag. When a car loses energy to air drag, what really happens is that innumerably many little air molecules get stirred up or knocked around by the car, and acquire some of the car's energy. Even when I do something directed and useful like lifting a box, I am also creating lots of heat and giving it off to my environment. Recall that humans are only around 25% efficient. The rest becomes heat. In that sense, humans always exert dissipative forces when they move things around or pedal bicycles.
When dealing with sliding friction while dragging a box across a floor, the molecules in the floor and on the underside of the box heat up a bit and acquire some more energy. That heat means that very many little molecules vibrate more energetically - taking energy from the box as a whole.
Using the present formalism of energy and work terms to determine motion, we would have a notoriously difficult time calculating any of these dissipative work terms explicitly. The reason for this is that as you know, work involves a path integral of force and displacement vectors. Evaluating such integrals for the dissipative forces is difficult or impossible because generally the force terms are path dependent and/or velocity dependent in a very complex way that lends itself better to just solving as we've done in the past, by Newton's laws. Furthermore, it's difficult via energy considerations to even find the path that a particle or system may take while subject to both conservative fields and dissipative forces.
Problems We Can Do
In the energy formalism of this chapter, there is one class of problem that we can easily solve - and in this case much easier than using forces. In the event that we know all the parameters necessary to calculate we may easily equate that with the work done by dissipative forces.
EXAMPLE: A skier of mass 90kg starts at the top of a slope during a giant slalom race. They start from rest and descend vertically by 700m. They cross the finish line going 40m/s. How much energy is lost to drag and sliding friction together?
SOLUTION: It is precisely this sort of problem where this methodology shines. At first it'd seem we need more information - like the path specifics and the speed during the run since the drag depends on it. However, all we need to do is note that whatever loss of potential energy fails to show up as kinetic energy must be lost to friction. Read that again if you didn't get it. It's important. In other words, in the absence of dissipative forces, we always expect , which means as we lose potential energy, we should gain kinetic energy. If this sum is non-zero, there must be some work done by non-conservative forces on the system - in this case work by sliding friction of the skis on the snow and air drag on the skier. So we can find the amount of energy lost by: (assuming that the initial speed is zero). This leads to -558kJ. The fact that the non-conservative work is negative means energy was taken from the system, or lost to the non-conservative forces of drag and sliding friction.
Non-Conservative Positive Work
I want to be sure that you don't think non-conservative work terms must always be negative. Consider a sail boat. Suppose a sailboat is sitting in a lake on a calm day when a wind suddenly arises. The boat goes from rest to 4m/s as the wind blows. Clearly the wind force, which is non-conservative, is doing positive work on the sailboat. If the wind keeps propelling the sailboat for the next hour as it traverses the lake, how much work did the wind do? We can't know. That's because there was also the non-conservative hydrodynamic drag on the boat as it moves through the water. That work IS negative. All we can say is that at the end of the hour, if the boat is still moving at the same 4m/s, the boat has no change in K, and therefore the work done by the air on the sails is exactly equal and opposite to the work done by hydrodynamic forces so that the net work done is zero - leading to no change in K.
As another example, humans, as mentioned earlier, always do work in a non-conservative way. If our muscles were conservative we'd climb a mountain on foot and perhaps feel tired, but on descending the same mountain we'd recover all of the energy we spent climbing. We'd arrive back at our starting point having burned no more calories than we would have had we never hiked at all. Of course that does not happen. We spend energy on the way up the mountain and continue to spend it while descending. Where did it go? With the increase in body temperature, we were causing a little global warming. Hopefully exercise won't become illegal anytime soon due to its contribution to the melting of the polar ice caps!
As yet another example, the work done by gasoline engines is non-conservative, but positive. We can't run an internal combustion engine in reverse and recover gasoline. That's how you know it's non-conservative. What's burned is burned, just like exercising. Nonetheless, positive work is done by engines as cars drive along roads. Being non-conservative means that along with the work being done by the engine to propel the car down the road, there is also a lot of environmental heating going on. The majority of the energy of combustion, in fact, turns into nothing but heat.
In contrast to gasoline engines, the technology of electric motors actually allows for considerable recovery of energy. In other words, cars that employ electric motors powered by batteries will drain the batteries while picking up speed as the motors do positive work, but will replenish the batteries while slowing back down and doing negative work. The reason they can accomplish this is that an electric motor is one and the same thing as an electric generator. The details of this will have to wait until second semester, but being true it certainly is a good argument for using an electric motor in vehicle traffic that does a lot of repeated speeding up and slowing down - like traffic in southern California where I am writing this. Keep in mind that the vehicle must still overcome air drag, which is energy permanently lost to the environment. However, about 85% of kinetic energy due to the car driving down the road can be recovered while slowing down and using the motors as electric generators. The irrecoverable 15% is due to heating in wires, batteries, etc, that is lost to the environment for good.
Contrast this to a normal gasoline-powered car that turns 100% of its kinetic energy to heat (in the brake discs) as it comes to a stop. That energy is gone for good. So getting 85% back is great! That energy recovery is the motivation for driving fully electric vehicles.
There is also a class of vehicle (the hybrid) that has just enough battery capacity to drive you for a few miles with a modest electric motor/generator. The idea is that there is a small-capacity battery that recovers energy upon braking just like in the electric car, and then that stored energy powers the electric motor when the car wishes to pick up speed again. At steady cruise the vehicle depends on a normal gasoline engine. So the hybrid technology makes up for the least efficient aspect of a gasoline vehicle - the total loss of energy to the environment upon braking. Furthermore, there are no awkward range considerations in a hybrid vehicle when, for instance, on a long road trip. Range, on the other hand, is currently one major drawback for fully electric vehicles. I expect both range and charging time (the other major drawback) to only improve in the coming years.