Question

A cylinder is placed on a rough inclined surface of inclination '. Minimum value of coefficient of static friction between the cylinder and the surface so that the cylinder rolls without slipping is

Answer 1

Hey there! First of all such a good question! Had to use so many concepts to get here…

Here goes,

This is a diagram of the situation. Now, next will be a free body diagram of the Solid Cylinder.

Now, by looking at the situation, we can say that the normal force will be balanced by Mgcosβ, while friction will apply in the opposite direction of the motion. We know that rolling is a motion of the body made possible due to torque, so let's calculate the torque due to all the forces here.

(consider radius of the cylinder to be ‘r’ and point of rotation to be the Centre of the mass of the sphere)

Torque due to Normal force and Mg (and all its components) will be zero (because they pass through the point of rotation). But, torque due to friction would be =(f*r) (force by perpendicular distance).

Also, we know that

Torque = Moment of inertia*Angular acceleration

Thus we get

τ=Iα

But τ=f*r and I(for solid cylinder is) = 1/2 M*r² and also α=a/r

Thus, we get, after substituting the values for f, I, and a;

μ(Mgcosβ) * r = 1/2 Mr² * gsinβ/r

Solving which gives us

μ=1/2 tanβ