### Problems

## 14.1 Fluids, Density, and Pressure

Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb.). What is the volume in liters of this much mercury?

What is the mass of a deep breath of air having a volume of 2.00 L? Discuss the effect taking such a breath has on your body’s volume and density.

A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces $89.0\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{3}$ of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes’ principle.)

Suppose you have a coffee mug with a circular cross-section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.

A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.

A trash compactor can compress its contents to 0.350 times their original volume. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?

A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?

What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.

The tip of a nail exerts tremendous pressure when hit by a hammer because it exerts a large force over a small area. What force must be exerted on a nail with a circular tip of 1.00-mm diameter to create a pressure of $3.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}\text{?}$ (This high pressure is possible because the hammer striking the nail is brought to rest in such a short distance.)

A glass tube contains mercury. What would be the height of the column of mercury which would create pressure equal to 1.00 atm?

The greatest ocean depths on Earth are found in the Marianas Trench near the Philippines. Calculate the pressure due to the ocean at the bottom of this trench, given its depth is 11.0 km and assuming the density of seawater is constant all the way down.

What pressure is exerted on the bottom of a gas tank that is 0.500-m wide and 0.900-m long and can hold 50.0 kg of gasoline when full?

A dam is used to hold back a river. The dam has a height $H=12\phantom{\rule{0.2em}{0ex}}\text{m}$ and a width $W=10\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$ Assume that the density of the water is $\rho =1000\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}.$ (a) Determine the net force on the dam. (b) Why does the thickness of the dam increase with depth?

## 14.2 Measuring Pressure

Find the gauge and absolute pressures in the balloon and peanut jar shown in Figure 14.12, assuming the manometer connected to the balloon uses water and the manometer connected to the jar contains mercury. Express in units of centimeters of water for the balloon and millimeters of mercury for the jar, taking $h=0.0500\text{m}$ for each.

Assuming bicycle tires are perfectly flexible and support the weight of bicycle and rider by pressure alone, calculate the total area of the tires in contact with the ground if a bicycle and rider have a total mass of 80.0 kg, and the gauge pressure in the tires is $3.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}$.

## 14.3 Pascal's Principle and Hydraulics

How much pressure is transmitted in the hydraulic system considered in Example 14.3? Express your answer in atmospheres.

What force must be exerted on the master cylinder of a hydraulic lift to support the weight of a 2000-kg car (a large car) resting on a second cylinder? The master cylinder has a 2.00-cm diameter and the second cylinder has a 24.0-cm diameter.

A host pours the remnants of several bottles of wine into a jug after a party. The host then inserts a cork with a 2.00-cm diameter into the bottle, placing it in direct contact with the wine. The host is amazed when the host pounds the cork into place and the bottom of the jug (with a 14.0-cm diameter) breaks away. Calculate the extra force exerted against the bottom if he pounded the cork with a 120-N force.

A certain hydraulic system is designed to exert a force 100 times as large as the one put into it. (a) What must be the ratio of the area of the cylinder that is being controlled to the area of the master cylinder? (b) What must be the ratio of their diameters? (c) By what factor is the distance through which the output force moves reduced relative to the distance through which the input force moves? Assume no losses due to friction.

Verify that work input equals work output for a hydraulic system assuming no losses due to friction. Do this by showing that the distance the output force moves is reduced by the same factor that the output force is increased. Assume the volume of the fluid is constant. What effect would friction within the fluid and between components in the system have on the output force? How would this depend on whether or not the fluid is moving?

## 14.4 Archimedes’ Principle and Buoyancy

What fraction of ice is submerged when it floats in freshwater, given the density of water at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ is very close to $1000\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$?

If a person’s body has a density of $995\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$, what fraction of the body will be submerged when floating gently in (a) freshwater? (b) In salt water with a density of $1027\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$?

A rock with a mass of 540 g in air is found to have an apparent mass of 342 g when submerged in water. (a) What mass of water is displaced? (b) What is the volume of the rock? (c) What is its average density? Is this consistent with the value for granite?

Archimedes’ principle can be used to calculate the density of a fluid as well as that of a solid. Suppose a chunk of iron with a mass of 390.0 g in air is found to have an apparent mass of 350.5 g when completely submerged in an unknown liquid. (a) What mass of fluid does the iron displace? (b) What is the volume of iron, using its density as given in Table 14.1? (c) Calculate the fluid’s density and identify it.

Calculate the buoyant force on a 2.00-L helium balloon. (b) Given the mass of the rubber in the balloon is 1.50 g, what is the net vertical force on the balloon if it is let go? Neglect the volume of the rubber.

What is the density of a woman who floats in fresh water with $4.00\text{\%}$ of her volume above the surface? (This could be measured by placing her in a tank with marks on the side to measure how much water she displaces when floating and when held under water.) (b) What percent of her volume is above the surface when she floats in seawater?

