How does mechanical energy work?

[Mr Davis 97]

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?

 

[paisiello2]

The roller coaster or the pendulum are simply geometrically constrained but are still governed by the potential gravitational energy that they have at any instance of time.

Making assumptions that there is no friction and all collisions are perfectly elastic is not very realistic but the conservation principle still holds.

 

[gleem]

For a conservative force as gravity the sum of the PE and the KE is constant. Thus (neglecting friction of course) you can say a change in PE plus the corresponding change in KE is equal to zero. Thus ΔPE = - ΔKE. If the PE decreases the KE increases by the same amount. In a gravitational field those changes depend only on the radial distance moved. Any solely lateral or sideways displacement does not produce a change in PE and thus the KE is not affected.

 

[DrClaude]

To add to gleem's answer, from the point of view of conservation of energy, in the absence of friction, air resistance, and inelastic collisions, there is no other form of energy into which mechanical energy will be converted, so it is conserved by itself. You can only exchange potential energy for kinetic energy and vice versa.

 

[CWatters]

...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.

Conservation of Energy says that any instant..

KE + PE = Constant ...... (1)

So a roller coaster trades KE and PE back and forth keeping the total energy constant.

If you start a coaster at height h and just let go it starts with PE = mgh and KE=0. Plug that into eqn1 and you get...

constant = mgh.

Lets say you want to calculate the velocity when the coaster has rolled down to new height h'

At that point the PE remaining = mgh' so eqn 1 becomes..

KE + mgh' = mgh
or
KE = mg (h - h')

In the case of a roller coaster KE also equals mv²/2 so you can write..

mv²/2 = mg (h-h')

Mass cancels and you can rearrange what's left to give an equation for the velocity some point h'..

v = SQRT{2g(h-h')}

 

[CWatters]

PS...

In the case of a roller coaster KE also equals mv²/2

Perhaps I should add that this is bit of a simplification. For example the wheels of the coaster might behave like flywheels so some of the KE maybe stored in the rotating mass of the wheels.