# B Generalized version of work-energy theorem

#### donaldparida

I know that for rigid bodies only the work-energy theorem states that the net work done on the body equals the change in kinetic energy of the body since a rigid body has no internal degrees of freedom and hence no other forms of energy such as potential energy. Is there a most generalized form of work energy theorem that is valid for rigid as well as non rigid bodies and for conservative as well as non-conservative force?

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#### Doc Al

Mentor
This might not be what you're looking for, but it applies to rigid or non-rigid bodies, since it deals with center of mass quantities:
$$F_{net}\Delta x_{cm}=\Delta (\frac{1}{2}m v_{cm}^2)$$

#### Shreyas Samudra

This might not be what you're looking for, but it applies to rigid or non-rigid bodies, since it deals with center of mass quantities:
$$F_{net}\Delta x_{cm}=\Delta (\frac{1}{2}m v_{cm}^2)$$
No
We can apply work energy theorem for a rigid body or a point particle substituting the body(center of mass) in former case(applying work energy theorem for rigid body) we may have terms like rational kinetic energy,...
But in the later one(applying work energy theorem for centre of mass of the rigid body) we may have terms for work done due to those forces which did zero work on the rigid body(due to no movement of point of contact)

Willy nilly we get the same result

It all matters on what you choose your system to be
What might be potential energy in one case can be work done by external force in another case

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#### Doc Al

Mentor
Yes! (The theorem is a consequence of Newton's 2nd law.)

We can apply work energy theorem for a rigid body or a point particle substituting the body(center of mass) in former case(applying work energy theorem for rigid body) we may have terms like rational kinetic energy,...
One can certainly apply conservation of energy, but I thought the question was asking for something different.

But in the later one(applying work energy theorem for centre of mass of the rigid body) we may have terms for work done due to those forces which did zero work on the rigid body(due to no movement of point of contact)
Exactly! Those terms have the appearance of work, but do not reflect actual work done (in the first law of thermo sense). Nonetheless, the theorem is valid.

#### Shreyas Samudra

I know that for rigid bodies only the work-energy theorem states that the net work done on the body equals the change in kinetic energy of the body since a rigid body has no internal degrees of freedom and hence no other forms of energy such as potential energy. Is there a most generalized form of work energy theorem that is valid for rigid as well as non rigid bodies and for conservative as well as non-conservative force?
Hey
What happened??

#### donaldparida

@Doc Al, I think that your equation will work out for rigid and non-rigid objects but why is there no account of change in other forms of energy(potential energy, heat energy) in the equation? Another question:Does Fnet include the non conservative forces as well or only the conservative forces(is Fnet the resultant of the net conservative force and the net non-conservative force)?

#### Doc Al

Mentor
I think that your equation will work out for rigid and non-rigid objects but why is there no account of change in other forms of energy(potential energy, heat energy) in the equation?
Don't confuse this equation with the much more general conservation of energy.

Another question:Does Fnet include the non conservative forces as well or only the conservative forces(is Fnet the resultant of the net conservative force and the net non-conservative force)?
Fnet includes all forces acting on the body, whether conservative or not.

#### donaldparida

Don't confuse this equation with the much more general conservation of energy.

#### Shreyas Samudra

Video 1 (work energy theorem-18:00 to the end)
Video 2 (work energy theorem-full video)

He' a great prof. (simply awesome)
BUT YOU HAVE TO UNDERSTAND INDIAN ACCENT !!

#### Doc Al

Mentor
Sure. The equation I gave is derived from Newton's 2nd law, using the properties of the center of mass. The left hand side looks like a work term, but is not. (It's sometimes called center of mass work or pseudowork.) The work terms used in conservation of energy equations require the forces to be combined with the displacement of their point of application, not the displacement of the center of mass. And conservation equations can have all sorts of internal energy quantities.

