April 23rd, 2015, 06:37 AM  #1 
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus  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 freefalling 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?

April 23rd, 2015, 07:49 AM  #2  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,554 Thanks: 596 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
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 IntroPhysics 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 nonconservative forces acting on the system from outside but we usually just add the source of the nonconservative 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 4Momentum. Dan  
April 23rd, 2015, 08:06 AM  #3 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 1,928 Thanks: 628 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
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 counterintuitive. 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. 
April 24th, 2015, 10:16 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 16,775 Thanks: 1236  

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