Chapter 9

Suppose you use the tip of one finger to support a 1.0-kg object. If your finger has a diameter of \(2.0 \mathrm{~cm}\), what is the stress on your finger?

A 5.0 -m-long rod is stretched \(0.10 \mathrm{~m}\) by a force. What is the strain in the rod?

A 250-N force is applied at a \(37^{\circ}\) angle to the surface of the end of a square bar. The surface is \(4.00 \mathrm{~cm}\) on a side. What are (a) the compressional stress and (b) the shear stress on the bar?

A metal wire \(1.0 \mathrm{~mm}\) in diameter and \(2.0 \mathrm{~m}\) long hangs vertically with a \(6.0-\mathrm{kg}\) object suspended from it. If the wire stretches \(1.4 \mathrm{~mm}\) under the tension, what is the value of Young's modulus for the metal?

When railroad tracks are installed, gaps are left between the rails. (a) Should a greater gap be used if the rails are installed on (1) a cold day or (2) a hot day? Or (3) does the temperature not make any difference? Why? (b) Each steel rail is \(8.0 \mathrm{~m}\) long and has a cross-sectional area of \(0.0025 \mathrm{~m}^{2}\). On a hot day, each rail thermally expands as much as \(3.0 \times 10^{-3} \mathrm{~m}\) If there were no gaps between the rails, what would be the force on the ends of each rail?

A cylindrical eraser of negligible mass is dragged across a paper at a constant velocity to the right by its pencil. The coefficient of kinetic friction between eraser and paper is \(0.650 .\) The pencil pushes down with \(4.20 \mathrm{~N}\). The height of the eraser is \(1.10 \mathrm{~cm}\) and its diameter is \(0.760 \mathrm{~cm} .\) Its top surface is displaced horizontally \(0.910 \mathrm{~mm}\) relative to the bottom. Determine the shear modulus of the eraser material.

If you dive to a depth of \(10 \mathrm{~m}\) below the surface of a lake, (a) what is the pressure due to the water alone? (b) What is the absolute pressure at that depth?

A 75.0 -kg athlete performs a single-hand handstand. If the area of the hand in contact with the floor is \(125 \mathrm{~cm}^{2}\) what pressure is exerted on the floor?

The gauge pressure in both tires of a bicycle is \(690 \mathrm{kPa}\). If the bicycle and the rider have a combined mass of \(90.0 \mathrm{~kg}\), what is the area of contact of each tire with the ground? (Assume that each tire supports half the total weight of the bicycle.)

In a sample of seawater taken from an oil spill, an oil layer \(4.0 \mathrm{~cm}\) thick floats on \(55 \mathrm{~cm}\) of water. If the density of the oil is \(0.75 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3},\) what is the absolute pressure on the bottom of the container?

What is the fractional decrease in pressure when a barometer is raised \(40.0 \mathrm{~m}\) to the top of a building? (Assume that the density of air is constant over that distance.)

To drink a soda (assume same density as water) through a straw requires that you lower the pressure at the top of the straw. What does the pressure need to be at the top of a straw that is \(15.0 \mathrm{~cm}\) above the surface of the soda in order for the soda to reach your lips?

During a plane flight, a passenger experiences ear pain due to a head cold that has clogged his Eustachian tubes. Assuming the pressure in his tubes remained at 1.00 atm (from sea level) and the cabin pressure is maintained at 0.900 atm, determine the air pressure force (including its direction) on one eardrum, assuming it has a diameter of \(0.800 \mathrm{~cm}\).

Here is a demonstration Pascal used to show the importance of a fluid's pressure on the fluid's depth (vFig. 9.36): An oak barrel with a lid of area \(0.20 \mathrm{~m}^{2}\) is filled with water. A long, thin tube of cross- sectional area \(5.0 \times 10^{-5} \mathrm{~m}^{2}\) is inserted into a hole at the center of the lid, and water is poured into the tube. When the water reaches \(12 \mathrm{~m}\) high, the barrel bursts. (a) What was the weight of the water in the tube? (b) What was the pressure of the water on the lid of the barrel? (c) What was the net force on the lid due to the water pressure?