#### donaldparida

So these equalities are valid for point particles, rigid bodies and non-rigid bodies, Right?
ΔK.E. = Wnet = Fnet . dcentre of mass = (Fnet conservative + Fnet non-conservative + Fnet external) . dcentre of mass = Wnet conservative + Wnet non-conservative + Wnet external
And the potential energy is included in Wnet conservative. Correct?But is there any proof that Wnet conservative=-ΔP.E.?
Also i could not understand one thing: do conservative forces change the K.E. and non-conservative forces change the P.E. or the opposite or do both the types of forces can change both the types of energy and what type of energy can external forces change.

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#### Doc Al

Mentor
So these equalities are valid for point particles, rigid bodies and non-rigid bodies, Right?
The center of mass equation that I gave is valid for all of these.
ΔK.E. = Wnet = Fnet . dcentre of mass = (Fnet conservative + Fnet non-conservative + Fnet external) . dcentre of mass = Wnet conservative + Wnet non-conservative + Wnet external
Careful about setting the center of mass "work" term equal to the total change in energy. Don't confuse that equation with the more general conservation of energy.
Also i could not understand one thing: do conservative forces change the K.E. and non-conservative forces change the P.E. or the opposite or do both the types of forces can change both the types of energy and what type of energy can external forces change.
All forces contribute to the change in the KE of the center of mass.

Here's what's tricky, and where the center of mass equation is useful. Imagine jumping into the air. Your KE obviously has changed, despite no external work being done on you. The center of mass equation still holds.

#### donaldparida

@Doc Al , I know that the work done by conservative forces is independent of the path taken, i.e., the work done by conservative forces in moving an object from an initial point to a particular final point is the same for different paths taken. Thus the potential energy of the object at the particular fixed point is same for different paths taken to reach the particular final point from the initial point and so it is wise and useful to use the concept of potential energy in case of the work done by conservative forces.
In case of non-conservative forces some difficulty arises because non-conservatives forces are path dependent. The work done by non-conservative forces is dependent on the path taken,i.e., the work done by non-conservative forces in moving an object from an initial point to a particular final point is different for different paths taken. Thus the potential energy of the object at the particular final point is different for different paths taken to reach the particular final point from the initial point and so it is not wise and useful to use of the concept of potential energy in case of the work done by non-conservative forces. Also mechanical energy is conserved in case of conservative forces while in case of non-conservative forces mechanical energy is not conserved. So far correct.Right?

Now i recently understood the reason why W net conservative = -ΔP.E.(which had been troubling me for long). The work done by conservative forces have been defined like this, so that the law of conservation of mechanical energy is complied to when no non-conservatives are acting on the body(In which case the law of conservation of mechanical is not complied to).Correct?

Also isn't it correct that both conservative forces and non-conservatives increase the potential energy of the body(although the concept of potential energy is not used for non-conservative forces) but we let the conservative forces account for all the changes in potential energy?

So in conclusion, All the types of forces(conservative, non-conservative and external forces) increase both the kinetic and potential energy of the body but we let the only the conservative force account for all the change in the potential energy.Right?

#### Doc Al

Mentor
Thus the potential energy of the object at the particular final point is different for different paths taken to reach the particular final point from the initial point and so it is not wise and useful to use of the concept of potential energy in case of the work done by non-conservative forces.
Potential energy is only defined for conservative forces.

Also mechanical energy is conserved in case of conservative forces while in case of non-conservative forces mechanical energy is not conserved. So far correct.Right?
True.

So in conclusion, All the types of forces(conservative, non-conservative and external forces) increase both the kinetic and potential energy of the body but we let the only the conservative force account for all the change in the potential energy.Right?
Not really. A change in potential energy can only involve conservative forces. If you think otherwise, perhaps you can give an example of what you mean.

