Understanding Mechanics: Work, Momentum, and External Forces Explained

[justagirl]

Hey, can any of you clarify a few things for me? I'd greatly appreciate it, thanks!

If the work done on an object is by conservative forces, does that mean the momentum is conserved? I know if there aren't any external forces, then momentum is conserved. Does conservative forces mean no external forces?

Also, is the work-kinetic energy theorem still valid for nonconservative forces?

Lastly, if there are no external forces in the x direction but there is in the y direction, then it's possible that the y component of the momentum is not constant right?

Thanks!

 

[arildno]

1.A conservative force means (simply put) that there exist a potential energy function from which we may derive the force.
(The potential energy function is a function of the position of the object)
The work done on an object between two positions by a conservative force, is given by the difference of the potential energy function evaluations at these points.

2. Momentum may well change, even if the conservative force exerts no net work!
Example:
Shoot a ball straight up with velocity V (we consider the force of gravity as the only force acing on the ball). Some time later, the ball returns with velocity -V.
This is consistent with no work done, since the kinetic energy of the ball is unchanged.
However, the ball's momentum have changed..
In general, change in kinetic energy is due to net work, whereas change in momentum is due to net impulse.

3.You should not confuse conservative forces with external forces; they are completely different concepts.
4. "Also, is the work-kinetic energy theorem still valid for nonconservative forces?"
Written appropriately, yes.

 

[MiGUi]

Conservative means that there is not any kind of disipative process such as heat or friction. Conservative don't implies external. Not neccesarily.

Think what you said: if there are not external forces, then the momentum is conserved. But why?

The definition of momentum is $$\vec{p} = m \vec{v}$$. If momentum is conserved then its module is constant with time, so:

$$\frac{dp}{dt} = \frac{d(mv)}{dt} = m \frac{dv}{dt} = ma = f$$

So, if f are external forces and are equal to zero, the momentum is conserved. But the second you said is not correct, conservative is not the same as external.

The work-kinetic energy theorem is valid if the force verifies the second Newton law.

Let be $$\delta W = \vec{F} d \vec{r}$$. If we want to know the whole work, we have to integrate:

$$\int_A^B \vec{F} d \vec{r} = \int_A^B m \frac{d \vec{v}}{dt} d \vec{r} = \int_A^B m \frac{d \vec{r}}{dt} d \vec{v} = m \int_A^B \vec{v} d \vec{v} = \frac{1}{2}mv^2 = T$$

And finally, your third question.

Imagine that you have a ball moving in one direction. If wind blows, the ball may slow down and its momentum will be affected, but is the vertical component of momentum affected at any time? The answer is no, because the whole momentum was horizontal.

This example answers your question. The y component is constant but the x one is not. That is not neccesary for none of them to be equal to zero. A external force may change only one component.

 

[arildno]

Just for the record:
There are no disagreements between MiGui's approach and my own, as far as I can see.

 

[MiGUi]

Yes arildno, but when I begin to write your message was not there, I posted only six minutes after you. You beat me this time ;)

 

[arildno]

I just wanted to save the thread starter some confusion if he/she thought we might have differing opinions.
Since we have not, having both approaches present highlights different aspects of the same idea(which, IMO, is beneficial)

 

[justagirl]

Thanks for your help! I just need one more clarification.

So I understand that external forces are not the same as conservative forces, so then it's FALSE to state that if the work is done by conservative forces, then momentum is conserved?

 

[arildno]

Yes, momentum is not necessarily conserved if the work done is due to conservative forces alone.
(It CAN be conserved, but not necessarily so).