Question

a uniform cylinder of mass 10 kg and radius 1m rolls without slipping on a rough inclined plane of inclination θ .the minimum value of coefficient of friction μ is

Answer 1

Given : a uniform cylinder of mass 10 kg and radius 1m rolls without slipping on a rough inclined plane of inclination θ.

To find : moment of inertia of cylinder about an tangential line, I = 1/2 mR² + mR²

= 3/2 mR²

we know, torque = Iα

⇒F. R = 3/2 mR²α

⇒mgsinθ . R = 3/2 mR²α

⇒gsinθ = 3/2 Rα

we know linear acceleration, a = Rα

⇒gsinθ = 3/2 a

⇒a = 2gsinθ/3 ..........(1)

for translational motion,

mgsinθ - friction = ma

⇒mgsinθ - μmgcosθ = m × 2gsinθ/3 [ from eq (1).]

⇒mgsinθ - 2mgsinθ/3 = μmgcosθ

⇒mgsinθ/3 = μmgcosθ

⇒μ = tanθ/3

Therefore minimum value of coefficient of friction, μ = tanθ/3.