A solid cylinder of mass m is kept in balance on a fixed incline of angle α$$=37$$∘ with the help of a thread fastened to its jacket. The cylinder does not slip. What force F is required to keep the cylinder in balance when the thread is held vertically ?In what direction should the thread be pulled to minimise the force required to hold the cylinder? What is the magnitude of this force? The values of minimum coefficient of static friction between cylinder and incline in the case when F is minimum.

Question

A solid cylinder of mass m is kept in balance on a fixed incline of angle α$$=37$$∘ with the help of a thread fastened to its jacket. The cylinder does not slip. What force F is required to keep the cylinder in balance when the thread is held vertically ?In what direction should the thread be pulled to minimise the force required to hold the cylinder? What is the magnitude of this force? The values of minimum coefficient of static friction between cylinder and incline in the case when F is minimum.

Infinity Learn · Accepted Answer

Balancing torque about C:$$Fr=fr$$ ⇒ $$f=FBalancing$$ force in the direction of incline:Fsin⁡α$$+f=mgsin$$⁡α⇒ $$F35+F=mg35$$⇒$$F=3mg8Let$$ force is applied at angle θ as shown with incline again f $$=$$ FFcos⁡θ$$+f=mgsin$$⁡α⇒ Fcos⁡θ$$+F=3mg5$$⇒ $$F=3mg5(1+$$$$cos$$⁡θ$$)F$$ is minimum when $$cos$$⁡θ$$=1$$⇒ θ$$=0$$∘⇒ $$Fmin=3mg5$$[$$1+1$$]$$=3mg10F+f=mgsin$$⁡α⇒ $$f+f=mgsin$$⁡α $$($$∵$$F=f)$$⇒ $$f=mgsin$$⁡α$$2fl$$−μ$$N=$$μmgcos⁡αFor no slipping, fe≥f⇒ μmgcos⁡α≥mgsin⁡α$$2$$⇒ μ≥$$tan$$⁡α$$2=3/42=38$$

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