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#### donaldparida

For example:When a non-conservative force moves an object from one point to another point there is indeed an energy change of the body and do you think of this:
And what do you think of this:The work done by conservative forces have been defined like this, so that the law of conservation of mechanical energy is complied to when no non-conservatives are acting on the body(In which case the law of conservation of mechanical is not complied to)

#### Doc Al

Mentor
For example:When a non-conservative force moves an object from one point to another point there is indeed an energy change of the body
Right. But if there is any change in PE it is due to the action of conservative forces. (By definition, essentially.)

And what do you think of this:The work done by conservative forces have been defined like this, so that the law of conservation of mechanical energy is complied to when no non-conservatives are acting on the body(In which case the law of conservation of mechanical is not complied to)
Sounds OK to me.

#### donaldparida

@Doc Al,If non-conservative forces do not change potential energy then why is it equal to ΔM.E.(ΔK.E.+ΔP.E.)?

#### donaldparida

@Doc Al ,In addition to the above question can you also tell that whether the work done by forces on the point of application equal to the work done on the center of mass of a body(Can we substitute the work done by the forces on the point of application by the center of mass)?

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#### Doc Al

Mentor
@Doc Al,If non-conservative forces do not change potential energy then why is it equal to ΔM.E.(ΔK.E.+ΔP.E.)?
Perhaps we're getting lost in semantics. Can a non-conservative force change the the PE of a system? Sure. But there must be conservative forces acting--else there would be no PE at all.

#### Doc Al

Mentor
@Doc Al ,In addition to the above question can you also tell that whether the work done by forces on the point of application equal to the work done on the center of mass of a body(Can we substitute the work done by the forces on the point of application by the center of mass)?
No, definitely not. The work done by forces on the point of application is the "real" work. A force can produce center of mass "work" yet do zero work against the point of contact. (Consider the example of jumping that I gave before.) And you can also have forces that do real work, yet no center of mass "work".

#### donaldparida

Oh now i got it!Thank you

#### donaldparida

No, definitely not. The work done by forces on the point of application is the "real" work. A force can produce center of mass "work" yet do zero work against the point of contact. (Consider the example of jumping that I gave before.) And you can also have forces that do real work, yet no center of mass "work".
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@Doc Al , Is it correct that for rigid bodies only, if a force act on it the work done on the center of mass is equal to the work done on the point of application of force?
Also i do not see how we can apply work energy theorem for non-rigid bodies since in general the work done on the center of mass is not equal to the work done on the point of application of force.
Also can a conservative force change the potential energy of a rigid body when it is lifted up?

#### jbriggs444

Homework Helper
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@Doc Al , Is it correct that for rigid bodies only, if a force act on it the work done on the center of mass is equal to the work done on the point of application of force?
One could come up with contrived examples of non-rigid bodies for which a particular force just happens to do "real work" equal to the "center of mass" work. This would require that the movement of the point of application in the direction of the applied force was equal to the movement of the center of mass in the direction of the applied force. [For example, one might hoist a bucket of swirling water from a well]

Also i do not see how we can apply work energy theorem for non-rigid bodies since in general the work done on the center of mass is not equal to the work done on the point of application of force.
You can locate the center of mass of a non-rigid body. You can apply a force to a non-rigid body. You can determine how much speed the center of mass will gain or lose based on the force applied and the distance the center of mass moves while the force is applied.

Also can a conservative force change the potential energy of a rigid body when it is lifted up?
It would be more correct to say that the potential energy of a rigid (or non-rigid) body is embodied by the conservative force. They are two ways of looking at the same thing. The conservative force acts to drain energy from the body as it is lifted up and acts to provide energy as it is allowed to fall back down. We look at the energy that is provided on the way back down and think of it as potential energy that was already present when the object was at its high point.

#### Doc Al

Mentor
@Doc Al , Is it correct that for rigid bodies only, if a force act on it the work done on the center of mass is equal to the work done on the point of application of force?
No, not even for rigid bodies is that true in general. Imagine an extended body, such as a rod. You can push the end of the rod, doing work on it, but some of that work will go into rotational energy, not center of mass energy.

@jbriggs444 gave an excellent response to your question; I encourage you to study it.