saturday morning question - torsion

huangyhg

saturday morning question - torsion
ok, lets say i have a stanely 25' tape measure.  it doesn't have to be stanely, but just pretend it is.  the tape cross-section is a little concave, essentially creating a shallow "u" section.
now, let's say that i extend that tape measure out, while holding it in my hand, and effectively creating a cantilever of the tape since the base is still in my hand. if i extend the tape out 3 inches, and rotate the base 90 deg in my hand, the free end of the tape rotates 90 deg.  however, if i carefully extend the tape out 10 feet still maintaining a cantilever (without the tape "buckling"), i can rotate the base of the tape measure 90 deg and the end of the tape doesn't rotate at all.
question 1:  what is the mechanism behind this phenomena?
question 2:  how do you free body this?
   
  
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nert
check out our whitepaper library.
just tried it - cool!
i suspect it has something to do with the downward curvature helping keep the cg of the cantilever as low as possible. my tape did twist a little at the end but nowhere near as much as the end i was holding. good thought provoker.
old ca se
i've attached a picture for additional clarification, because this can be interpreted different ways
   
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nert
there is a certain amount of torsion induced into the end of the tape when twisted. the opposing torsion is distributed along the length of the tape. the longer the tape, the more resistance to the induced twist at the end.
i think of the opposing torsion as one-half the width across the short face of the tape times the weight of tape per length. the more rigid the tape, the more torsion produced into the end when twisted. and therefore, a longer length of tape must be extended in order to accumulate enough to resist that torsion.
inertia4u:
while the same equations don't apply just think back to your old mm equations for torsion of a circular shaft...
angle of twist = torque*length/(j*g)
note that the angle of twist is proportional to the length...same idea applies to your tape except the equations are modified due to the cross section (and probably due to the rotational large displacement)..otherwise the concepts are the same...
as for a fbd it is just the same as any other fbd...cut a section and put the internal/external forces on the diagram....
ed.r.
what is resisting the twist? the fbd would have an applied torque at the end and gravity acting along the length of tape.  
i wouldn't even attempt to apply structural mechanics to this problem, however, if i re  
asixth:
exactly what i was thinking as i read the thread;  a star for you!
cheers,
ys
b.eng (carleton)
working in new zealand, thinking of my snow covered home...
i think asixth has got it.  same thing happens to i beams, short length just rotates.  when you have a long length and try and rotate it by hand (say to prime it) its less stiff torsionally (or more flexible torsionally)  so the longer the   
{sighs}
this is an elastic stability problem. case 1 is an unbuckled cantilever and case 2 is essentially a cantilever that has undergone severe lateral torsional buckling.