The large vessel shown in the figure contains oil and water. A body is submerged at the interface of oil and water such that $$45$$ percent of its volume is in oil while the rest is in water. The density of the body is _______________ $$kg/{m^3}.$$

The specific gravity of oil is $$0.7$$ and density of water is $$1000$$ $$kg/{m^3}.$$

Acceleration due to gravity $$g = 10\,m/{s^2}$$

Consider a frictionless, massless and leak-proof plug blocking a rectangular hole of dimensions $$2R \times L$$ at the bottom of an open tank as shown in the figure. The head of the plug has the shape of a semi-cylinder of radius $$R.$$ The tank is filled with a liquid of density $$\rho $$ up to the tip of the plug. The gravitational acceleration is $$g.$$ Neglect the effect of the atmospheric pressure.

The force $$F$$ required to hold the plug in its position is

A spherical balloon with a diameter of $$10m,$$ shown in the figure below is used for advertisements. The balloon is filled with helium $$\left( {{R_{He}} = 2.08\,\,\,\,kJ/kg.K} \right)$$ at ambient conditions of $${15^ \circ }C$$ and $$100\,\,kPa.$$ Assuming no disturbances due to wind , the maximum allowable weight (in Newton) of ballon material and rope required to avoid the fall of the balloon $$({R_{air}} = 0.289kJ/kg.K)$$ is _____________.

A hinged gate of length $$5$$ $$m,$$ inclined at $${30^ \circ }$$ with the horizontal and with water mass on its left, is shown in the figure below. Density of water is $$1000$$ $$kg/{m^3}$$. The minimum mass of the gate in kg per unit width (perpendicular to the plane of paper), required to keep it closed is