# Understanding the work-kinetic energy theorem

• fog37
In summary: KE while increasing the gravitational PE. If the system is defined as the spacecraft +Earth, then there is only one external force which is the rocket's propulsion force. This net external force increases the KE of the system and the mechanical energy of the system must change. In summary, the work-kinetic energy theorem states that the work done by any force, whether conservative or non-conservative, is equal to the change in kinetic energy of the body. This theorem does not consider potential energy, which cannot be associated with non-conservative forces. When the rocket propulsion on a spacecraft lifts it vertically upward, the propulsion force increases kinetic energy while gravity tries to decrease it and increase gravitational potential energy. The net work done on the system is
fog37
Hello,
the work-kinetic energy theorem only considers kinetic energy KE and not potential energy PE.
This theorem states that the work done by any force, be it conservative or non conservative, is equal to the change in kinetic energy of the body. I know that potential energy cannot be associated to nonconservative forces because the potential energy would be multi-valued and ambiguous in that case.
• The portion of the kinetic energy change due to the work done by conservative forces is equal to the negative change of PE, correct?
• What happens when the rockets on a spacecraft lift the craft vertically upward? The nonconservative force of the rockets change (increase) the kinetic energy of the spacecraft changes but also the gravitational potential energy increases (due to the increase in height). How do we reconcile the fact that this nonconservative force (rocket propulsion) changes the gravitational PE? If it does not, what force is providing the energy that gets stored in the gravitational PE?
Thanks,
fog37

fog37 said:
The portion of the kinetic energy change due to the work done by conservative forces is equal to the negative change of PE, correct?
OK.

fog37 said:
How do we reconcile the fact that this nonconservative force (rocket propulsion) changes the gravitational PE?
Not quite sure what you think needs reconciliation. The two forces (propulsion and gravity) both do work on the rocket. The work done by gravity can be expressed as the negative change of PE. I don't see how the work-energy theorem is impacted by that.

Statements regarding Conservation of energy are highly dependent on the choice of the system. In your case, if you think of the rocket as the system, then the propulsion + gravity are external agents. The change in kinetic energy of the rocket is equal to the net work done by the external forces of propulsion and gravity. Note that as the rocket lifts off, the work done by gravity is negative.

Ok, thank you to both.

It all depends on how we define the system, i.e. what is part of it and what is outside of it.

If the system is the spacecraft alone, then then there are two external forces on it: the rocket's propulsion force and the gravitational force of attraction. In this case the propulsion force increase KE while the gravitational force F_g tries to decrease KE while increasing the gravitational PE. Overall, net work is done by the system by the external forces. The mechanical energy of the system must change.

If the system is defined as the spacecraft +Earth, then there is only one external force which is the rocket's propulsion force. This net external force increases the KE of the system. The mechanical energy of the system must therefore change. What about the increase of gravitational PE? Gravity is an internal force in this case and the spacecraft and Earth are getting farther from each other so the potential energy between two increases. the work done by gravity to decrease KE is equal to the negative change in PE...Is that correct?

Any corrections? I think I am close to getting this...

thanks!
fog37

fog37 said:
If the system is the spacecraft alone, then then there are two external forces on it: the rocket's propulsion force and the gravitational force of attraction.
Right.

In this case the propulsion force increase KE while the gravitational force F_g tries to decrease KE while increasing the gravitational PE.

Let us first make sure we all understand what the system is: it is that part of the spacecraft that is not converted to burning ejected exhaust gas.

The gravitational PE is increasing, but it is unfortunate to say "the gravitational force F_g tries to decrease KE while increasing the gravitational PE". It is not the gravitational force that increases the PE, but the fact that the rocket is ascending, which is due to the propulsion force that is great enough to make it ascend.

Overall, net work is done by the system by the external forces. The mechanical energy of the system must change.
I think you intended to write on the system. Then yes, sum of works of external forces equals, with very good approximation, change of total mechanical energy of the system. It is an approximation because the concept of mechanical energy does not in general satisfy conservation of energy - some part of the work will end up as internal energy of the bodies (rocket engine/ spacecraft chassis vibrations that eventually make the spacecraft hotter).

If the system is defined as the spacecraft +Earth, then there is only one external force which is the rocket's propulsion force. This net external force increases the KE of the system. The mechanical energy of the system must therefore change.

