Conceptual problem with definition of work

[issacnewton]

Hi

I was reading from Serway in the chapter of "Energy and energy transfer". In this chapter the
author is introducing the concept of system and environment and then work. He also talks about kinetic energy and work-kinetic energy theorem and then conservation of energy theorem. I have attached a snapshot from the book. He says in bold letters that work is an energy transfer. Technically speaking, it should be "mechanical energy transfer" because
when the work done on the system is positive, its the mechanical energy that is transferred to the system. I know that this is chapter of work and author probably means that but later in the chapter, author also talks about other mechanisms of 'energy transfer' like heat, radiation. so the beginning students can become confused.

Also another related question regarding the work done. now its defined as

$$W=\int \, \vec{F}\cdot \vec{dr}$$

in one dimension case when the object is moving left on positive x axis, and if there is
constant force $$\vec{F}=-F \hat{i}$$ should $$\vec{dr}$$ be
$$-dx\,\hat{i}$$ or $$dx\, \hat{i}$$.

I think it should be $$-dx\,\hat{i}$$ since differential position vector is the difference
between the final position vector and initial position vector. But if we do this, and if the object is moving from x₂ to x₁ (x₂>x₁)
then we get

$$W\,=\,\int_{x_2}^{x_1} (-F\hat{i})\cdot(-dx\, \hat{i})$$

which is

$$W\,=F(x_1-x_2)$$ and this is negative. but since the external force is applied
to the system of particle, work should be positive since its mechanical energy
is increasing... so am I taking differential vector wrong ?

thanks

 

[K^2]

Technically speaking, in any form of energy transfer, there is work being done. So if you go down to microscopic level, total energy change is exactly equal to work done. However, it's often inconvenient to discuss motion of individual particles, so the net work done by random movement of particles is usually denoted as heat flow. It's still a type of work, but we tend to neglect that part and treating heat separately for convenience. Similar argument can be made for radiation, and any other form of energy flow you can think of.

 

[issacnewton]

yes , that's right. I forgot about the microscopic picture.

what do you think about my second question. is my reasoning correct ?

 

[issacnewton]

can anybody shed light on the second question ?

 

[zackptaylor]

Your second problem is happening because of the way you set up the integral. By putting the bounds of the integral from x2 to x1, you have already ensured that the motion of the particle is to the left (-x direction). You can either specify the direction using the sign of dr and running the integral from "smaller" to "larger" values. Or you can set dr as positive and use the limits to define the direction. You have done both, hence the sign error.

 

[issacnewton]

I see where I made mistake. work is defined using a line integral over vector fields. I looked up the definition of line integral over vector fields and I got correct answer. I rigorously used that. this shows the importance of math in physics