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Re: bending of object thrown into pool

This is a response to all of the excellent comments in last night's
digest about my bending query:

What if you shoot the arrow upward from below the water surface? Does
it speed up? (Robert Cohen)

Good point, it wouldn't. That seems a pretty damning argument against
a Snell's law analog.

I saw references to "throwing" and "launching" and other
highly dynamic verbs.

If you actually _throw_ a stick into a pool, you must
first worry about how it *behaves* in absolute terms,
before you worry about how it appears.

The behavior will be verrry complicated. In addition
to plain old buoyancy there will be hydrodynamic
forces such as lift and drag, which in turn depend on
the angle of attack and the sideslip angle etc., not
to mention dependence on Reynolds number etc. etc. etc.

In contrast if you have a static stick just sticking
into a quiet pool, it is straightforward to determine
the appearance by ray-tracing. The result is a
combination of Snell's law and projective geometry. (John Denker)

I did in fact mean the object is *moving*, crossing the boundary from
one fluid to another. But focus on the bending as it crosses the
interface, not the longer term effects of drag and so on. Perhaps
start with the simpler case of a golf ball.

I'm assuming that it's the object's trajectory, and not the object
itself, that is bending. Think of a toy car with wheels that don't
turn. If you roll the car from a tile floor onto a carpet, at an
oblique angle to the boundary, the car will turn. This is because
the wheels on one side are traveling at a different speed than the
wheels on the other side. You might get the same effect by sliding
a piece of two-by-four across the tile floor. It'll tend to spin
around when it hits the carpet.

Nice examples!

A ball, on the other hand, has only one contact point and so would
be less likely to turn as it crosses the boundary.

Agreed, size and shape of the object are important.

Should "would be less likely to" be replaced by the more categoric
"would not" in this sentence?

All of the above assumes that there is no lip or crack or other
place that causes bouncing at the interface.

Yes. In particular, wouldn't an object thrown into a pool also tend
to "bounce" off the surface and hence might be expected to bend away
not toward the surface, assuming it penetrates at all? Again, someone
should actually try this, I bet the book writer did not!

In the case of tossing stuff into a pool, the trajectory will be
altered by the water resistance.

This is not obvious to me. What about your above example of the ball?
Is a difference in fluid viscosities *alone* enough to bend the
trajectory, in the absence of surface tension, asymmetric shape of
the object, etc?

None of this is the same as Snell's Law, in which ALL of the
trajectory change occurs at the interface between two media. For a
physical object traveling in a medium, the trajectory is
continuously altered by the medium, i.e. friction, air resistance,
or water resistance. (Vickie Frohne)

I don't see this either. If we assume drag oppositely directed to
velocity and that the object is cylindrically symmetric, it can
travel straight. Also, isn't your above example of the toy car
analogous to Snell's law? If I think of the axle as a wavefront, it
turns for what seems like the same basic reason as light does, namely
different points along the same line are traveling at different
speeds.

According to the PSSC version of Newton's corpuscular model, light
particles bend in the same direction as predicted by Snell's law, the
explanation being that attractive forces in the interface between the
less optically dense medium and the more dense medium cause the normal
component of the velocity of the corpuscles to be greater after they
have passed into the more dense medium, while the tangential component
of velocity (and momentum) remains the same.

But does "attractive forces in the interface" mean "surface tension"?
In that case, I still don't see why the object should bend toward the
normal. If the idea is that the object is pulled toward the interface
as it approaches it, shouldn't it also be pulled back as it recedes
from it upon transmission? Kind of like crossing on a skateboard from
a flat, down a half-pipe, and back up to the *same* flat (assuming
the action is purely in the interface)?

A little right triangle
trigonometry
for a velocity diagram shows that sin(theta1)/sin(theta2)=v(2)/v(1),
where 1 refers to the medium of the incident corpuscles and 2 to that
of the refracted corpuscles. If medium 1 is vacuum,
sin(theta1)/sin(theta2)=v(2)/c. Thus the index of refraction is
n=v(2)/c, just the reciprocal of the value
according to the wave theory.

