Class content > Energy: The Quantity of Motion
Prerequisites
Starting with the Newtonian framework for describing motion -- objects, interactions, Newton's laws -- and asking the question, "what part of Newton's 2nd law tells us how an object changes its speed (irrespective of direction)", we were led to introduce the concept of kinetic energy -- ½mv2. The work-energy theorem (derived from Newton's 2nd law) led us to introduce the idea of potential energy -- energy connected to interaction forces. The concept of a potential energy lets us see interaction forces as a way to "store" or "release" kinetic energy.
At this point, we have determined expressions to quantify potential energy due to gravity, electricity, and springs. To calculate the potential energy associated with an interaction force between two objects we first treated a single object as our system and considered how much work the interaction force does on our object of interest via the work-energy theorem. Then we changed our perspective and considered both objects as our system. In that case the interaction force between the two objects within the same system allows the system to "store" potential energy in the interaction force.
For example, the kinetic energy of an object thrown upward is "stored" in the greater displacement of the object from the earth and then released when the object falls back down. The kinetic energy of an object hitting a spring is "stored" in the squeeze of the spring. As one positive charge approaches another fixed charge head on it will slow and "store" its kinetic energy in its closeness to the fixed charge.
We were able to do a quantitative calculation for three of the forces we have studied: gravity, electricity, and springs. We focused on cases where all potential energy can get transferred into kinetic energy of only one of the interacting objects. Since we can interpret normal and tension forces as a kind of very stiff spring force (see Young's modulus), we could define potential energies for these as well and we will when it turns out to be convenient.
Finally, what about our other forces? Our resistive forces?
Conservative vs. non-conservative forces
It turns out that we cannot define potential energies associated with resistive forces. We can see this at the macro level by remembering the condition that we needed to turn work into a potential energy: the work done by the force had to be reversible, i.e. energy stored by the force needs to be able to recreate the KE it had drained, that is, if a force decreased the KE in one direction, it had to restore it when the object moved back in the opposite direction. Whats more, the work only depended on the starting and ending positions and not on what happened in between. This way, if you reversed the direction, you would reverse the sign of the work and get the PE back when you reversed. Forces for which this works are called conservative.
This requirement fails for the resistive forces of friction, viscosity, and drag. For two relatively moving objects that exert resistive forces on each other, the forces always point to oppose the relative motion. As a result, the work is negative -- the force and displacement are in opposite directions (and cos 180 = -1) so it devours KE. If you try to reverse the motion, the forces reverse too and the work is still negative. It continues to devour KE instead of restoring it. Forces like this -- whose work can't be written as a change in a potential energy -- are called non-conservative. The key marker of non-conservative forces is velocity dependence. The force doesn't only depend on the object's position, but on its velocity as well.
The simplest example is the friction force acting on a block sliding on a table. If we give it a push the block will slide, but the friction from the table will slow it down and bring it to a stop. If we push it back the other way, the friction will still slow it down -- not speed it up, restoring the energy it stole. So the work done by these kind of forces cannot be written as a change in a potential energy. For friction, the dependence of the force on velocity is not its magnitude, but only its direction. That's enough.
A work-energy theorem with Potential Energy
Potential Energy made things nice and gave us a good way to think about motion, so we would like to take advantage of this even if there are non-conservative forces in our system. Fortunately, we can still include non-conservative forces via the work energy theorem. What we need to do is consider our system to include only interactions that we can describe as PE, and separating the non-conservative interaction forces as interactions that cross the system boundary. A system schema for a situation like this is shown below. Objects A through D interact through conservative forces, but the interaction of object A with object E is non-conservative, so Object is is considered outside our system of interest.
In this case we define our system by keeping all the conservative forces inside the boundary (which we can do now that we know how to calculate the PE for these interactions), and leaving the non-conservative force outside the system. We write the work done by conservative forces as a PE and leave the work done by non-conservative forces as work.
The kinetic energy here should be the sum of the kinetic energies of all objects within the system. Since in many cases you will encounter many of the other object are stationary (such as the three cases described earlier with gravity, spring forces with a fixed spring, and interactions with fixed charges) we often only have to consider the kinetic energy of a single object.
A conservation law
This equation does provide us with a conservation law and the conditions when it can be used:
The principle of conservation of mechanical energy: When an object feels conservative forces from other objects whose position can be considered fixed and when the resistive forces felt by that object can be ignored, the work-energy theorem takes the form of a conservation law:
where U represents the sum of all potential energies -- gravitational, electric, and spring.
OR
When there are only conservative forces acting within a system, the energy within that system is conserved.
When is it useful?
This way of describing motion as an exchange between kinetic and potential energies is particularly useful when we can ignore resistive force and when we are comparing an initial and a final condition without worrying about what happens in between the initial and final time, or how long it takes to get from the initial to the final condition. If you're sliding down a (nearly) frictionless slide and want to know how fast you are going when you get to the bottom of the slide-- but don't care how long it takes you -- the conservation of mechanical energy will tell you. If you throw a ball straight up at a certain speed and want to know how high it goes -- but don't care when it gets to the top-- the conservation of mechanical energy will tell you.
What's really going on with resistive forces?
What is the conservation of mechanical energy really about? We know that total energy is conserved. How is this theorem about "mechanical energy" different? What's going on with our resistive forces that seem to "drain" or destroy mechanical energy? The answer here is that resistive forces move mechanical energy -- coherent energy where the KE of motion is associated with momentum (KE = p2/2m) -- into internal or thermal energy. This is still KE, but it is now associated with random internal motion so there is no net momentum associated with it. This is discussed in more detail in the follow on, Mechanical energy loss -- thermal energy.
Short note about system-thinking
Now that we've seen that mechanical energy is conserved for systems that only have conservative interaction forces, we can also use the system schema to investigate how the energies flow even when only conservative force are present. Lets say we are interested in how much work object B does on the rest of the system. To study this, we move object B outside the system boundary (dashed line), as shown below. Energy conservation tells us that the only force that changes the energy within our dashed line system is the interaction with object B by connection with a spring.
Then we could write the energy conservation principle as:
This equation allows us see how much work object B does on the rest of the system, and where the work could be going into: kinetic or potential energy of the objects A,C, and D or their electrical or gravitational interactions.
Joe Redish 11/5/11
Vashti Sawtelle 11/19/12
Wolfgang Losert 11/20/12
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