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Coaster Dynamics
Physics Primer
Chapter 13
Friction and Nonconservative Work
In the previous chapter, the only forms of energy considered were kinetic and potential energy. While many meaningful analyses can be performed using just kinetic and potential energy, we must expand our model for the conservation of energy to include non-conservative forces -- otherwise, our roller coaster car would never stop!
Conservation of Energy and Nonconservative Work
Work is called conservative if it occurs without any energy loss. It might naturally follow, therefore, that nonconservative work would entail the loss of energy.
For instance, one of the most common nonconservative forces is friction. When an object moves with friction, some energy is "lost." For example, if you were to push a roller coaster car from rest up (but not over) a hill, and then let it roll back down, it will not return to the exact starting point -- instead, it will stop a little short of the starting location. If there was no friction, the work would be perfectly conservative, and the car would return to the exact starting point. The fact that it doesn't means that some energy must have been lost along the way. However, what really happened to the missing energy?
The energy associated with the friction was only "lost" in the sense that we are no longer able to keep tabs of it with our "model" of the system. In reality, the friction energy is not lost or destroyed, it is converted to other forms of energy -- primarily heat and sound (so the General Principle of Energy Conservation remains valid). However, our model of a roller coaster car doesn't include heat or sound... so we come up short in our "energy bookkeeping."
Obviously, friction (and air resistance, as well) must be included in our model of a roller coaster car in order to accurately simulate its behavior. Therefore, equation (12-1), which we previously used for the conservation of energy, must be modified slightly, as follows:
(13-1)
E =
KE +
PE + Wnc = 0
where Wnc is the total nonconservative work.
Nonconservative Forces and Work
There are many examples of nonconservative forces... however, the only ones we include in our roller coaster model are friction and air resistance.
The force of friction is a somewhat difficult phenomenon to model, as there are different forms of friction -- including static friction, sliding friction, and rolling friction.
If you place a flat object on a table with a rough surface and push lightly on it, the table will push back (in accordance with Newton's Third Law of Motion) and the object won't move. If you push hard enough, though, the object will "break free" and begin to slide.
The amount of force initially offered by the table is greater than the force that occurs after the object begins to slide. The initial force is from static friction (or sometimes called "sticktion"). It is caused by the extremely small protrusions on the interacting surfaces that tend to "lock" the surfaces in place. Because of this, the force required to break this "sticktion" bond is relatively large, compared to the force of sliding friction. The magnitude of static friction is difficult to quantity... and relatively little experimental data is available to provide for accurate mathematical models of static friction.
Once the surfaces begin sliding motion, the friction force decreases... and perhaps surprisingly, remains relatively independent of the speed of the sliding motion. To a reasonable approximation, sliding friction is modeled as follows:
(13-2) Fs = (N)(cs)
where Fs is the force of sliding friction, N is the normal force (usually equal to the weight), and cs is the coefficient of sliding friction. The coefficient of sliding friction is a dimensionless number, usually obtained from experimental data for the surfaces of interest.
For a rolling surface, such as a wheel, the model of rolling friction is of the same form as for sliding friction. However, the coefficients of rolling friction are typically much lower than those for sliding friction. The model typically used for rolling friction is:
(13-3) Fr = (N)(cr)
where Fr is the force of rolling friction (newton), N is the normal force (newton), and cr is the coefficient of rolling friction (dimensionless).
Air resistance is also a nonconservative force that should be included in an accurate roller coaster model. In older roller coasters, air resistance had little affect. However, with today's high speed designs, air resistance becomes significant as the speeds become large. The model typically used for air resistance is:
(13-4) Fd = (cd)(A)(d)(v²/2)
where Fd is the force due to air resistance (newtons), cd is the drag coefficient (dimensionless), A is the projected area of the object (m²), d is the air density (kg/m³), and v is the object's velocity (m/s²).
In a properly designed roller coaster, the car will reach the end of the track with near-zero velocity. In this case, essentially all the initial energy is consumed by the nonconservative work of friction and air resistance. Thus, from equation (13-1), we get the following relationship for the total nonconservative work:
(13-4) Wnc = Wf + Wd = PE0 + KE0
where Wnc is the total nonconservative work, Wf is the work done by rolling friction, Wd is the work done by air resistance, PE0 is the initial potential energy, and KE0 is the initial kinetic energy.
Typically, the initial potential energy, PE0, is equal to the potential energy at the top of the lift hill. Usually, the speed of the roller coaster at the top of the lift hill is very low (just enough to push the car over the edge). This means that the initial kinetic energy is usually very small compared to the initial potential energy. Thus, KE0, can typically be neglected in any calculations.
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