VDU_Physics_2_006

## Physics 2.-  Fluid Statics

Javier Montenegro Joo

jmj@VirtualDynamicsSoft.com

Fluid Statics.-

(1) An open U-shaped tube has branches of section  A and, it contains a liquid of a certain density.  Determine the volume of another liquid having a density which is half the density of the first liquid, which must be poured in a branch of the tube so that the first liquid rises ho  in the other branch.  Both liquids are immiscible.

Solution.-

The figure shows the two acts of this problem:  In the first act appears only the first liquid. In the second act, the second liquid has been poured in the left branch of the tube, being h  the height of this second fluid. For the first liquid to raise a height ho in the right hand branch, the interface level must descend  ho  in the left branch. Under these conditions, at the level of the interface, the pressure is the same in both branches:

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(2)    A 0.50 kg object weighs 1.50 N in water and 0.70 N in petroleum. Determine the density of the petroleum.

Solution.-

The upthrust or buoyancy “E” is the real weight minus the apparent weight of the object in the fluid:

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(3) A long hose containing two immiscible liquids is vertically bent as shown in the sketch. A piston is making a pressure Po at the left extreme of the hose, while the extreme at the right side is closed with a stopper.  Determine the pressure at the stopper.

Solution .-

Note that when varying the pressure Po exerted by the piston at the left extreme of the hose, the pressure changes along the entire system, hence in order to find the pressure at any point, the connected pressures starting at the left extreme must be considered.  Bear in mind that at the level of two contiguous interfaces the pressure is the same. If the piston were retired the pressure at its position would be that of the atmosphere.

(4) In the figure, a container contains liquid of density D. (a) Determine the pressures at points   a, b, c, and d.  (b) Demonstrate that the pressure at  point b is greater than that at point a. (c) Demonstrate that if the pressures at points  a and b  were equal, then the liquid would be at the same level at both points.

Solution.-

·         The sketch below shows the container and the interfaces I1 and I2.

·         At the place where the container is open, the pressure is the atmospheric pressure Po.

·         Pressures are always positive or zero. Negative pressures are impossible.

·         At the apparently empty regions a and b, there is air exerting pressures Pa  and  Pb., respectively.

·         At points a and b, if the corresponding pressures Pa and Pb were higher, the levels of the liquid would be lower at those points and, vice versa.

·         If points a and b were at the same level, then Pa = Pb.

·         At the bottom of the tank, the pressures are the same: Pc = Pd.

·         At the level of each interface, the pressures are the same.

(5) From an unidentified flying object flying at 500 m altitude drops a mechanical piece made of an extraterrestrial metal. The terrestrials find that the weight of the piece is 25 N.  When the piece is hung from a spring scale, also known as Roman scale, and immersed completely in water, the scale shows that the piece apparently weights 15 N.  Determine the density of the metallic piece

Solution.-

·     The buoyancy B is the force the liquid exerts on the immersed body and, it is given by the difference between the real weight of the body and its apparent weight.

·     According to Archimedes, the buoyancy is given by the weight of the displaced fluid.

(6) In the figure, a balloon full of gas at a pressure PG is connected to a U-shaped tube which contains two liquids, A and B, whose densities are shown. Determine:  (a) the height H (b) the pressure of the gas in the balloon for liquid B to be leveled in both branches of the U-tube.

Solution .-

At the level of the interface between the gas and the liquid B, the pressure is the same. On the other hand, if liquid B is leveled in both branches of the U-tube:   H = 0.

(7) A hollowed ball is floating right beneath the surface of a liquid of known density D. The volume of the ball is Vb and its density is 4D. Determine: (a) The force applied by the liquid on the ball, (b) The mass of the ball (c) The volume of the cavity inside the ball.

Solution.-

·     The force applied by the liquid on the ball is the push-up force, also known as buoyancy B and, it is equal to the weight of the displaced fluid.

·     Since the hollowed ball is a shell and it is completely immersed in the liquid, the buoyancy equals the weight of this shell.

·     The total volume of the ball Vb includes the volume of the shell and that of the cavity.

(8) Consider a compact cylindrical block of radius R and height H, made of a material of density D. In order to study the block, it is placed in a liquid, where the block floats with 25% of its volume above the surface. Determine: (a) The density of the liquid (b) The density of the liquid if the block were a cube of side L and its density were two times higher.

Solucion.-

Since the block is floating on the liquid, the weight of the block is equilibrated by the buoyancy (up-thrust), and the buoyancy B is equal to the weight of the displaced fluid.

[9] A compact ball having a radius of 30 cm and a density of  600 kg/m3  is released at 10m depth in a tank full of water. Taking into consideration the weight of the ball but disregarding the friction between the ball and the water, determine: (a) velocity of the ball when reaching the surface of the water (b) time for the ball to reach the surface (c) Acceleration of the ball. (d) Height the ball jumps out of the surface (e) Time the ball is in the air.

Solution.-

While the ball is in the water it experiments a push up or buoyancy, this implies an upward acceleration. When the ball reaches the surface of the water it has a certain velocity v  and the ball jumps out of the water with that velocity, thus the ball reaches some height H in the air and this takes some time t. The total time the ball spends in the air is the time spent to go upwards and the time spent to fall back to the water, both times are equal.

[10] The figure shows a vessel containing gas, water and oil. Determine the pressures at points 1, 2, 3 and 4.

Solution.-

At the level of the interface the pressures are equal.