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On 02/03/2012 04:58 PM, LaMontagne, Bob wrote:
Here is a simple problem I proposed to my general physics students
that has generated a lot of discussion by them despite the problem's
simplicity. I will throw it out here for your amusement.
Lock the wheels of a long skateboard so it won't move. Now take a 1
kg mass and drop it from a short distance onto the skateboard with a
horizontal speed of 2 m/s. The mass slides along the surface of the
board and stops after a half meter.
Now unlock the wheels and repeat the experiment. How far does the
mass now slide relative to the surface of the skateboard before it
stops sliding? Take the mass of the skateboard to be 10 kg. How far
does it slide relative to the ground?
Y'all can probably guess what I'm going to say:
Whenever you see a new problem, you should check to see whether or
not it is well-posed. If you've been warned that people have been
struggling with the problem, you should check extra carefully.
Today's problem turns out to be seriously underspecified.
In accordance with the usual rules for dealing with such problems,
we should pick at least one of the possible meanings and analyze it.
Ideally we should map out the entire solution-set, i.e. the entire
ensemble of possible meanings, but that is not always feasible.
http://www.av8n.com/physics/ill-posed.htm
1) Here is one reasonable interpretation: There are those who say
that in introductory physics class, it is traditional to neglect
friction, unless otherwise specified. In today's question, friction
is not mentioned, so the most reasonable assumption is that the block
slides without friction for 0.5 m and then hits a peg. It latches
onto the peg.
(The peg isn't mentioned in the problem-statement, but friction
isn't mentioned either. We have to assume /something/.)
The sliding behavior (before hitting the peg) is exactly the same
whether or not the wheels are locked. The behavior after hitting
the peg of course depends in the obvious way on whether or not the
wheels are locked.
2) Here is another reasonable interpretation: Again, we have to assume
something, so let's assume the force of friction is some function of
velocity ... in particular, the relative velocity between the block and
the skateboard. The form of this function is not known, although for
reasons of symmetry (and other good reasons) we assume the function goes
to zero when the relative velocity is zero. We emphatically assume the
friction is not a function of position ... which means there is /no peg/
in this scenario.
Again y'all won't be surprised by what comes next: There is a nice
_scaling argument_ that allows us to almost-completely solve the problem,
even without knowing the functional form of the friction/velocity
relationship.
We can restate the problem more elegantly by not locking the wheels,
but rather by analyzing how the behavior depends on the mass of the
skateboard. The case of locked wheels is indistinguishable from the
case of a verrry massive skateboard.
The parameter of interest turns out to be α, which I define to be the
actual mass of the block divided by the reduced mass. In the special
case we were asked to consider, α is 1.1, but it's just as easy to solve
for arbitrary values of α.
Let x_fixed, v_fixed, and a_fixed represent the position, velocity, and
acceleration of the block (relative to the lab frame) in the case where
the skateboard is infinitely massive.
Similarly let x_rel, v_rel, and a_rel represent the position, velocity,
and acceleration of the block relative to the skateboard. These functions
implicitly depend on the mass of the skateboard, i.e. on α.
I claim:
x_rel(t) = (1/α) x_fixed(α t) [3]
v_rel(t) = v_fixed(α t) [4]
a_rel(t) = α a_fixed(α t) [5]
These equations are manifestly correct in the limit of a very massive
skateboard (α = 1).
The velocity equation [4] is particularly interesting. It tells us that
the system goes through the same sequence of velocities, no matter what
the value of α; it just goes through the sequence quicker when α is
large. This is a scaling law; we are using "scaled time".
The acceleration equation [5] is consistent with the velocity equation,
as you can see by differentiating and applying the chain rule. It is
also consistent with the equations of motion, in particular Newton's
third law, which says that the force on the skateboard must be equal
and opposite to the force on the block. Hence (using the second law)
the relative acceleration must be greater than the absolute acceleration
of the block by a factor of α.
The position equation [3] is consistent (by differentiation) with the
other equations ... and is unique up to an uninteresting additive
constant.
These equations answer all parts of the question except:
How far does it slide relative to the ground?
This part of the question is not scalable. The answer is infinitely
sensitive to the functional form of the friction/velocity relationship.
The block could travel anywhere from 0.5 m to infinity, depending on
as-yet unspecified details.
In particular, sometimes sliding friction is approximated as being
independent of velocity (for nonzero velocity), but this is usually a
terrible approximation. By way of contrast, note that lubricants tend
to operate at very low Reynolds number, and produce drag that scales
roughly like velocity to the first power. As a further contrast, at
high Reynolds numbers, fluid dynamic drag scales approximately like
velocity squared. Or maybe the skateboard uses eddy-current damping.
Who knows. The problem is seriously underspecified.
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