The mechanical energy of the system changes because we know both kinetic and potential energy increase. If kinetic energy increased but potential energy decreased at the same rate, the mechanical energy would stay the same.
What about the increase of gravitational PE? Gravity is an internal force in this case and the spacecraft and Earth are getting farther from each other so the potential energy between two increases. the work done by gravity to decrease KE is equal to the negative change in PE...Is that correct?

It is problematic to assign intention to work, or to force. In this case, the gravity force acts in direction that decreases the effect of the propulsion, which is to increase kinetic energy. But gravity could also increase this effect, if the rocket was up side down.

Sum of the works done by gravity forces indeed equals minus the change of gravitational PE, provided we include both the work done on the rocket and work done on the Earth (which is very small and can be neglected).

fog37 said:
If the system is the spacecraft alone, then then there are two external forces on it: the rocket's propulsion force and the gravitational force of attraction. In this case the propulsion force increase KE while the gravitational force F_g tries to decrease KE while increasing the gravitational PE. Overall, net work is done by the system by the external forces. The mechanical energy of the system must change.

We write the Work-Energy Theorem as ##W=\Delta KE##. If, as you wish, we say that ##W=W_c+W_{nc}## then we have ##W_c+W_{nc}=\Delta KE## (Eq 1).

If instead you wish to replace ##W_c## with ##-\Delta PE## then you can do that and you get ##W_{nc}=\Delta KE+\Delta PE## (Eq 2).

So you use Eq 1 if you want to deal with the conservative force in terms of the work done by it, or you use Eq 2 if you want to deal with the conservative force in terms of the change in potential energy associated with it.

Just keep in mind that as your studies progress into thermodynamics you will find that these relations, although they are perfectly valid dynamical relations, they are not valid energy relations.

Thank you Mister T and everyone else.

Mister T, I am curious about what do you mean when you say "...that these relations, although they are perfectly valid dynamical relations, they are not valid energy relations..." ? Could you give me a few more details?

Fundamentally, I think there were just two types of energies: kinetic and potential. All forms of energy can be reframed as either type.

All forces in nature are conservative. Even friction at the microscopic level. Non-conservative forces are tension, propulsion forces, the force that a human arm exerts to lift and object, etc.

fog37 said:
Mister T, I am curious about what do you mean when you say "...that these relations, although they are perfectly valid dynamical relations, they are not valid energy relations..." ? Could you give me a few more details?

Let's say you do 10 J of work on a block as you push it at a constant velocity across a horizontal surface. (For example, you exert a force of 10 N as it moves a distance of 1.0 m.)

Now, since it moves at a constant speed ##\Delta KE## is zero. If you explain this by using the work-energy theorem, which is a perfectly valid dynamical relation, you say that the friction force does -10 J of work on the block. Thus the net work done is zero and the change in kinetic energy is zero, and all is well.

But from the perspective of an energy relation you have a problem, because the block and surface got warmer, and that increase in temperature requires an energy source. The explanation is that the internal energy of the block and surface increase by 10 J because you do 10 J of work on the system when you push the block. Thus it is not valid, from an energy perspective, to say that the friction force does -10 J of work on the block, even though it is a perfectly valid claim from a dynamical perspective.

If you look at a few introductory college-level physics textbooks you will find that they take different approaches to the way they introduce this topic.

There is also a discussion of this issue in the physics education literature.

## What is the work-kinetic energy theorem?

The work-kinetic energy theorem is a principle in physics that states that the change in an object's kinetic energy is equal to the net work done on the object.

## How is the work-kinetic energy theorem related to work and energy?

The work-kinetic energy theorem shows the relationship between the work done on an object and the resulting change in its kinetic energy. It demonstrates that work is directly proportional to the change in kinetic energy.

## What is the formula for the work-kinetic energy theorem?

The formula for the work-kinetic energy theorem is:
W = ΔKE = KEf - KEi = ½mvf2 - ½mvi2
Where W is the work done, ΔKE is the change in kinetic energy, KEf and KEi are the final and initial kinetic energies, and m and v represent the mass and velocity of the object, respectively.

## How can the work-kinetic energy theorem be applied in real-world situations?

The work-kinetic energy theorem can be used to analyze the motion of various objects, such as a moving car or a swinging pendulum. It can also be used to calculate the power needed to perform certain tasks, such as lifting a heavy object.

## What are the limitations of the work-kinetic energy theorem?

The work-kinetic energy theorem assumes that there are no external forces acting on the object, and that the forces are constant and in the same direction. It also does not take into account non-conservative forces, such as friction, which can affect the object's kinetic energy. Additionally, the theorem only applies to systems where the net work done is equal to the change in kinetic energy.

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