A thin walled hemispherical shell of mass m and radius R is pressed against a smooth vertical wall. Through a very small aperture at its top, water of density ρ is filled in it completely. Minimum magnitude of force is to be applied to the shell for liquid not to escape from it. The weight of hemispherical shell is m g and total mass of water is M=(2/3) π R3 ρ ⋅ The net external force is M g √(α/β)+(1+(m/M))2 Find (α+β).

Question

A thin walled hemispherical shell of mass m and radius R is pressed against a smooth vertical wall. Through a very small aperture at its top, water of density ρ is filled in it completely. Minimum magnitude of force is to be applied to the shell for liquid not to escape from it. The weight of hemispherical shell is m g and total mass of water is M=(2/3) π R3 ρ ⋅ The net external force is M g √(α/β)+(1+(m/M))2 Find (α+β). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

FH=(ρ g R) π R2=ρ g π R3=(3 M g/2) Fv=(M+m) g F text net =√[(M+m) g]2+((3 M g/2))2 F text net =M g √(9/4)(1+(m/M))2

FH​=(ρgR)πR2=ρgπR3=23Mg​ Fv​=(M+m)g Fnet ​=[(M+m)g]2+(23Mg​)2​ Fnet ​=Mg49​(1+Mm​)2​

$F_{H} = (ρ g R) π R^{2} = ρ g π R^{3} = \frac{3 M g}{2}$
$F_{v} = (M + m) g$
$F_{net} = [(M + m) g]^{2} + (\frac{3 M g}{2})^{2}$
$F_{net} = M g \frac{9}{4} (1 + \frac{m}{M})^{2}$