Q. A sphere of mass m rolls without slipping on an inclined plane of inclination theta. Find the linear acceleration of the sphere and the force of friction acting on it. What should be the minimum coefficient of static friction to support pure rolling ?

The Equation of linear motion of centre of mass is given by
Fx= maxmgsinθ-f=ma
The Equation of  rotational  motion of centre of mass is given by
Γx=Ια ; where Γ is total torque, I is momen.t of inertia about c.m. and αis the angular acceleration of spherefr=25mr2αfr=25mr2ar   , as we know a=αr.f=25ma      ...........(1)putting this value in above equation, we getmgsinθ-25ma=maa=57gsinθusing equation (1) we have f=25mgsinθ. Thus acceleration and frictional force are 57gsinθ and 25mgsinθ respectively.
Now for pure rolling
μN>fμmgcosθ>27mgsinθμ>27tanθ. Thus the minimum coeficient of static friction is 27tanθ

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