Equilibrium and Force on Lower Hemisphere

[Vibhor]

Homework Statement

Homework Equations

The Attempt at a Solution

Using the fact that the sphere is in equilibrium in the 'y' direction , the force exerted by the floor N = 2mg .

Now , if I consider the lower hemisphere then , vertical force exerted by top hemisphere + weight of lower hemisphere = Normal reaction from the floor .

This means , vertical force exerted by top hemisphere on lower = mg

Should this be the total force exerted on the lower hemisphere ?

Thanks

 

[TSny]

Yes, the sphere as a whole is "in equilibrium in the y direction" since the center of mass of the sphere has no acceleration.

However, does the center of mass of the lower hemisphere have any acceleration?

 

[Vibhor]

However, does the center of mass of the lower hemisphere have any acceleration?

It has centripetal acceleration ω²r ( r=3R/8) directed towards the center of the sphere i.e in +y direction (upwards)

Taking + y to be upwards , 2mg - mg +R = mω²r , gives R = 5mg/8 downwards i.e option A)

Is that ok ??

 

[TSny]

It has centripetal acceleration ω²r ( r=3R/8) directed towards the center of the sphere i.e in +y direction (upwards)

Taking + y to be upwards , 2mg - mg +R = mω²r , gives R = 5mg/8 downwards i.e option A)

Is that ok ??

Yes, I believe that's all correct.

 

[Vibhor]

Yes, I believe that's all correct.

Thank you