19. Fluid Mechanics

A large warehouse is 100 m wide, 100 m deep, 10 m high:
a) What is the total weight of the air inside the warehouse?
b) How much pressure does the weight of the air apply on the floor?

A tall cylindrical beaker 10 cm in radius is placed on a picnic table outside. You pour 5 L of an 8,000 kg/m³ liquid and 10 L of a 6,000 kg/m³ liquid into. Calculate the total pressure at the bottom of the beaker. (Use g=10 m/s².)

A wooden cube, 1 m on all sides and having density 800 kg/m³ , is held under water in a large container by a string, as shown below. The top of the cube is exactly 2 m below the water line. Calculate the difference between the force applied by water to the top and to the bottom faces of the cube (Hint:calculate the two forces, then subtract. Use g=10 m/s².)

The deepest known point on Earth is called the Mariana Trench, at ~11,000 m (~36,000 ft). If the surface area of the average human ear is 20 cm2, how much average force would be exerted on your ear at that depth?

(II) A wind blowing over the flat roof of a house causes the roof to lift off the house. If the house is 6.2 m x 12.4 m in size, estimate the weight of the roof. Assume the roof is not nailed down.

(II) What is the lift (in newtons) due to Bernoulli’s principle on a wing of area 82 m² if the air passes over the top and bottom surfaces at speeds of 280 m/s and 150 m/s , respectively?

During each heartbeat, approximately 70 cm³ of blood is pushed from the heart at an average pressure of 105 mm-Hg. Calculate the power output of the heart, in watts, assuming 70 beats per minute.

(III) A steel cable is to support an elevator whose total (loaded) mass is not to exceed 3100 kg. If the maximum acceleration of the elevator is 1.8 m/s² , calculate the diameter of cable required. Assume a safety factor of 8.0.

(II) An iron bolt is used to connect two iron plates together. The bolt must withstand shear forces up to about 3600 N. Calculate the minimum diameter for the bolt, based on a safety factor of 7.0.

(II) At depths of 2000 m in the sea, the pressure is about 200 times atmospheric pressure (1 atm = 1.0 x 10⁵ N/m²). By what percentage does the interior space of an iron bathysphere’s volume change at this depth?

(II) (a) What is the maximum tension possible in a 1.00-mm-diameter nylon tennis racket string?

(II) What is the maximum tension possible in a 1.00-mm-diameter nylon tennis racket string? (b) If you want tighter strings, what do you do to prevent breakage: use thinner or thicker strings? Why? What causes strings to break when they are hit by the ball?

(I) What is the difference in blood pressure (mm-Hg) between the top of the head and bottom of the feet of a 1.75-m-tall person standing vertically?

(II) While exploring a sunken ocean liner, a small robot submarine measured the absolute pressure at the level of the ship to be 413 atmospheres.

(b) Calculate the magnitude of the force due to the water on a viewing port of the submarine at this depth if the viewing port has a surface area of 0.0100 m² .

(III) Water stands at a height h behind a vertical dam of uniform width b.

(a) Use integration to show that the total force of the water on the dam is F = 1/2 ρgh²b .

(II) Assume the supports of the uniform cantilever shown in Fig. 12–79 (m = 2900 kg) are made of wood. Calculate the minimum cross-sectional area required of each, assuming a safety factor of 9.0.

<IMAGE>

(II) How high would the level be in an alcohol barometer at normal atmospheric pressure?

(III) A beaker of liquid accelerates from rest, on a horizontal surface, with acceleration α to the right.

(c) How does the pressure vary with depth below the surface?

Intravenous transfusions are often made under gravity, as shown in Fig. 13–59. Assuming the fluid has a density of 1.00 g/cm³, at what height h should the bottle be placed so the liquid pressure is

(a) 52 mm-Hg

<IMAGE>

Intravenous transfusions are often made under gravity, as shown in Fig. 13–59. Assuming the fluid has a density of 1.00 g/cm³ (c) If the blood pressure is 75 mm-Hg above atmospheric pressure, how high should the bottle be placed so that the fluid just barely enters the vein?

<IMAGE>

A bicycle pump is used to inflate a tire. The initial tire (gauge) pressure is 210 kPa (30 psi). At the end of the pumping process, the final pressure is 310 kPa (45 psi). If the diameter of the plunger in the cylinder of the pump is 2.5 cm, what is the range of the force that needs to be applied to the pump handle from beginning to end?

Estimate the total mass of the Earth’s atmosphere, using the known value of atmospheric pressure at sea level.

When a person drives or hikes to a higher altitude, and even more during descent, volume changes of trapped air in the middle ear can cause ear discomfort until the middle-ear pressure and exterior pressure are equalized.

(a) If a rapid descent at a rate of 7.0 m/s or faster commonly causes ear discomfort, what is the maximum rate of increase in atmospheric pressure (that is, dP/dt) tolerable to most people?

When you ascend or descend a great deal when driving in a car, your ears “pop,” which means that the pressure behind the eardrum is being equalized to that outside. If this did not happen, what would be the approximate force on an eardrum of area 0.20cm² if a change in altitude of 1120 m takes place?

(I) A marble column of cross-sectional area 1.4m² supports a mass of 22,000 kg.

(c) By how much is the column shortened if it is 8.6 m high?

Parachutists whose chutes have failed to open have been known to survive if they land in deep snow. Assume that a 75-kg parachutist hits the snow with an area of impact of .030m² at a velocity of 55 m/s, and that the ultimate strength of body tissue is 5 x 10⁵ N/m². Assume that the person is brought to rest in 1.0 m of snow. Show that the person may escape serious injury.

A 25-kg object is being lifted by two people pulling on the ends of a 1.15-mm-diameter nylon cord that goes over two 3.00-m-high poles that are 4.5 m apart, as shown in Fig. 12–93. How high above the floor will the object be when the cord breaks?

<IMAGE>

Suppose a 65-kg person jumps from a height of 3.0 m down to the ground.

(e) Estimate the stress and determine if the tibia will break in a stiff-legged landing ( d = 1.0 cm).

<IMAGE>

(III) Show that inside a soap bubble, there must be a pressure ∆P in excess of that outside equal to ∆P - 4γ/r , where r is the radius of the bubble and γ is the surface tension. [Hint: Think of the bubble as two hemispheres in contact with each other, one on the left and one on the right; and remember that there are two surfaces to the bubble, outside and inside. Note that this result applies to any kind of membrane, where 2γ is the tension per unit length in that membrane.]