Fluids, Buoyancy, Archimedes

1. Calculate the pressure at the bottom of a trench due to ocean water, given that its depth is 11.0 Km and assuming that the density of ocean water is 1.025 g/cm³.

2. The flow of a particular oil would be streamlined if the Reynold's number was 2500 or less. The oil is to flow through a pipeline consisting of a 5 inch diameter pipe. The specific gravity of this oil measured to be 0.86. The 20 degree Celsius viscosity of the oil is .084 poise.

a. What can be the maximum avg. speed of the oil in the pipeline in order to avoid turbulence?

b. How would your answer change if the temp of the oil was to rise above 20 degree celsius?

3. A large building is tightly closed, the pressure inside being the same as that outside, 988 mbars. Suddenly a strong wind starts to blow across the face of this building, peaking at 23 meters per second. The dimensions of the windows in this building are 1.2 m by 2.15 m.

a. What will be the pressure difference across these windows as the wind blows?

b. What force will be acting on each window?

4. Given the density of a tennis ball as 0.084 g/cc and a diameter of 3.8 cm, what force can submerge the ball in water?

5. How is fluid flow through a hole in a tank related to the area of the hole?

6. A pipeline has an entrance diameter of 1 ft. and an exit diameter of 2 ft. Water flows into the entrance of the pipe at 100 ft/sec. The pressure at the entrance is 200 lb per square foot. Find the velocity and pressure at the exit. Assume that there is no friction. Assume that the density of water is about 62 lb per cubic foot.

7. The envelope and basket of a hot-air balloon have a combined weight of 2.45 kN and the envelope has a capacity (volume) of 2.18x10³ m³. When it is fully inflated, what should be the temperature of the enclosed air to give the balloon a lifting capacity (force) of 2.67 kN (in addition to the balloon's weight)? Assume that the surrounding air, at 20ºC, has a weight per unit volume of 11.9 N/m³ and a molecular mass of 0.028 kg/mole, and is at a pressure of 1.0 atm.

8. Air pressure is 1x10⁵ N/m², air density is 1.3 kg/m³ . If one blows carefully across the top of a straw sticking up 0.10 m from the liquid in a soft drink can, it is possible to make the soft drink rise half way up the straw and stay there. How fast must the air be blown across the top of the straw?

9.A submarine will experience a pressure of 10.0 N per square millimeter at what depth?

10. What is the lift (in Newtons) due to Bernoulli's principle on a wing of area 78 m² if the air passes over the top and bottom surfaces at speeds of 260 m/s and 150 m/s, respectively?

11. A body floats in a liquid A with its 9/10th of its volume immersed in it ,while in a liquid B with its 3/5th of volume immersed in it. Compare the densities of the liquids A and B.

12. A scuba diver can withstand pressures up to 4 atmospheres without risk of getting the bends. What is the maximum safe diving depth?

13. A water main of radius 10 cm runs from a reservoir on a hill down to a town, where houses are connected to it by supply pipes of 1cm radius. When one house only switches on a tap, the flow rate 0.25 L/s. What is the average velocity of water in the mains, and in the household's supply pipe? Where is the pressure highest? And lowest?
(A) At the top of the hill;
(B) In the mains outside in the street;
(C) At the tap which is switched on.

14.

14. The Environment Agency measures the average velocity of flow in a river to be v = 1.1 m/s where it is h = 0.5 m deep and w = 5 m wide.
(A) What is its flow rate?
(B) Just downstream, in a narrower part of the river it is 2m wide and 1 m deep. What is the flow rate and the average velocity here?
(C) The concentration of a toxic pollutant coming from a paint factory at this flow rate is 50 percent of the EC safe limit. During a drought, the river is observed to slow down to v’ = 0.8 m/s and is shallower at h’ = 0.3 m, though still w = 5 m wide. What is the new flow rate? Is the EC limit now exceeded?

15. A thin spherical shell of mass 0.500 kg and diameter 0.180 m is filled with alcohol (= 806 kg/m³). It is then released from rest on the bottom of a pool of water. Find the acceleration of the alcohol filled shell buoyancy lifts it toward the surface of the water.

16. A small child is asleep in a stationary train. A helium balloon is tied with a length of light string to her wrist and otherwise floats freely in the still air. As the train accelerates forward, what happens to the balloon?

Answers

1. Given the density of ocean water is uniform at 1.025 g/cm³ = 1025 kg/m³, the weight of water is 1.005 x 10⁴ N/m³ .

The pressure of water at this depth (the weight per square meter) would be

(1.005 x 10⁴ N/m³)(11.0 x 10³ m) = 1.105 x 10⁸ N/m² = 1.105 x 10⁵ kPa

(about 16000 psi)

For solutions to all the problems on this page click here.

The lifting force required = 2.45 kN + 2.67 kN = 5120 N

weight of 2.18x10³ m³ air at 20ºC = 11.9 N/m³ * 2.18x10³ m³ = 25942 N

From Archimedes' principle,

Lifting force = [weight of 2.18x10³ m³ air at 20ºC] - [weight of 2.18x10³ m³ hot air]

Substituting calculated values,

5120 N = 25942 N - [weight of 2.18x10³ m³ hot air]

weight of 2.18x10³ m³ hot air = 20822 N

From the ideal gas law, PV is proportional to WT

Since P and V are constant,

W₁T₁ = W₂T₂

(W₁ / W₂)T₁ = T₂

(25942 N / 20822 N)(293 K) = 365 K

365 K = 92ºC

For solutions to all the problems on this page click here.