Re: are normal reaction and tension conservative ?

[Chetan <chetanreddy@ETH.NET>]

John.S. Denker wrote:

> For homework, come up with an example along the following lines:
>   -- We have an object at the lower end of a long rope.
>   -- I move the upper end of the rope.
>   -- The result is spectacularly non-conservative.


Dear Mr.Denker,

Suppose a man is pulling a body of mass 'm' vertically up with an
accerleration 'a'.
Then the tension in the rope is T1 = m(g+a) where g=aceleration due to
gravity. After time 't' the man suddenly stops pulling. The body then rises
through some height and returns to the position at the moment the pull in
the string became zero after a*t. The work done by the rope till this moment
is  m(g+a)h - {m (at)^2}/2 where h= (a t^2)/2 . The body comes to rest at
this moment. If now the body is lowered to the initial position with the
same magnitude of accerleration 'a' downward. the tension now is  T2 =
m(g-a).the work done in this part is      -m(g-a)h  where h is the same as
before.
The total work done by tension is
             W =  2mah - {m (at)^2}/2
               =  m(a^2)(t^2)/2

This is nonzero. And the tension spectacularly non-conservative when the
duration of time for which initial acceleration is provided is long.


> Notes:
>  == You can assume the rope is non-stretchy if you like.
> == You can constrain me to moving the upper end purely vertically if you
> like.
I seem to have done this(i am not sure about the non-stretchy part)




Let me thank you all for being so encouraging.


Chetan.