Net work done on Earth-object system and potential energy

[NoahCygnus]

My book says:
"Let us derive an expression for the potential energy associated with an object at a given location above the surface of earth. Consider an external agent lifting an object of mass $m$ from an initial height $y_{i}$above the ground to a final height $y_{f}$. We assume the lifting is done slowly with no acceleration so the applied force from the agent can be modeled as being equal to in magnitude to the gravitational force on the object. The work done by the external agent on the earth-object system as the object undergoes this upward displacement is given by : $W_{net} = \vec{Fapp}\cdot \Delta \vec{r} = mg\hat{j}\cdot[(y_{f} - y_{i})\hat{j}] = mgy_{f} - mgy_{i}$

Where this result is the net work done on the system because the applied force is only force by the environment. The equation represents a transfer of energy into the system and the energy appears in a different form called potential energy. Therefore we call the quantity $mgy$ as the gravitational potential energy $U_{g}$. $U_{g} \equiv mgy$.
Therefore $W_{net} = \Delta U_{g}$."

What I don't understand is , why has the author stated that only the external force does net work done on the system? Why didn't the author include the work done due to gravity in the net work? Shouldn't the net work done on the system be $W_{net} = W_{app} + W_{g}$ , which is the sum of all the works done by both external and internal forces?

 

[jtbell]

Why didn't the author include the work done due to gravity in the net work?

The effect of gravity is already included via the potential energy.

Alternatively, you can ignore the concept of potential energy and include the work done by gravity explicitly.

When you use potential energy, one really should use a term like "net non-conservative work" ($W_{\rm{nc}}$) instead of "net work", in order to avoid confusions like this.

 

[NoahCygnus]

The effect of gravity is already included via the potential energy.

Alternatively, you can ignore the concept of potential energy and include the work done by gravity explicitly.

When you use potential energy, one really should use a term like "net non-conservative work" ($W_{\rm{nc}}$) instead of "net work", in order to avoid confusions like this.

Got it. One more question, work done on a system is the change in its energy, and according to work-kinetic energy theorem, net work done by all the forces , both conservative and non conservative, external or internal is equal to change in kinetic energy. In the above situation we see there is no change in kinetic energy, so net work done on the system is 0. But there is a change in potential energy, due to the work done by gravity ,even though the net work done is 0. Does that mean work-kinetic energy theorem is not a general statement and does not account for changes in the other types of energy by the individual work done by the forces?

 

[jtbell]

Correct, the more general version is $\Delta E = W_{\rm{nc}}$ where:

• $\Delta E$ is the change in the object's mechanical energy (kinetic + potential): $\Delta E = \Delta K + \Delta U$
  
• $W_{\rm{nc}}$ is the net work done by non-conservative forces (forces that don't have potential energy associated with them)

 

[NoahCygnus]

Correct, the more general version is $\Delta E = W_{\rm{nc}}$ where:

• $\Delta E$ is the change in the object's mechanical energy (kinetic + potential): $\Delta E = \Delta K + \Delta U$
  
• $W_{\rm{nc}}$ is the net work done by non-conservative forces (forces that don't have potential energy associated with them)

So in a system, the net work done by all the forces may be zero, but there can still be changes in the other forms of energy except the kinetic energy?

 

[jtbell]

but there can still be changes in the other forms of energy except the kinetic energy?

Yes, if you have more than one form of potential energy. Consider a mass hanging on a spring.

 

[NoahCygnus]

Yes, if you have more than one form of potential energy. Consider a mass hanging on a spring.

But it can be internal energy too, as in case of a block that we push on a table having friction with a constant speed, the net work done is zero, as suggested by work kinetic energy theorem, but still there is a change in internal energy caused by the work done by the frictional force, am I correct?

 

[jtbell]

Sure, in that case you have to consider the internal energy of both the block and the surface it's sliding on. They both warm up, right?

I think this is getting beyond the scope of the usual work-energy theorem. It's normally applied in situations where we can ignore the effects of the object's internal structure. This situation is sliding () over towards thermodynamics.

 

[NoahCygnus]

Sure, in that case you have to consider the internal energy of both the block and the surface it's sliding on. They both warm up, right?

I think this is getting beyond the scope of the usual work-energy theorem. It's normally applied in situations where we can ignore the effects of the object's internal structure. This situation is sliding () over towards thermodynamics.

You have been a big help. Finally I can rest. Thank you very much.