Re: [Phys-l] Conservation of Energy Question

[John Denker <jsd@av8n.com>]

On 04/30/2011 07:09 AM, Michael Barr wrote:
> I the last issue of the Vernier Caliper they proposed a pretty good question
> to their blog.  They have a lab called Energy of a Tosses Ball.  You put a
> motion sensor on the table, hold a ball over it, throw is straight up, then
> catch it.  You get a PE and KE graph after you input the mass. When you have
> the software add KE + PE to get total energy you should get a horizontal
> line, and you do in many cases.  When the person doing the lab used a beach
> ball the line was curved (concave down).  So, it showed total energy going
> up, then down.
>
> So, the blog question was why.  There were the main possibilities people
> wrote in.

> 3. Spinning (To get these results the ball would have to start out spinning,
> slow down until max height, and then speed up its spin on the way down, so
> that is out too)

We agree that spinning doesn't explain the observations.


> 1. Air Friction (seems right to me but no one gave a great explanation)

Not a chance.  Friction, strictly speaking, would produce a 
monotonic decrease of system energy, due to energy flowing out
across the boundary of the system ... very unlike the observed
increase then decrease.


> 2. Buoyancy (Would be zero for a beach ball due to similar air pressure
> inside the ball, so, not an issue)

This is a major correction, and surely explains /most/ of the
observed effect.  If you don't correct for buoyancy, you will
seriously mis-estimate the mass.

Note that the buoyancy affects he PE differently from the KE.  
This should be obvious if you consider the limiting case of a 
neutrally-buoyant helium balloon, which has no net PE at all, 
but still has plenty of mass.  Note that in order to maintain
sanity and to communicate clearly, it helps to distinguish
  weight = plain old weight             = m g
  reduced weight                        = m g - buoyancy
where
  mass = plain old mass = inertial mass = m

This is perhaps even more obvious if you consider the motion of
an upwardly-buoyant balloon, thrown initially downwards.

Note that an ordinary platform scale measures the reduced weight, 
and you must correct for buoyancy in order to determine the actual
weight (and thence the actual mass).  OTOH there is such a thing 
as an inertial balance that is immune to buoyancy if properly used:
  http://www.labscientificequipments.com/inertial-balance-kit-5496.html


4.  If you look closely, you will discover another way in which 
the mass can be mis-estimated, in addition to the aforementioned
buoyancy.  I'm talking about _fluid dynamics_ and in particular
about the _virtual mass_.  (In particle physics and solid-state
physics this would be called the _dressed mass_ but in fluid
dynamics it is more commonly called virtual mass.)

When you move an object through a fluid, there is KE in the object
itself and also KE in the flowing fluid.  The fluid _must_ flow in
order to get out of the way of the object.  This is not a frictional
effect, as you can see from the fact that you put KE into the fluid
at the start of the motion and you get it all back at the end of
the motion.  [There may be frictional effects also, but those are
not what we are talking about in this item (4).  The usual procedure
is to account for the non-frictional effects first, and then see
what's left.]

The equations for non-frictional flow ("potential flow") around
a sphere are worked out in many places
  http://www.google.com/search?q=sphere+%22potential+flow%22+%22virtual+mass%22+

Note that you will never see the virtual mass on an ordinary
platform scale, because the fluid is guaranteed to be neutrally
buoyant ... since it is immersed in more fluid.