Ah, that's what was opposite in Newton's theory, thanks for the reminder.

Although Roemer had estimated the speed of
light in vacuum (outer space) in 1676, the direct measurement of the
speed of light in a medium such as water had to wait until the middle
of the nineteenth century. So there was no way to decide on the basis of
which speed was greater in Newton's day. (I think Carl's assertion is
correct once it is known that light travels slower in the more dense
medium.)

Okay.

PSSC models corpuscles going from a less dense medium to a more dense
medium with a rolling ball moving on a higher horizontal to a lower
horizontal surface, the surfaces connected by a ramp. (The direction
could be reversed to model a corpuscle going from a more dense to a less
dense medium.)

This is not the same as the attractive force being only in the
interface. In fact, it relies on a uniform gravitational field in
both media. Shut off gravity and this model no longer works. (The
problem now is how to keep the ball in contact with the surface.
Okay, replace the surfaces with an S-shaped tube in deep space.)

There are photographs of this in Chapter 14 of the 2nd ed. of _PSSC
Physics_ and Chapter 5 of _College Physics, Physical Science Study
Committee_. This is actually a basis for a PSSC experiment, "The
'Refraction" of Particles" to see if they change direction according to
Snell's law. It is in the PSSC _Laboratory Guide_ at least through the
4th ed. I could not find the corpuscular model of refraction in either
the text or lab guide for the latest (7th) edition from around 1991.

I think the demise of PSSC was a great loss to the teaching of physics.
It brought the idea of models to my attention long before I heard of the
more recent Modeling Instruction -- particularly the contrast of the
models of Newton and Huygens for light. (Hugh Logan)

I would like to see such photos. Maybe I can track down one of these
references. Did they smooth off the sharp corners getting onto and
off of the ramp?

I've always associated the question of changes in direction in terms
of a boundary, not in terms of the actualy media, at least for
"corpuscles".

For example, consider a golf ball rolling (without friction) at a
steady speed on a level surface. Now suppose the ball obliquely
hits a ramp to a higher, but also level surface. Once it reaches
the top surface, it will be a) travelling at a slower, steady speed,
and b) heading farther away from the normal. The higher the ramp
and the slower the ball is going at the end, the more it bends. If
the ramp is high enough, you observe "total external reflection".

This is the opposite of how waves behave. (Tim F)

As Hugh points out, this depends on which side of the ramp you
associate with which medium.

According to Eugene Hecht's _Optics_, 4th ed., p. 141, the derivation of
Snell's law on the basis of Newton's corpuscular law was actually first
published by Rene' Descartes in 1637. Hecht gives a modern, quantum
mechanics version of this for photons which he regards as being "a bit
simplistic," but of some pedagogical value. Equating the tangential
components of the momenta of the incident and refracted photons as
Descartes did for Newtonian corpuscles,
p(1)*sin(theta1)=p(2)*sin(theta2). Recognizing that the momentum of a
photon is p=h/lambda, this becomes

[h/lambda1]*sin(theta1)=[h/lambda2]*sin(theta2) .

Multiplying through by c/f, where f is the frequency, and using
f*lambda=v, one gets

[h/v(1)]*sin(theta1)=[h/v(2)]*sin(theta2)

or sin(theta1)/sin(theta2)=v(1)/v(2), the correct result (upside down
from the Newton-Decartes result.) Thus

sin(theta1)/sin(theta2)=n(2)/n(1) where the index of refraction is
defined as in the wave theory (n=c/v).

This latter "derivation" was first pointed out to me by my major
professor, Dr. Earle K. Plyler of FSU, in connection with some ideas he
had about photons -- in 1965, before I saw it in any text.

According to Hecht, the fact that light travels slower in the denser
medium was probably first inferred from experiment by Thomas Young in
1802, from the fact that the measured wavelength was shorter in the more
optically dense medium -- before the definitive experiments of Foucault
in 1850 (with a rotating mirror and a long column of water.) (Hugh Logan)

Thanks Hugh for a very helpful historical primer! Carl
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/