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Re: [Phys-l] block sliding on a skateboard

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf
Of LaMontagne, Bob
Sent: Saturday, February 04, 2012 5:13 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] block sliding on a skateboard



-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Saturday, February 04, 2012 2:37 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] block sliding on a skatebo

..............................................................
................ snip


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.


[LaMontagne, Bob] Under the usual freshman physics
assumptions (friction constant and independent of velocity -
no pegs or other gimmicks - no wind, etc.) the block actually
travels a little LESS than 0.5 m relative to the ground while
it is sliding on the skateboard. As JD states, the forces on
the block and skateboard are equal and opposite - but since
the skateboard is accelerating, the block comes to the same
speed as the skateboard a little sooner than if the
skateboard couldn't roll.

Please correct my thinking on this if it is wrong. I will
need to revisit this with my students (big mea culpa).

Bob at PC

I agree with you Bob. Based on the conditions you gave, the force on the block is the same while it is sliding whether or not the wheels are locked, and it stops sliding sooner when they are not locked because in that case, it stops sliding upon achieving the velocity of the skateboard which by that time has a magnitude greater than zero. I get
x_s = (1/α)(2-1/α)d
for the distance traveled by the block relative to the ground where α is the α defined by John Denker and d is the distance the block slides when the wheels are locked. x_s is less than d for any α > 1 and goes to d in the limit as α goes to 1.
For the values given in the original problem it yields
x_s = .496 m

Jeff


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