Definition of potential energy

[Leo Liu]

Potential energy is generally a function of position vector $\vec r$ and it is defined as $\int_i^f \vec F(\vec r)d\vec r=-U(\vec r) \bigg| _{i}^{f}=U(\vec r_i)-U(\vec r_f)$, where the force is conservative. Using the fact that the integral of force is also the definition of work, I obtain: $$K_f-K_i=U_i-U_f\implies K_i+U_i=K_f+U_f\equiv E_{mech}$$

What I would like to know are the reasons to introduce potential energy if work-energy theorem is sufficient for solving many problems. Any input will be appreciated!

 

[Dale]

I don’t understand the conflict you see between the two.

 

[Lnewqban]

... What I would like to know are the reasons to introduce potential energy if work-energy theorem is sufficient for solving many problems.

Perhaps the following definition?

https://en.m.wikipedia.org/wiki/Potential_energy

"The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall."

 

[Leo Liu]

"The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall."

Okay, I think I get it. But how would you explain where the energy of a planet orbiting a star comes from, as there was no force having brought the planet from the CM of the star to its current orbit?

 

[kuruman]

Mechanical energy conservation is derived from the work energy theorem by moving the work done by the conservative force (with a sign change) to the other side of the equation. You are completely correct when you say that the work energy theorem is sufficient to solve many problems without invoking mechanical energy conservation. However, if you use mechanical energy conservation, you can solve the same many problems without having to do any work integrals because the potential energy function has already done them for you.

 

[Lnewqban]

Okay, I think I get it. But how would you explain where the energy of a planet orbiting a star comes from, as there was no force having brought the planet from the CM of the star to its current orbit?

I believe most planets came from their stars.
Nevertheless, even if a free-flying planet or meteorite has been attracted by the gravitational field of a sufficiently close star, it will tend to escape from it following its original rectilinear trajectory, working against the existing field of gravity potential of the star.
If its original kinematic energy was not high enough, it will remain trapped and will continue orbiting instead.

 

[PeroK]

But how would you explain where the energy of a planet orbiting a star comes from, as there was no force having brought the planet from the CM of the star to its current orbit?

Where does any energy come from?

Where did the star and planet come from?

 

[Leo Liu]

Where does any energy come from?

Where did the star and planet come from?

It comes from the central force and our definition.
I think people usually define r=$\infty$ to have 0 potential energy; the closer the planet is to the star, the lower the potential it has. In this case, the planet is keeping a minimum distance from the star. So the $\Delta U$ from the orbit to the star is positive, implying that the energy at the orbit is higher.

"The formation and evolution of the Solar System began 4.5 billion years ago with the gravitational collapse of a small part of a giant molecular cloud.[1] Most of the collapsing mass collected in the center, forming the Sun, while the rest flattened into a protoplanetary disk out of which the planets, moons, asteroids, and other small Solar System bodies formed."
Well, the explanation above is just a hypothesis. No one really knows.
Ref: https://en.wikipedia.org/wiki/Forma...re thought,orbit around the central protostar.

 

[sophiecentaur]

how would you explain where the energy of a planet orbiting a star comes from

There are two issues here. The total Energy that a planet (or any other body) in orbit (around anything) can be treated as being 'just there' and you can work from that for predictions of future behaviour.

How the situation actually arrived is always pretty complicated but it has to arise due to interactions between more than just the two bodies (planet and star). A body, cruising around in space cannot suddenly be captured by a star it passes near because its Total Energy is greater than its PE, relative to the star. If it gets close, it will go faster and faster and leave again in a hyperbolic orbit. The body has to lose some of its energy, possibly by going past another body (say a big planet) and then it can be going in an appropriate direction at an appropriate speed (= Velocity) at a particular distance to end up in orbit. You will have heard about Slingshot Orbits, used by spacecraft to pinch some orbital energy from another planet and send them way out from the Sun. Same thing; not random but played for.

Jupiter is huge and it's responsible for the formation of the orbits of asteroids. Some of Saturns moons are responsible for the maintenance of the rings by 'shepherding' the dust and rock particles into temporarily stable orbits. But all orbits are potentially unstable because Jupiter or another big object can arrive at a position that destabilises things. I imagine Astronomers would not be aware if an object had been thrown out of Solar orbit because it's not there any more and too far away to spot a small object.

 

[A.T.]

But how would you explain where the energy of a planet orbiting a star comes from, as there was no force having brought the planet from the CM of the star to its current orbit?

Energy conservation doesn't apply to the big bang.

 

[Dale]

But how would you explain where the energy of a planet orbiting a star comes from, as there was no force having brought the planet from the CM of the star to its current orbit?

Why would the planet have started at the CM of the star?

Your system starts with a certain amount of energy, momentum, and angular momentum. From that point on the energy is conserved, but you have to start with a valid system. The planets don't start at the CM of the star so asking about the force moving it from the CM to its orbit doesn't make sense.

 

[sophiecentaur]

you have to start with a valid system

Agreed and there a many 'valid' systems that do not involve long lasting elliptical orbits.

 

[anorlunda]

The starting point for a solar system is a cloud of gas. So you have to track changes in energy and momentum starting from that.

 

[sophiecentaur]

The starting point for a solar system is a cloud of gas. So you have to track changes in energy and momentum starting from that.

That takes care of everything but extra solar guests which choose to stay.