Question

A uniform solid sphere of mass `1 kg` and radius 10 cm is kept stationary on a rough inclined plane by fixing a highly dense particle at `B`. Incination of plane is `37^@` with horizontal and `AB` is the diameter of the sphere which is parallel to the plane, as show in figure. Calculate

a. mass of the particle fixed at `B`
b. minimum required coefficient of friction between sphere and plane to keep sphere in equilibrium.

Answer 1

Let us assume the mass of the particle of be `m`. Let us draw the free body diagram of the system of sphere and particle. Since system is in stastic equilibrium, torque of forces acting on the system should be zero. Taking torque about the point `A` of all of the forces acting on the system,
`(Mg sin 37^@)+(mgsin37^@)R`
`=(mgcos37^@)R`
`implies m=3kg`

Considering forces normal the plane.
`N=Mgcos37^@+mgcos37^@=32N`
The friction force between sphere and plane is static nature. Now considering forces along the plane.
`f=mgsin37^@+mgsin37^@=24N`
But `flemuN` where `mu` is coefficient of friction which gives
`mugef/N` or `mu_("min")=0.75`