Consider a cuboidal vessel (2R$$ × $$2R$$ × $$R) of mass m with a hemispherical cavity of radius R, which is kept on a horizontal smooth surface as shown in the figure. The vessel has very small hole of $$cross-sectional$$ area “a”. Now water of density ρ is poured into space developed between the horizontal surface and vessel through a hole at top very slowly. When the height of water level is h, the vessel lifts off the surface and liquid leaks through space generated. (R$$ $$=$$ $$2m$$, ρ $$=$$ $$103$$ $$kg/m3$$, m $$=$$ $$9$$π $$kg) . If the value of h is $$5k$$ cm, find the value of k. (Atmospheric$$ pressure $$=$$ $$105$$ $$N/m2$$ $$).

Question

Consider  a  cuboidal  vessel  (2R$$ × $$2R$$ × $$R)   of mass m with  a hemispherical  cavity  of  radius  R, which is kept  on  a  horizontal smooth surface as shown in the figure. The vessel has very small hole of $$cross-sectional$$ area “a”. Now water of  density  ρ   is  poured  into  space  developed  between the horizontal surface and vessel through a hole at top very slowly. When the height of water level is h, the vessel lifts off the surface and liquid leaks through space generated. (R$$ $$=$$ $$2m$$, ρ $$=$$ $$103$$ $$kg/m3$$, m $$=$$ $$9$$π $$kg) . If the value of h is $$5k$$ cm, find the value of k.    (Atmospheric$$ pressure $$=$$ $$105$$  $$N/m2$$ $$).

Infinity Learn · Accepted Answer

$$mg=N+FBSince$$,$$N=0$$⇒$$mg=FB$$       [here $$FB=Vimmersed$$ρwaterg]⇒$$mg=$$[π$$R2h$$−∫$$0h$$π$$(R2$$−$$z2)dz$$]ρg⇒$$mg=$$ρg[π$$R2h$$−π[$$R2h$$−$$h33$$]]⇒$$h3=3m$$πρ$$=3$$×π×$$9$$π×$$103$$⇒$$h=310=0.3m=30m$$⇒$$5k=30$$⇒$$k=6$$

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