A cylindrical body of cross - sectional area A, height $$H$$ and density $${\rho _s},$$ is immersed to a depth $$h$$ in a liquid of density $$\rho ,$$ and tied to the bottom with a string. The tension in the string is

Air flows through a venture and into atmosphere. Air density is $$\rho $$; atmospheric pressure is $$'\,{P_a}';$$ throat diameter is exit diameter is $$'{D_t}';$$ and exit velocity is $$U.$$ the throat is connected to a cylinder containing a frictionless piston attached to a spring. The spring constant is $$'k'.$$ the bottom surface of the piston is exposed to atmosphere. Due to the flow, the piston moves by distance $$x.$$ assuming incompressible frictionless flow, $$x$$ is

A water container is kept on a weighing balance. Water from a tap is falling vertically into the container with a volume flow rate of $$'Q'$$; the velocity of the water when it hits the water surface is $$'U'$$. At a particular instant of time the total mass of the container and water is $$'m'.$$ The force registered by the weighing balance at this instant of time is

A syringe with a frictionless plunger contains water and has at its end a $$100$$ $$mm$$ long needle of $$1$$ $$mm$$ diameter. The internal diameter of the syringe is $$10$$ $$mm.$$ Water density is $$1000\,\,kg/{m^3}.$$ . The plunger is pushed in at $$10$$ $$mm/s$$ and the water comes out as a jet.

Assuming ideal flow, the force $$F$$ in Newton’s required on the plunger to push out the water is