Q.

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.

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a

mg/2

b

3mg/4

c

3mg/8

d

5mg/8

e

3mg/10 parallel to incline

f

vertical 3mg/8

g

horizontal mg/5

h

3mg/5 perpendicular to incline

i

3/4

j

3/8

k

3/16

l

None of these

answer is , , .

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Detailed Solution

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⁡α2fl−μN=μmgcos⁡αFor no slipping, fe≥f⇒ μmgcos⁡α≥mgsin⁡α2⇒ μ≥tan⁡α2=3/42=38
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