Is internal energy really decreasing in situation 2?

1. Mar 16, 2017

vcsharp2003

1. The problem statement, all variables and given/known data
I am trying to understand the law of conservation of energy from a very general perspective but coming across some issues. I am using the equation mentioned at end of this post, which is true for a system with no heat flows into or out of the system.

In two situations described below,if the applied force and friction force are both non-conservative then using the general equation below, the change in internal energy of block should be negative of the sum of work done by applied force and friction force. This makes sense in situation 1 where internal energy increases but not in situation 2 where I am getting that internal energy of block decreases. I am sure I am making a mistake somewhere but cannot see it.

SITUATION 1 ( without applied force)
A block of mass 1 kg moving at 10 m/s on a rough horizontal surface with a coefficient of friction of 0.25 is brought to rest by friction. We will focus from initial instant when it's speed is 10 m/s to when it comes to rest.

SITUATION 2 ( with an applied force)
A force of 200 N is applied to a block of 1 kg mass on a rough horizontal surface with a coefficient of friction of 0.25. This causes the block to move in the direction of force. We will focus on first 10 m of the block's motion.

Question
Does this reasoning for situation 2 sound correct if it's assumed that no heat flows into and out of the block?

2. Relevant equation

General equation for conservation of energy without heat flows

3. The attempt at a solution

SITUATION 1 ( without applied force)
The work done by friction force in this case is .5 x 1 x 102 = 50 J.
If we use the equation given at end of this post then the work done by non-conservative force of friction is converted to internal energy of the block i.e. internal energy change = - (-50) = + 50 J which causes internal energy of block to rise.

SITUATION 2 ( with an applied force)
For the first 10 m, work done by the applied force = + 200 x 10 = + 2000 J and work done by friction force is = - 1 x 10 X .25 = - 2.5 J.
So, net work done by non-conservative forces is 2000 - 2.5 = + 1997.5 J. If the equation given below is true then change in internal energy is - 1997.5 J which means the internal energy of the block has decreased by the end of this 10 m distance.

Last edited: Mar 16, 2017
2. Mar 16, 2017

TSny

For situation 2 you didn't calculate the work done by friction correctly. (Either you didn't calculate the normal force correctly or you forgot to include the distance.)

Have you taken into account the change in KE for situation 2?

3. Mar 16, 2017

Staff: Mentor

The problem is with the equation $\Delta U_{int}=-W_{nonconservative}$. Where did this equation come from all of a sudden? The correct form of the first law for a closed system with Q = 0 is $$\Delta (KE)+\Delta (PE)+\Delta U=-W$$

4. Mar 16, 2017

vcsharp2003

Hi,
I looked up that equation from University of Rochester website at this link under the section titled 8.6. Conservation of energy
: http://teacher.pas.rochester.edu/phy121/LectureNotes/Chapter08/Chapter8.html.

5. Mar 16, 2017

vcsharp2003

Hi,
Yes, you are correct. I did not calculate situation 2 correctly. Thanks for pointing that out. But even after this correction, it still doesn't make sense.

New calculations are as below.
But if this equation is true as we have seen, then it means the internal energy of block has decreased by 1975 J. The question is what has happened to this LOST internal energy of the block?

6. Mar 16, 2017

TSny

You have two non-conservative forces in your situation 2: The applied force of 200 N and the friction force. As in section 8.5 of your link, you can start with the work-energy theorem: Wnet = ΔKE. Thus,
Wapp + Wfric + Wgrav = ΔKE. You don't have any work done by gravity, but I included it for generality.

Note that the change in internal energy is due to the work done by friction: Wfric = -ΔUint. The work done by gravity can be written as Wgrav = -ΔUg. So, you get

Wapp = ΔKE + ΔUg +ΔUint

If you take the block, table, and earth as the "system", then this says that the work done on the system by the applied force equals the change in the total energy of the system.

7. Mar 16, 2017

vcsharp2003

So, it seems that the equation mentioned under section 8.6 is missing an important piece of information even though it's always correct mathematically. It should have been like what you mentioned where internal energy is only affected by friction and not by a non-friction applied force. While the equation under section 8.6 makes mathematical sense, your equation seems to make more conceptual sense.

For the block as a system, if I use your equation then the KE of block increases and so does it's internal energy. I am not getting how to put the conservation law in words for situation 2, so it all makes sense. Or may be it's incorrect to take just block as the system.

Last edited: Mar 16, 2017
8. Mar 16, 2017

Staff: Mentor

Let's focus on Situation 1. With all due respect to your reference, let's take a step back to freshman physics. What is your equation for the force balance on the block in Situation 1?

9. Mar 16, 2017

vcsharp2003

Equation of motion in horizontal direction:
The force on block in horizontal direction is friction force = 0.25 x 1 x 10 = 2.5 N = 1 x horizontal acceleration. i.e. 2.5 = 1 x a

Equation of motion in vertical direction:
Also, force in vertical direction on block is (normal reaction from table - weight of block) = 1 X 10 - R = 1 x 0 i.e. R = 10, since there is no acceleration in vertical direction for the block.

Last edited: Mar 16, 2017
10. Mar 16, 2017

Staff: Mentor

I was thinking more of $$ma=m\frac{dv}{dt}=-f$$where f is the magnitude of the friction force and "a" is the acceleration of the block in the +x direction. Do you agree with this? If so, what do you get if you multiply both sides of this equation by v = dx/dt, where "v" is the velocity of the block in the +x direction?

11. Mar 16, 2017

vcsharp2003

Yes, I agree with the equation. If both sides of the equation are multiplied by v then
$$mv\frac{dv}{dt}=-fv$$

12. Mar 16, 2017

Staff: Mentor

So, $$mv\frac{dv}{dt}=-f\frac{dx}{dt}$$If I integrate this from 0 to time t, I get:
$$m\frac{v^2}{2}-m\frac{v_0^2}{2}=-f(x-x_0)$$where $v_0$ is the velocity at $x_0$. What is your physical interpretation of the left- and right sides of this equation?

13. Mar 16, 2017

vcsharp2003

My interpretation is
Change in KE of block = Work done by the block against friction

Last edited: Mar 16, 2017
14. Mar 16, 2017

vcsharp2003

I think the equation mentioned in section 8.6 is only true if there is no applied force but only dissipative forces like friction or some other force that causes dissipation (i.e. conversion to heat by increasing internal energy). So, it seems that the context of equation has not been mentioned clearly by the author. What you showed in your last post is the correct equation when applied forces exist in addition to friction.

15. Mar 16, 2017

vcsharp2003

It appears that the context of the equation under section 8.6 has not been clearly mentioned by the author of the notes under that link. Your equation is correct in when both applied force and friction force exists, but the equation I quoted is only true if friction force is there without any applied force.

Could you please let me know what does W in your equation represent? Is it the net work done by non-conservative forces or by all forces?

16. Mar 16, 2017

Staff: Mentor

Right. So, from mechanics considerations, we've shown that:$$\Delta (KE)=-W$$ If we combine this with the equation:$$\Delta (KE)+\Delta (PE)+\Delta U=-W$$we obtain:$$\Delta U=0$$

17. Mar 16, 2017

Staff: Mentor

My W includes all forces.

18. Mar 16, 2017

Staff: Mentor

I haven't yet looked at what TSny has done with you on Situation 2, but, if you like, I can help you work through Situation 2 using the same methodology that we have used in Situation 1. Any interest?

19. Mar 16, 2017

vcsharp2003

So, you are saying that no change in internal energy occurs in situation 1? I thought friction will cause dissipation which will produce heat in the block i.e. increase of internal energy happens.

20. Mar 16, 2017

Sure.