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

A two dimensional fluid element rotates like a rigid body. At a point with in the element, the pressure is $$1$$ unit. Radius of the Mohr's circle, charactering the state of stress at the point, is

The pressure gauges $${G_1}$$ and $${G_2}$$ installed on the system show pressures of $${P_{G1}} = 5.00$$ bar and $${P_{G2}} = 1.00$$ bar. The value of unknown pressure $$P$$ is