A man has a mass of 80 kg and a density of $955{\text{kg/m}}^{3}$ (excluding the air in his lungs). (a) Calculate his volume. (b) Find the buoyant force air exerts on him. (c) What is the ratio of the buoyant force to his weight?

A simple compass can be made by placing a small bar magnet on a cork floating in water. (a) What fraction of a plain cork will be submerged when floating in water? (b) If the cork has a mass of 10.0 g and a 20.0-g magnet is placed on it, what fraction of the cork will be submerged? (c) Will the bar magnet and cork float in ethyl alcohol?

What percentage of an iron anchor’s weight will be supported by buoyant force when submerged in salt water?

Referring to Figure 14.20, prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes’ principle). You may assume that the buoyant force is ${F}_{2}-{F}_{1}$ and that the ends of the cylinder have equal areas$A$. Note that the volume of the cylinder (and that of the fluid it displaces) equals $({h}_{2}-{h}_{1})A$.

A 75.0-kg man floats in freshwater with 3.00% of his volume above water when his lungs are empty, and 5.00% of his volume above water when his lungs are full. Calculate the volume of air he inhales—called his lung capacity—in liters. (b) Does this lung volume seem reasonable?

## 14.5 Fluid Dynamics

What is the average flow rate in ${\text{cm}}^{3}\text{/s}$ of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?

The heart of a resting adult pumps blood at a rate of 5.00 L/min. (a) Convert this to ${\text{cm}}^{3}\text{/s}$. (b) What is this rate in ${\text{m}}^{3}\text{/s}$?

The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions. On average, the river has a flow rate of about 300,000 L/s. At the gorge, the river narrows to 20-m wide and averages 20-m deep. (a) What is the average speed of the river in the gorge? (b) What is the average speed of the water in the river downstream of the falls when it widens to 60 m and its depth increases to an average of 40 m?

(a) Estimate the time it would take to fill a private swimming pool with a capacity of 80,000 L using a garden hose delivering 60 L/min. (b) How long would it take if you could divert a moderate size river, flowing at ${5000\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}\text{/s}$ into the pool?

What is the fluid speed in a fire hose with a 9.00-cm diameter carrying 80.0 L of water per second? (b) What is the flow rate in cubic meters per second? (c) Would your answers be different if salt water replaced the fresh water in the fire hose?

Water is moving at a velocity of 2.00 m/s through a hose with an internal diameter of 1.60 cm. (a) What is the flow rate in liters per second? (b) The fluid velocity in this hose’s nozzle is 15.0 m/s. What is the nozzle’s inside diameter?

Prove that the speed of an incompressible fluid through a constriction, such as in a Venturi tube, increases by a factor equal to the square of the factor by which the diameter decreases. (The converse applies for flow out of a constriction into a larger-diameter region.)

Water emerges straight down from a faucet with a 1.80-cm diameter at a speed of 0.500 m/s. (Because of the construction of the faucet, there is no variation in speed across the stream.) (a) What is the flow rate in ${\text{cm}}^{3}\text{/s}$? (b) What is the diameter of the stream 0.200 m below the faucet? Neglect any effects due to surface tension.

## 14.6 Bernoulli’s Equation

Suppose you have a wind speed gauge like the pitot tube shown in Figure 14.32. By what factor must wind speed increase to double the value of *h* in the manometer? Is this independent of the moving fluid and the fluid in the manometer?

If the pressure reading of your pitot tube is 15.0 mm Hg at a speed of 200 km/h, what will it be at 700 km/h at the same altitude?

Every few years, winds in Boulder, Colorado, attain sustained speeds of 45.0 m/s (about 100 mph) when the jet stream descends during early spring. Approximately what is the force due to the Bernoulli equation on a roof having an area of $220{\text{m}}^{2}$? Typical air density in Boulder is $1.14{\text{kg/m}}^{3}$, and the corresponding atmospheric pressure is $8.89\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}{\text{N/m}}^{2}$. (Bernoulli’s principle as stated in the text assumes laminar flow. Using the principle here produces only an approximate result, because there is significant turbulence.)

What is the pressure drop due to the Bernoulli Effect as water goes into a 3.00-cm-diameter nozzle from a 9.00-cm-diameter fire hose while carrying a flow of 40.0 L/s? (b) To what maximum height above the nozzle can this water rise? (The actual height will be significantly smaller due to air resistance.)

(a) Using Bernoulli’s equation, show that the measured fluid speed *v* for a pitot tube, like the one in Figure 14.32(b), is given by $v={\left(\frac{2{\rho}^{\prime}gh}{\rho}\right)}^{1\text{/}2}$, where *h* is the height of the manometer fluid, ${\rho}^{\prime}$ is the density of the manometer fluid, $\rho $ is the density of the moving fluid, and *g* is the acceleration due to gravity. (Note that *v* is indeed proportional to the square root of *h*, as stated in the text.) (b) Calculate *v* for moving air if a mercury manometer’s *h* is 0.200 m.

