question form

Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

** **

** **

** **

Thanks for pointing this out. That link is now fixed, and here's a copy:

First we give a brief intuitive explanation:

Friction acts in the direction opposite motion both going up and coming

down, dissipating some of the original KE.  So the automobile ends up

moving more slowly at the end. 

Looking at the energy changes in more detail:

Intuitively:

• We can think about what happens in the process of going up and

  coming back down.  In the absence of friction the initial KE would

  convert to an equal amount of PE gain, all of which would then converted

  back to KE on the return. 

• With friction present, however, the automobile doesn't climb as high

  as it would if friction was not present.  There will be less PE to

  loses, and additional energy will be lost to friction on the way back

  down.

A little more rigorously:

• As the automobile coasts along the incline, in whichever direction,

  it will experience a force of rolling friction in the direction opposite

  its motion.  The force it exerts against friction is in the

  direction of its motion, so it does positive work. 

• The automobile will continue coasting up the incline until its KE

  becomes zero (i.e., until it comes to rest for an instant). 

• Since it does positive work against friction, its total change in PE

  as it travels up the incline will be less than its original KE. 

It will then coast back down the incline, losing PE and again doing

work against friction. 

When it returns to its starting position, its loss of PE on the way

down have been equal to the PE gain it experienced going up.  T

he PE loss on the way down is therefore less than its original KE,

so even without the friction loss on the way down it would would end up

with less KE than when it started. 

The frictional force on the way down once again opposes motion, so

some of the PE loss is dissipated against friction, leaving it with even

less KE. 

Now let's put this specifically into the context of the work-energy theorem,

which can be stated in either of two equivalent ways:

• `dPE + `dKE + `dW_by_noncons = 0, where `dW_by_noncons is the work done

  BY the sytem (in this case the automobile) against nonconservative forces,

  or equivalently

• `dW_on_nc = `dKE + `dPE.  where `dW_on_nc is the work done ON the

  system by the nonconservative frictional force

We can use either formulation, depending on whether we want to think in

terms of `dW_by_noncons (as in the first formulation) or `dW_on_noncons (as in

the second formulation):

• If we want to view this situation from the point of view of the

  automobile, it is doing positive work against the frictional force so `dW_by_noncons

  is positive and this will lead us to the first formulation.

• If we want to view this from the point of the forces acting on the

  automobile, the frictional force exerted on it is doing negative work. 

  So `dW_on_noncons is negative and this will lead us to the second

  formulation.

To answer the question, it isn't necessary to consider anything but the fact

that the automobile ends up where it started, and in the process does positive

work against friction (or alternatively that friction does negative work on it):

Between the instant it starts up the incline and the instant it returns

to its initial point, its PE change is zero, and it does positive work

against the frictional force. 

• Using the formulation

`dPE + `dKE + `dW_by_noncons = 0

of the work-energy theorem, since `dW_by_noncons is positive and `dPE

is zero we have

0 + `dKE + `dW_by_noncons = 0, which we rearrange into the form

`dKE = - `dW_by_noncons. 

The right-hand side is negative, so `dKE is negative.  The

automobile ends up moving more slowly than when it started.

• Using alternative formulation of the work-energy theorem we have

`dW_on_nc = `dKE + `dPE. 

`dW_on_nc is negative, and `dPE is again zero, so 

`dKE = `dW_on_nc,

which is negative. 

We again conclude that the automobile ends up moving more slowly

than when it started.

question form

Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

** **

** **

** **

You wouldn't be required to do this query, since none of the questions pertain to your course.