plz help me out in this mechanics question

Question

three  solid hemispheres of radii 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 symetrically on top of them. the normal reaction between the top sphere and any hemisphere assuming the system to be in static equilibrium is

Answer 1

Is that the end of the problem?  They want you to write an expression, in terms of given quantities and constants, for the normal force exerted on the sphere by each hemisphere?  I will proceed with that idea, and, if I am incorrect, please let me know.

 

I am not sure how much you have to prove here about the spatial relationship of the spheres, but the main thing we know is that all the hemispheres and the sphere have the same radius.  This may be tricky to see without a diagram, but this makes a "kissing spheres"-type configuration, with the center of each hemisphere and the center of the sphere on the vertices of a pyramid where each face is an equilateral triangle -- in other words, a regular tetrahedron.  We can go into the details of that, if needed.

 

Whenever two spheres (or the curved surfaces of hemispheres) are in this kind of tangential contact, a line can be drawn from the center of one sphere, through the point of contact, to the center of the other sphere.  At exactly the point of contact, the line is perpendicular to both spherical surfaces (running, as it does, along the radial directions of the spheres).

 

Since the normal force of a surface must always be exerted perpendicular to that surface, the normal force of each hemisphere below on the sphere above must run along this line between the center of that hemisphere and the center of the sphere.  Because that line is a side, running toward the apex, of the regular tetrahedron, it is elevated at 60 degrees above the horizontal.

 

So, at the point of contact, the normal force of each lower hemisphere is exerted perpendicular to the spherical surface, and thus along a line that is elevated at 60 degrees above the horizontal.  This will help determine how to write an expression for the normal force in terms of given quantities and constants.

 

I think looking in the vertical direction, using Newton's 2nd law on the supported sphere, will be most instructive.  In that direction, the forces on the sphere are the force of gravity downward, which I usually define as negative, and then the three vertical components of the normal forces from the hemispheres.  My drawing tells me that those vertical components are upward, and the symmetry of the situation (with the equal-radii spheres and hemispheres) tells me that all three components should be equal.  And, with the system in equilibrium, as defined, I know the overall acceleration (and thus the acceleration component in the vertical direction) has to be zero.  So Newton's 2nd law in the vertical becomes:

 

F_netz = 3N_z-mg = 0 

 

where N_z is the vertical component of each normal force from a hemisphere below.

 

Because the overall normal force lies along a line elevated 60 degrees above the horizontal, my drawing tells me that the relationship between N_y and the overall normal for N on each sphere is:

 

N_z/N = sin(60^o)

 

Thus, N_z= Nsin(60^o)

 

So, in terms of the overall normal force N (that we are trying to find a n expression for in terms of given quantities and constants), the Newton's second law equation above becomes:

 

F_netz = 3(Nsin(60^o)) - mg = 0

 

sin(60^o) = (√3)/2, from trigonometry, so the above becomes:

 

F_netz =  (3N√3)/2 - mg = 0 --> (3N√3)/2 = mg

 

From there, it is a hop and a skip of algebra to solve for an expression for N in terms of given quantities and constants.  If you would like to go into more detail, or check an answer, just let me know.

 

And, of course, if I misinterpreted the problem, please let me know that, too!

 

Hope this helps!