Q A sphere of mass m rolls without slipping on an inclined plane of inclination theta Find the - Physics - System Of Particles And Rotational Motion

Question

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 ?

Kulshekhar Mehta · Accepted Answer

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The Equation of linear motion of centre of mass is given by
∑Fx= max⇒mgsinθ-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 sphere⇒fr=25mr2α⇒fr=25mr2ar , as we know
a=αr ⇒f=25ma (1)putting this value in above equation, we
getmgsinθ-25ma=ma⇒a=57gsinθusing equation (1) we
have f=25mgsinθ Thus acceleration and frictional force are
57gsinθ and 25mgsinθ respectively
Now for pure rolling
μN>f⇒μmgcosθ>27mgsinθ⇒μ>27tanθ
Thus the minimum coeficient of static friction is 27tanθ

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 ?