Change in the potential energy of a star after explosion

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

I think : the question means that almost all of the potential energy gets used into the explosion.

If this is true then the potential energy gets reduced by $\frac { GM^2} R$ or if the star just gets transformed into a bigger sphere, then its final potential energy will be $\frac { GM^2} R_f$ , where $R_f$ depends on the final density. Since the final density is unknown, I can solve the question this way.

So, the correct opinion is (e).

Is this correct?

 

[haruspex]

almost all of the potential energy gets used into the explosion.

All of what GPE? Isn't GPE conventionally taken as zero at infinity, and negative everywhere else?
i do not think they mean GPE is used to create the explosion.

 

[Pushoam]

All of what GPE?

All of what GPE? - total amount of GPE

Isn't GPE conventionally taken as zero at infinity, and negative everywhere else?

So, does the question mean that the distance between particles become almost $\infty$ after explosion? This means that there is no star after explosion. In this case, the correct option is (e). Right?

 

[PeroK]

If some particles are moved further apart, does their total GPE increase or decrease?

 

[haruspex]

All of what GPE? - total amount of GPE

Taken literally, having no GPE would correspond to all having collapsed to an infinitely dense point. Compared with that state, all other states have infinite GPE. As I posted, convention is to say that zero GPE is the maximum, i.e. all matter scattered infinitely far from all other matter. GPE for any other configuration is relative to that, so negative. This is why it is meaningless to speak of all the GPE in the system being used somehow.

 

[Pushoam]

If some particles are moved further apart, does their total GPE increase or decrease?

Potential energy of two particle system is given by $\frac { -GMm}R$ , which is negative. So, as R increases, magnitude of potential energy decreases, but potential energy itself increases.

How is this going to help?

O.k.

So, the potential energy should increase by $GM^2/R$ .

Since there is no option like this, my assumption that potential energy of a sphere of mass M is $GM^2/R$ may be wrong.

Now, the potential energy of a sphere with uniform charge density $\rho$ and total charge q is calculated by U = $½\int_{\tau }\rho V d\tau$ ...(1)

Here, V ( $\vec r$ ) is potential due to the whole configuration at $\vec r$.

$V(\vec r) = \frac { kq}{2R} \left( 3 - \frac { r^2} {R^2} \right)$ ...(2)

Now, the gravitational potential will be = $V(\vec r) = \frac { -Gq}{2R} \left( 3 - \frac { r^2} {R^2} \right)$ ...(3)

Putting (3) into (1) and doing the integration,

$U = \frac { -3GM^2}{5R}$

So, energy gets increased by $\frac { 3GM^2}{5R}$. The correct option is (c).
Is this correct?

 

[Pushoam]

Taken literally, having no GPE would correspond to all having collapsed to an infinitely dense point.

I didn't understand this.
No GPE will mean : A) there is one particle with mass M, that is the infinitely dense point particle. I think this is what you mean here.

B) All the constituent particles are infinitely away from each other i.e. not interacting with each - other. Since, the star explodes instead of collapsing, here we have to consider this case.

Compared with that state, all other states have infinite GPE.

I didn’t get this. In case of infinitely dense particle, potential energy is not well defined as there is no other particle.

In all other states, the system will have some size. It will be either a system of particles or continuous mass distribution. So, in other states, potential energy of the system is well – defined and finite.

As I posted, convention is to say that zero GPE is the maximum, i.e. all matter scattered infinitely far from all other matter. GPE for any other configuration is relative to that, so negative.

This I understand.

This is why it is meaningless to speak of all the GPE in the system being used somehow.

O.k.

What I have understood is: in this problem, since the GPE is negative, GPE is not that kind of energy which could be used. Some other form of energy has to be given to the star to get it exploded.

From where does this energy come?

Does the kinetic energy of constituent particles get converted into potential energy ?

 

[Pushoam]

Mistake : The potential energy of a solid sphere is not $\frac {-GM^2} R$. It is $\frac {-3GM^2}{5 R}$.

 

[jbriggs444]

All of what GPE?

All of what GPE? - total amount of GPE

Taken literally, having no GPE would correspond to all having collapsed to an infinitely dense point.

I didn’t get this. In case of infinitely dense particle, potential energy is not well defined as there is no other particle.

You are speaking of GPE as if it were a fixed positive quantity -- an amount of energy available to fuel an explosion, for instance. That view does not hold up under scrutiny. @haruspex is trying to take that viewpoint to its logical conclusion to show why it does not hold up.

If there is such a thing as "all of the GPE" (a positive quantity) then there must be such a thing as "none of the GPE" (zero potential energy). Naturally, that would have to be the configuration in which potential energy is at its minimum. The configuration in which potential energy is minimized is with all the mass concentrated in an infinitely dense central point. But if that potential energy is zero then the potential energy at any other point would need to be infinite.

You point out that this does not make sense. That the potential energy at infinite density is undefined. Indeed so. That is the point. A reference to "all of the GPE" involves an implicit claim that the potential energy for an infinitely dense object is well defined. This is a an example of reductio ad absurdum. If a claim leads to an absurd consequence then that claim must be false. Therefore there is no such thing as "all of the GPE" in the sense that phrase was originally invoked.

You are now asking about the energy that powers the explosion:

From where does this energy come?

The answer is that we are not told. Furthermore, it does not matter. Perhaps there was a burst of nuclear fusion. Whatever the cause, the star has exploded and is now spread out in a much less dense configuration.

 

[Pushoam]

You are speaking of GPE as if it were a fixed positive quantity -- an amount of energy available to fuel an explosion, for instance.

I read my original post again. Yes, the original post does speak like the above.
I took GPE to be a positive quantity, which is wrong.
GPE of this system is a fixed negative quantity. And negative of this GPE should be given to the system to explode it.
Is this correct?

 

[jbriggs444]

I read my original post again. Yes, the original post does speak like the above.
I took GPE to be a positive quantity, which is wrong.
GPE of this system is a fixed negative quantity. And negative of this GPE should be given to the system to explode it.
Is this correct?

Correct.

However, I would quibble a bit about the word "fixed". Potential energy is not a fixed absolute quantity. Instead, it is a relative quantity. It is relative to an arbitrary reference that one decides to consider as the zero point. Only differences in potential energy are physically meaningful. "Fixed absolute" potential energies depend on a choice of reference level and are physically meaningless unless that reference level is specified.

If you re-read the original problem statement, you can see that the author was careful never to speak of potential energy in the absolute sense. Instead the problem asks about the difference in potential energy.

 

[Pushoam]

GPE of this system is a fixed negative quantity

So, the more correct statement is :
The difference between the GPE of this system ( before explosion) and after explosion is a fixed negative quantity.Right?

 

[jbriggs444]

So, the more correct statement is :
The difference between the GPE of this system ( before explosion) and after explosion is a fixed negative quantity.Right?

Ummm. [I think we are in complete agreement and just dealing with how to phrase things]

It is a fixed quantity. Whether you consider it negative or positive depends on which you are subtracting from which. The word "difference" may not be the best choice to make that distinction clear.

The change in GPE going from initial state to final state is a fixed positive quantity.

The GPE of the initial state relative to the final state is a fixed negative quantity.