Q1: What is the force required (in Newtons) to hold a spherical balloon stationary in water at a depth of H from the air-water iterface? The balloon is of radius 0.1 m and is filled with air.

ANS:A -

To find the force required to hold a spherical balloon stationary in water at a depth HHH from the air-water interface, we need to consider the buoyant force acting on the balloon due to the displaced water. Here are the steps to calculate the force:

1. Buoyant Force Calculation:
   • The buoyant force FbF_bFb​ acting on the balloon is equal to the weight of the water displaced by the balloon.
   • The displaced volume of water VdisplacedV_{\text{displaced}}Vdisplaced​ by the balloon is the volume of the spherical balloon submerged in water.
2. Volume of the Balloon:
   • The volume VballoonV_{\text{balloon}}Vballoon​ of a spherical balloon is given by: Vballoon=43πr3V_{\text{balloon}} = \frac{4}{3} \pi r^3Vballoon​=34​πr3 where rrr is the radius of the balloon.
3. Submerged Volume:
   • The volume submerged VsubmergedV_{\text{submerged}}Vsubmerged​ depends on the depth HHH of submersion.
   • For a spherical cap submerged to depth HHH, the submerged volume can be found using geometric formulas, but a simpler approach is to use the volume of the whole sphere and subtract the volume above the depth HHH.
4. Buoyant Force Formula:
   • The buoyant force FbF_bFb​ is given by the weight of the displaced water: Fb=ρwater⋅g⋅VsubmergedF_b = \rho_{\text{water}} \cdot g \cdot V_{\text{submerged}}Fb​=ρwater​⋅g⋅Vsubmerged​ where:
     • ρwater\rho_{\text{water}}ρwater​ is the density of water,
     • ggg is the acceleration due to gravity,
     • VsubmergedV_{\text{submerged}}Vsubmerged​ is the volume of water displaced by the submerged part of the balloon.
5. Density and Gravity Values:
   • Density of water, ρwater=1000\rho_{\text{water}} = 1000ρwater​=1000 kg/m³ (assuming standard conditions).
   • Acceleration due to gravity, g=9.81g = 9.81g=9.81 m/s².
6. Calculations:
   • Radius of the balloon, r=0.1r = 0.1r=0.1 m.
   • Volume of the whole balloon, Vballoon=43π(0.1)3V_{\text{balloon}} = \frac{4}{3} \pi (0.1)^3Vballoon​=34​π(0.1)3 m³.
   • Submerged volume VsubmergedV_{\text{submerged}}Vsubmerged​ can be calculated geometrically or approximated based on HHH.

Without the exact submerged volume VsubmergedV_{\text{submerged}}Vsubmerged​, I can't provide the exact force in Newtons. To find VsubmergedV_{\text{submerged}}Vsubmerged​, one typically uses the spherical cap volume formula or numerical methods to approximate the volume of the sphere below the depth HHH. Once VsubmergedV_{\text{submerged}}Vsubmerged​ is determined, you can calculate FbF_bFb​ using the formula above to find the force required to hold the balloon stationary in water at depth HHH.