Re: [Phys-l] time to bottom of ramp

[Rick Tarara <rtarara@saintmarys.edu>]

As Patricia notes, the key here is really the average velocity over the 
trip.  With the incline that will be .5 x sqrt(2gH).  With the other path it 
is likely to be higher since a significant section of the trip is done at 
the highest speed sqrt(2gH).  Which gets there first depends, I think, on H 
and L but depending on the details on the geometry here, it might also be 
possible to show that the curved path always wins (which is my intuition 
here) since the crude drawings don't seem to show that the overall path of 
the curved trip is that much longer (if longer at all) than the ramp.  My 
intuition seems opposite of Patricia's since I would judge that the longer L 
is compared to H, the larger percentage of the trip on the curved path is at 
the highest speed making that path more likely to be the shortest time.

Rick

----- Original Message ----- 
From: <fizix29@aol.com>
To: <phys-l@carnot.physics.buffalo.edu>
Sent: Thursday, October 26, 2006 1:04 PM
Subject: Re: [Phys-l] time to bottom of ramp


> |
>  \
>   `-------
> |----L----|
>
> The circular ramp looks something like this.  My poor attempt at ASCII art
>
> Justin Parke
> Oakland Mills High School
> Columbia, MD
>
>
> -----Original Message-----
> From: fizix29@aol.com
> To: phys-l@carnot.physics.buffalo.edu
> Sent: Thu, 26 Oct 2006 12:47 PM
> Subject: [Phys-l] time to bottom of ramp
>
>
> From Tipler, 5th ed., problem 7-8:
>
> "You are given two frictionless ramps and a block to slide down them.  One 
> ramp
> is in the form of an inclined plane with height H and length L.  The other 
> ramp
> is cut in the form of a partial arc of a circle, but also has height H and
> length L.  (my note:  according to the diagram in the problem what they 
> mean by
> length L is the horizontal length of the ramp, not the arc length.  Also 
> the
> circular ramp has a horizontal section; it looks like a capital J lying on 
> its
> side.)  You slide the block down each ramp, starting from rest, and 
> measure the
> time it takes to reach the bottom (my note:  I assume they mean "the end 
> of the
> ramp" since the mass will reach the bottom of the circular ramp before it
> reaches the end of the ramp) and the speed of the block upon getting 
> there.  You
> find that..."
>
> Answer provided by the publisher:
>
> "Because the block sliding down the circular arc travels a greater 
> distance (an
> arc length is greater than the length of the chord it defines) but arrives 
> at
> the bottom of the ramp with the same speed that it had at the bottom of 
> the
> inclined plane, it will require more time to arrive at the bottom of the 
> arc. "
>
> I disagree.
>
> Comments?
>
> Justin Parke
> Oakland Mills High School
> Columbia, MD
>
>
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