A container of water has a cross-sectional area of $A=0.1\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$. A piston sits on top of the water (see the following figure). There is a spout located 0.15 m from the bottom of the tank, open to the atmosphere, and a stream of water exits the spout. The cross sectional area of the spout is ${A}_{\text{s}}=7.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}{\text{m}}^{2}$. (a) What is the velocity of the water as it leaves the spout? (b) If the opening of the spout is located 1.5 m above the ground, how far from the spout does the water hit the floor? Ignore all friction and dissipative forces.

A fluid of a constant density flows through a reduction in a pipe. Find an equation for the change in pressure, in terms of ${v}_{1},{A}_{1},{A}_{2}$, and the density.

## 14.7 Viscosity and Turbulence

(a) Calculate the retarding force due to the viscosity of the air layer between a cart and a level air track given the following information: air temperature is $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$, the cart is moving at 0.400 m/s, its surface area is $2.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2},$ and the thickness of the air layer is $6.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}\text{m}$. (b) What is the ratio of this force to the weight of the 0.300-kg cart?

The arterioles (small arteries) leading to an organ constrict in order to decrease flow to the organ. To shut down an organ, blood flow is reduced naturally to 1.00% of its original value. By what factor do the radii of the arterioles constrict?

A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force and the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while the drag force is assumed to be given by Stokes Law, ${F}_{\text{s}}=6\pi r\eta v.$ Show that the terminal speed is given by $v=\frac{2{R}^{2}g}{9\eta}\left({\rho}_{\text{s}}-{\rho}_{1}\right)$, where *R* is the radius of the sphere, ${\rho}_{\text{s}}$ is its density, and ${\rho}_{1}$ is the density of the fluid, and $\eta $ the coefficient of viscosity.

Using the equation of the previous problem, find the viscosity of motor oil in which a steel ball of radius 0.8 mm falls with a terminal speed of 4.32 cm/s. The densities of the ball and the oil are 7.86 and 0.88 g/mL, respectively.

A skydiver will reach a terminal velocity when the air drag equals their weight. For a skydiver with a large body, turbulence is a factor at high speeds. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be ${F}_{\text{D}}=\frac{1}{2}\rho A{v}^{2},$ and setting this equal to the skydiver’s weight, find the terminal speed for a person falling “spread eagle.”

(a) Verify that a 19.0% decrease in laminar flow through a tube is caused by a 5.00% decrease in radius, assuming that all other factors remain constant. (b) What increase in flow is obtained from a 5.00% increase in radius, again assuming all other factors remain constant?

When physicians diagnose arterial blockages, they quote the reduction in flow rate. If the flow rate in an artery has been reduced to 10.0% of its normal value by a blood clot and the average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?

An oil gusher shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. Neglecting air resistance but not the resistance of the pipe, and assuming laminar flow, calculate the pressure at the entrance of the 50.0-m-long vertical pipe. Take the density of the oil to be $900\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$ and its viscosity to be $1.00({\text{N/m}}^{2})\cdot \text{s}$ (or $1.00\phantom{\rule{0.2em}{0ex}}\text{Pa}\cdot \text{s}$). Note that you must take into account the pressure due to the 50.0-m column of oil in the pipe.

Concrete is pumped from a cement mixer to the place it is being laid, instead of being carried in wheelbarrows. The flow rate is 200 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the pump is $8.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$. (a) Calculate the resistance of the hose. (b) What is the viscosity of the concrete, assuming the flow is laminar? (c) How much power is being supplied, assuming the point of use is at the same level as the pump? You may neglect the power supplied to increase the concrete’s velocity.

Verify that the flow of oil is laminar for an oil gusher that shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. The vertical pipe is 50 m long. Take the density of the oil to be $900\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$ and its viscosity to be $1.00({\text{N/m}}^{2})\cdot \text{s}$ (or $1.00\phantom{\rule{0.2em}{0ex}}\text{Pa}\cdot \text{s}$).

Calculate the Reynolds numbers for the flow of water through (a) a nozzle with a radius of 0.250 cm and (b) a garden hose with a radius of 0.900 cm, when the nozzle is attached to the hose. The flow rate through hose and nozzle is 0.500 L/s. Can the flow in either possibly be laminar?

A fire hose has an inside diameter of 6.40 cm. Suppose such a hose carries a flow of 40.0 L/s starting at a gauge pressure of $1.62\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$. The hose goes 10.0 m up a ladder to a nozzle having an inside diameter of 3.00 cm. Calculate the Reynolds numbers for flow in the fire hose and nozzle to show that the flow in each must be turbulent.

At what flow rate might turbulence begin to develop in a water main with a 0.200-m diameter? Assume a $20\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ temperature.