Question icon
Grade 10Mechanics

A massless rope is tossed over a wooden dowel of radius r in order to lift a heavy object of weight W off of the floor, as shown in Fig. The coefficient of sliding friction between the rope and the dowel is μ. Show that the minimum down- ward pull on the rope necessary to lift the object is
src=data:image/png;base64,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
(Hint: This problem required techniques from integral calculus.)
src=data:image/png;base64,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

Profile image of Hrishant Goswami
11 Years agoGrade 10
Answers icon

1 Answer

Profile image of Jitender Pal
11 Years ago
235-189_1.PNG
Take antilog,
235-2421_1.PNG