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36

Part of 2007 F = Ma

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2007 F = ma #36: Resultant Length of Rope

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11/21/2012
A point object of mass mm is connected to a cylinder of radius RR via a massless rope. At time t=0t = 0 the object is moving with an initial velocity v0v_0 perpendicular to the rope, the rope has a length L0L_0, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds TmaxT_{max}. Express your answers in terms of TmaxT_{max}, mm, L0L_0, RR, and v0v_0. [asy] size(200); real L=6; filldraw(CR((0,0),1),gray(0.7),black); path P=nullpath; for(int t=0;t<370;++t) { pair X=dir(180-t)+(L-t/180)*dir(90-t); if(X.y>L) X=(X.x,L); P=P--X; } draw(P,dashed,EndArrow(size=7)); draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy]What is the length (not yet wound) of the rope?
<spanclass=′latex−bold′>(A)</span> L0−πR <span class='latex-bold'>(A)</span>\ L_0 - \pi R
<spanclass=′latex−bold′>(B)</span> L0−2πR <span class='latex-bold'>(B)</span>\ L_0 - 2 \pi R
<spanclass=′latex−bold′>(C)</span> L0−18πR <span class='latex-bold'>(C)</span>\ L_0 - \sqrt{18} \pi R
<spanclass=′latex−bold′>(D)</span> mv02Tmax <span class='latex-bold'>(D)</span>\ \frac{mv_0^2}{T_{max}}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}