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Grade upto college level Mechanics

The handle of a floor mop of mass m makes an angle θ with the vertical direction; see Fig. Let src=data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAVCAIAAAA8SdJPAAAAqElEQVQ4jeWSyw3EMAhEpy4Koh6qoRmKyR5iFI8/aKPkFt8w8BhG4Hjv4YusMAHUN+EtVpgAYpGxK4W3WEOvK/6WNbIebDixFrKuMExQyWTWqpdaXQv3iMUruU4qXNvHmRuyxHJF6nKFWqJdz88wye4wmRSCUSLSDWwGTf2bTTtWN3X9crHdxYFLK1bmt+cLKq0u/HIoK8OMRqMvLVmdSc3Hyvun7wusH2gwaf0iY64pAAAAAElFTkSuQmCC be the coefficient of kinetic friction between mop and floor and src=data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAASCAIAAADKe1n0AAAAoklEQVQ4jeWUzRHFIAiEt64tiHqohmYoJu/gMyMGjU5yCzdw+Nj1D8ergaTmSkBskO7iXAlQveYmId3Fde0mWBaX4B5YzXCJuFWrCS62u3LD6hUXvZmgp5VSieuYHmeCqs4EopVuQvXDlaf0UpnjTECymV3cnkJM5tYjzpV3G/XnD04n4u6Gt8wUGHDzS+HKuriE619XF+2ZjjxkX8CD+BTuB6SM1MOdMxaNAAAAAElFTkSuQmCC the coefficient of static friction between mop and floor. Neglect the mass of the handle. (a) Find the magnitude of the force F directed along the handle required to slide the mop with uniform velocity across the floor. (b) Show that if θ is smaller than a certain angle θ0 the mop cannot be made to slide across the floor no matter how great a force is directed along the handle. What is the angle θ0?
src=data:image/png;base64,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

Profile image of Amit Saxena
11 Years agoGrade upto college level
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1 Answer

Profile image of Navjyot Kalra
11 Years ago
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When the mop slides down and moves with constant velocity, the acceleration ax should be equal to zero, that is, ax = 0.

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