Question

Three solid hemispheres of radeii R each are placed in contact with each other with their flat faces on a rough horizontal surface a sphere of mass M and radius R is placed symmetrically on top of them the normal reaction between the top sphere and any hemisphere assuming the system to be static​

Answer 1

Answer:

the normal reaction between the top sphere and any sphere is 2√3 Mg/9

Explanation:

The sphered will be placed as shown in figure.

Since all the hemispheres and the spheres are of radius R, joining their centres will form a regular tetrahedron (a pyramid made up of equilateral traingles)

A line drawn through the point of contact of the sphere and a hemisphere will pass through the centre of both the hemisphere and the sphere. Nornal reaction will act along this line. It will make 60° from the horizontal

Since, they are all symmetric, the normal reactions will be same

let the normal reactions through the point of contact between a hemisphere and the sphere be N

Then the vertical component of all the three normal reactions will balance out the weight of the sphere

⇒ 3Nsin 60° = Mg

⇒ 3N × √3/2 = Mg

⇒ N  = 2Mg/3√3

or N = 2√3 Mg/9