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One-fourth of a rope of length $l$ is hanging down over the edge of a frictionless table. The rope has a uniform, linear density (mass per unit length) $\lambda$ (Greek lambda), and the end already on the table is held by a person. How much work does the person do when she pulls on the rope to raise he rest of the rope slowly onto the table? Do the problem in two ways as follows. (a) Find the force that the person must exert to raise the rope and from this the work done. Note that this force is variable because at different times, different amounts of rope are hanging over the edge. (b) Suppose the segment of the rope initially hanging over the edge of the table has all of its mass concentrated at its center of mass. Find the work necessary to raise this to table height. You will probably find this approach simpler than that of part (a). How do the answers compare, and why is this so?

$\frac{\lambda g l^{2}}{32}$

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Indian Institute of Technology Kharagpur