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 hollow cylinder has length $$L,$$ inner radius $${{r_1}}$$, outer radius $${{r_2}}$$, and thermal conductivity $$k.$$ The thermal resistance of the cylinder for radial conduction is

Two cylindrical shafts $$A$$ and $$B$$ at the same initial temperature are simultaneously placed in a furnace. The surfaces of the shafts remain at the furnace gas temperature at all times after they are introduced into the furnace. The temperature variation in the axial direction of the shafts can be assumed to be negligible. The data related to shafts $$A$$ and $$B$$ is given in the following Table.

The temperature at the center-line of the shaft $$A$$ reaches $${400^ \circ }C$$ after two hours. The time required (in hours) for the center-line of the shaft $$B$$ to attain the temperature of $${400^ \circ }C$$ is ____________

Consider the radiation heat exchange inside an annulus between two very long concentric cylinders. The radius of the outer cylinder is $${R_0}$$ and that of the inner cylinder is $${R_i}$$ . The radiation view factor of the outer cylinder onto itself is