Please Answer the question in photo.

Question

Consider two uniform hemispheres with radius R and made of gold of density \(\rho_g\). The top half is nailed to a support and space between the hemispheres is filled with water as shown. Given that the surface tension of water is S and contact angle between gold and water is \(\theta\). The distance between the two hemispheres so that bottom half don't fall is (Assume radius of hemisphere >> distance between the two hemisphere)

Answer 1

Correct option is (1) \(\frac{4 S \cos \theta}{\rho_g g R}\)

Weight = Volume × Density × g.

The volume of a hemisphere is (2/3)πR³,

So Weight = (2/3)πR³ ρ_g g

Upward Force = Surface Tension × Length of contact line

The length of the contact line for the bottom hemisphere is 2πR,

So Upward Force = S × 2πR

Set the upward force equal to the weight and solve for the distance:

S × 2πR = (2/3)πR³ ρ_g g, leading to the distance being proportional to S cosθ.

The distance between the two hemispheres is \(\frac{4 S \cos \theta}{\rho_g g R}\).