# Work-energy theorem and Conservation of energy formula

We have $$W_{net}=\Delta K \quad(work-energy\; theorem)\tag{1}$$ And also $$W_{net}=\Delta K +\Delta U \quad(Conservation\;of \;energy \;formula)\tag{2}$$ How's that happening?

In proof for bernoulli's equation there's a place which they say:$\Delta W=\Delta E$which$\Delta W$ does not contain work of gravity and however it's related to external or internal energy but I can't understand? You can see the proof here: http://www.4physics.com/phy_demo/bernoulli-effect-equation.html

This is a nice example of why notation matters.

You know there are two types of forces: non-conservative and conservative forces.

Let's call $W_{cons}$ the work done by conservative forces. Let's call $W_{NC}$ the work done by the rest.

The work-energy theorem states that

$$W=\Delta E_k$$

However, this work is the total work. This can be splitted in two parts: $W=W_{cons}+W_{nc}$. So

$$\Delta E_k = W_{cons}+W_{nc}$$

Now, we define a quantity called $E_p$ such taht $W_cons=-\Delta E_p$. The minus sign is a convention, but it is important to keep it in mind. Hence

$$\Delta E_k = W_{cons}+W_{nc}$$ $$\Delta E_k = -\Delta E_p +W_{nc}$$ $$\boxed{ \Delta E_k + \Delta E_p = W_{nc} }$$

So your "net" work refers only to non conservative forces in your second equation.

$W net$ in the first equation is not the same as $Wnet$ in the second equation.

The first equation relates to $W=Fxd=\Delta KE$ where the work is done displacing the center of mass of the system. It relates to the external energy of the system with respect to an external frame of reference.

$Wnet$ in the second equation is the sum of the first equation plus the work done on the system to change it internal energy (missing is the possibility of Q). The most common type of the latter work is boundary work (expanding or contracting the boundaries of the system).

The complete form of the first law is

Q – W = ΔE = ΔU + ΔKE + ΔPE

Where

ΔE = Total energy change of the system, which is the sum of change in internal and external energy of the system.

ΔKE = Change in kinetic energy of the system as a whole. This relates to a change in the velocity of the center of mass. By the work energy principle:

F x d = ΔKE

ΔPE = Change in potential energy of the system as a whole, such as a change in elevation of the center of mass (change in gravitational potential energy).

Q and ΔU are as always.

W now includes both boundary work and work done on or by the system a whole.

• So , Is the second one($W_{net}=\Delta K +\Delta U$)counts both the net work done by external source and potential energy which that object already had? – Abbas Jul 31 '18 at 16:31
• And what is "Q" which you've referred to ? – Abbas Jul 31 '18 at 16:33
• @Abbas, the complete form of the first law comes from thermodynamics, where Q refers to the heat absorbed by a given system. – David White Jul 31 '18 at 17:55
• "Wnet in the second equation is the sum of the first equation plus the work done on the system to change it internal energy" Doesn't an external force intended to change internal energy? – Abbas Jul 31 '18 at 18:09