Conservation of mechanical energy

In summary, the conversation discusses a problem with the train of thought regarding the relationship between total energy, kinetic energy, and potential energy. The conversation also delves into the concept of potential energy and its inclusion in equations such as the 1st law of thermodynamics and the total mechanical energy equation. It is concluded that the two equations cannot be equated as they have different definitions of ΔE.
  • #1
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I have a problem with my train of thought.

I have narrowed it down to several premises, and I think that one of them must be wrong.

Premise 1

Δ total energy (E) = Δ kinetic energy (K) + Δ potential energy (U)

(assuming that there is no change in any of the other forms of energy in the system)


Premise 2

Δ E = q + w

Consider the system of a block.

The block is initially traveling upwards (+y direction), away from the Earth's surface.

At t=0 it has a kinetic energy of 1/2mv2 (v is non zero). It initially also has gravitational potential energy of 0.

The velocity eventually decreases to 0 (at t1). The whole time (from t=0 to t1), the Earth's gravitational attraction is exerting a force of mg on the block.

No other force is acting on the block, since it is already in motion at t=0.

It appears that the work done by the gravitational force on the block is -mgh.

But, ΔE = 0, because the kinetic energy has simply been transformed into potential energy.

So, how can work be non zero (-mgh)?

It seems that (looking at the block as the system)

ΔE = 0 = W= -mgh

(no heat transfer, so q=0, no friction, no other energy transfer)

Since this is impossible, I know that I did something wrong.

Thanks for your help!
 
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  • #2
Welcome to PF!

Hi moderate! Welcome to PF! :smile:

You've been double-counting :wink: …

Potential energy is just another name for (minus) work done (by a conservative field such as gravity). :biggrin:

From the PF Library …

Is potential energy energy?

There is confusion over whether "energy" includes "potential energy".

On the one hand, in the work-energy equation, potential energy is part of the work done.

On the other hand, in the conservation-of-energy equation (and conservation of course only applies to conservative forces), potential energy is part of the energy.​
 
  • #3
Hey - thanks for the quick response and the welcome!

Let me make sure I understand what you are saying.

For the 1st law of thermodynamics ("conservation-of-energy equation"), ΔE=q+W, potential energy is already accounted for (on the left side), so I do not need to add another work term?

Essentially:

ΔE=q+W

and ΔE=ΔK

therefore:

ΔK=q+W
ΔK=0 + (-mgh)
ΔK + mgh = 0

And, mgh = ΔU (this is what it is called by convention, "gravitational potential energy")

Correct?

This makes me slightly uncomfortable. Let me think about why.

Ok, so when I was taught that

ΔE=ΔK+ΔU, this can not be equated to the 1st law of thermodynamics?

The two terms with the same name (ΔE) in different equations include different components of what is called "mechanical energy"? One includes gravitational potential energy and kinetic energy, and the other doesn't.

In other words, "mechanical energy" is not some inherent form of energy, but it is made up of kinetic energy (which is "inherent" to the object), and the work due to gravity that was brought over from the other side of the ΔE=q+W equation?

This seems awkward. :uhh:
 
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  • #4
Now you're just confusing me. :redface:

The simple rule to remember is that you can't count anything twice.

So, since potential energy and work done are the same thing, you can't use both. :smile:
 
  • #5
Yeah, sorry, I posted before I had my thoughts in order. :blushing:

My main point can be summed up like this:

ΔE is a term that appears in two equations:

ΔE=q+W (1st law of thermodynamics) and ΔE=ΔK+ΔU (total mechanical energy)

However, ΔE is not the same in both cases. In one case (left equation) it includes only kinetic energy. In the right equation, it includes both kinetic and potential energy. So, the two can not be equated.

I hope that this was more clear.

(what you are saying makes sense)
 
  • #6
moderate said:
ΔE is a term that appears in two equations:

ΔE=q+W (1st law of thermodynamics) and ΔE=ΔK+ΔU (total mechanical energy)

However, ΔE is not the same in both cases. In one case (left equation) it includes only kinetic energy. In the right equation, it includes both kinetic and potential energy. So, the two can not be equated.

Yes, that's right! :smile:

'fraid you just have to live with it! :redface:​
 

1. What is the law of conservation of mechanical energy?

The law of conservation of mechanical energy states that the total amount of energy in a closed system remains constant. This means that energy cannot be created or destroyed, it can only be transformed from one form to another. In the case of mechanical energy, it refers to the sum of potential and kinetic energy in a system.

2. How is mechanical energy conserved?

Mechanical energy is conserved through the interplay of potential and kinetic energy. As an object moves in a gravitational field, it gains kinetic energy as it speeds up and loses potential energy as it moves to a lower position. These changes in energy balance each other out, resulting in the conservation of mechanical energy.

3. Can mechanical energy be converted into other forms of energy?

Yes, mechanical energy can be converted into other forms of energy, such as thermal energy or electrical energy. For example, when a moving object collides with another object, some of its kinetic energy is converted into thermal energy due to friction. This conversion does not violate the law of conservation of energy as the total energy in the system remains constant.

4. What are some examples of conservation of mechanical energy in everyday life?

One example of conservation of mechanical energy in everyday life is a pendulum. As the pendulum swings back and forth, it continuously converts between potential and kinetic energy, but the total amount of energy remains constant. Another example is a rollercoaster, where potential energy is converted into kinetic energy as the car moves down a hill, and then back into potential energy as it moves up the next hill.

5. What is the significance of conservation of mechanical energy in scientific research and technology?

The law of conservation of mechanical energy is a fundamental principle in physics and is essential in understanding and predicting the behavior of physical systems. It is also crucial in the development of technologies, such as energy conversion and storage systems, where the conservation of energy must be taken into account for efficient and sustainable designs.

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