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April 23rd, 2015, 07:37 AM   #1
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Question Mechanical energy conservation

I am a little confused about how mechanical energy conservation operates when it comes to things like predicting velocity. I know that if conservative forces are the only forces acting on a body, then we can say that mechanical energy is conserved. This is simple to see when we have lateral up and down motion, but when it comes to predicting the velocity of a pendulum or a roller coaster (neglecting all friction) I'm not sure how the law operates. For example, given the initial peak height of the roller coaster, I can predict the velocity at any point, despite the fact that there are various loops and curves. And for a pendulum, the motion is in an arc. Despite these complexities, the same equations used for these situations are used for simple free-falling situations. Could someone give me a deeper understanding of how these equations are able to make predictions about velocity and such in complex situations like riding a roller coaster?
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April 23rd, 2015, 08:49 AM   #2
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Originally Posted by Mr Davis 97 View Post
I am a little confused about how mechanical energy conservation operates when it comes to things like predicting velocity. I know that if conservative forces are the only forces acting on a body, then we can say that mechanical energy is conserved. This is simple to see when we have lateral up and down motion, but when it comes to predicting the velocity of a pendulum or a roller coaster (neglecting all friction) I'm not sure how the law operates. For example, given the initial peak height of the roller coaster, I can predict the velocity at any point, despite the fact that there are various loops and curves. And for a pendulum, the motion is in an arc. Despite these complexities, the same equations used for these situations are used for simple free-falling situations. Could someone give me a deeper understanding of how these equations are able to make predictions about velocity and such in complex situations like riding a roller coaster?
One of the most profound equations in all of Physics is the Law of Conservation of Energy. One of the more general forms of this equation can be written as $\displaystyle W_{nc} = \Delta E$, where $\displaystyle W_{nc}$ is the sum of the work done by non-conservative forces (which can't be put in terms of a potential energy), and $\displaystyle \Delta E$ is the change in total mechanical energy of the system. $\displaystyle \Delta E$ has two kinetic energy terms (linear and rotational) and a number of potential energy terms.

Until you get into some really advanced Physics any derivation of $\displaystyle W_{nc} = \Delta E$ pretty much gives a much more general result than the problem that was used to derive it. One subtly though...In the usual Intro-Physics problems it does not predict velocities, it predicts speeds. To get a velocity you are more or less stuck with using Kinematics and not energy methods.

There are a vast number of problems one can use energy conservation on. The reason is usually that we define a "system" to be closed, thus the energy content inside the system must be constant. We can loosen that a bit by use of non-conservative forces acting on the system from outside but we usually just add the source of the non-conservative force as part of the system anyway. Investigation of closed systems is a big field that not only includes Mechanics but also Thermodynamics.

There is another big conservation law: The Law of Conservation of Momentum. When you get to Relativity these two Laws are combined into one: The Law of Conservation of the 4-Momentum.

-Dan
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April 23rd, 2015, 09:06 AM   #3
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Topsquark's answer is great. I just wanted to point out that there is nothing so special about rotation or directions such that Newton's laws or energy formulae break down. If you take rotating systems or systems that involve changes in direction, everything should still hold as normal (and indeed it does!). Both linear and rotational "contributions" to energy terms must be considered.

This does mean that you sometimes get unusual consequences that might seem counter-intuitive. Gyroscopic motion is a good example of a situation that conserves energy and momentum and yet gives resultant motion that seems totally bizarre. Another example is rotating reference frames and the Coriolis effect.
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April 24th, 2015, 11:16 AM   #4
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Originally Posted by Mr Davis 97 View Post
I am a little confused about how mechanical energy conservation operates...
Are you including potential energy?
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