A square carpet of a mass of $$20$$ kg is dragged from one room into another as shown in the figure. The width of the corridor and the carpet is the same, i.e., $$2$$ m. The friction coefficient is $$0.1$$ at the end of the first room and $$0.2$$ at the other end, in the corridor it changes linearly from $$0.1$$ to $$0.2$$. The carpet presses the floor uniformly. Find the amount of work (in$$ $$J) that must be done to drag the carpet slowly from the first room to the other.

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Infinity Learn · Accepted Answer

Let the coefficient of friction for the corridor is given asμ(x)=0.1+(0.2$$−$$0.1)x2Let$$ us divide the carpet into infinitesimally thin vertical strips. The mass per unit width of the carpet beλ$$=20$$ $$kg2$$ $$m=10$$ kg mWork done in moving each strip $$(dl) through a distance dx is μ$$($$λ$$dl)gdx$$. Total work done in moving each strip from one end to the $$other=$$∫$$02$$ $$0.1+0.12x$$λ$$dlgdx=(0.3)$$λdlgTotal work done in moving the carpet $$=$$∫$$02$$ $$0.3$$λgdl⇒$$0.6$$×λ$$g=60$$ J

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