The door and the seals on an aircraft are subject to a tremendous amount of force during flight. At an altitude of \(10000 \mathrm{~m}\) (about \(33000 \mathrm{ft}\) ), the air pressure outside the airplane is only \(2.7 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}\) while the inside is still at normal atmospheric pressure, due to pressurization of the cabin. Calculate the force due to the air pressure on a door of area \(3.0 \mathrm{~m}^{2}\).

The pressure exerted by a person's lungs can be measured by having the person blow as hard as possible into one side of a manometer. If a person blowing into one side of an open tube manometer produces an \(80-\mathrm{cm}\) difference between the heights of the columns of water in the manometer arms, what is the gauge pressure of the lungs?

A hydraulic lift in a garage has two pistons: a small one of cross-sectional area \(4.00 \mathrm{~cm}^{2}\) and a large one of cross-sectional area \(250 \mathrm{~cm}^{2}\). (a) If this lift is designed to raise a 3500 -kg car, what minimum force must be applied to the small piston? (b) If the force is applied through compressed air, what must be the minimum air pressure applied to the small piston?

The Magdeburg water bridge is a channel bridge over the River Elbe in Germany ( \(\mathbf{F i g .} 9.38\) ). Its dimensions are length \(918 \mathrm{~m}\), width \(43.0 \mathrm{~m},\) and depth \(4.25 \mathrm{~m}\). (a) When filled with water, what is the weight of the water? (b) What is the pressure on the bridge floor?

A hypodermic syringe has a plunger of area \(2.5 \mathrm{~cm}^{2}\) and a \(5.0 \times 10^{-3}-\mathrm{cm}^{2}\) needle. (a) If a \(1.0-\mathrm{N}\) force is applied to the plunger, what is the gauge pressure in the syringe's chamber? (b) If a small obstruction is present at the end of the needle, what force does the fluid exert on it? (c) If the blood pressure in a vein is \(50 \mathrm{~mm} \mathrm{Hg},\) what force must be applied on the plunger so that fluid can be injected into the vein?

A funnel has a cork blocking its drain tube. The cork has a diameter of \(1.50 \mathrm{~cm}\) and is held in place by static friction with the sides of the drain tube. When water is added to a height of \(10.0 \mathrm{~cm}\) above the cork, it comes flying out of the tube. Determine the maximum force of static friction between the cork and drain tube. Neglect the weight of the cork.

(a) If the density of an object is exactly equal to the density of a fluid, the object will (1) float, (2) sink, (3) stay at any height in the fluid, as long as it is totally immersed. (b) A cube \(8.5 \mathrm{~cm}\) on each side has a mass of \(0.65 \mathrm{~kg}\). Will the cube float or sink in water? Prove your answer.

An object has a weight of \(8.0 \mathrm{~N}\) in air. However, it apparently weighs only \(4.0 \mathrm{~N}\) when it is completely submerged in water. What is the density of the object?

A solid ball has a weight of \(3.0 \mathrm{~N}\). When it is submerged in water, it has an apparent weight of \(2.7 \mathrm{~N}\). What is the density of the ball?

A wood cube \(0.30 \mathrm{~m}\) on each side has a density of \(700 \mathrm{~kg} / \mathrm{m}^{3}\) and floats levelly in water. (a) What is the distance from the top of the wood to the water surface? (b) What mass has to be placed on top of the wood so that its top is iust at the water level?

(a) Given a piece of metal with a light string attached, a scale, and a container of water in which the piece of metal can be submersed, how could you find the volume of the piece without using the variation in the water level? (b) An object has a weight of \(0.882 \mathrm{~N}\). It is suspended from a scale, which reads \(0.735 \mathrm{~N}\) when the piece is submerged in water. What are the volume and density of the piece of metal?

An aquarium is filled with a liquid. A cork cube, \(10.0 \mathrm{~cm}\) on a side, is pushed and held at rest completely submerged in the liquid. It takes a force of \(7.84 \mathrm{~N}\) to hold it under the liquid. If the density of cork is \(200 \mathrm{~kg} / \mathrm{m}^{3}\), find the density of the liquid.

An ideal fluid is moving at \(3.0 \mathrm{~m} / \mathrm{s}\) in a section of a pipe of radius \(0.20 \mathrm{~m}\). If the radius in another section is \(0.35 \mathrm{~m},\) what is the flow speed there?

(a) If the radius of a pipe narrows to half of its original size, will the flow speed in the narrow section (1) increase by a factor of \(2,\) (2) increase by a factor of 4 , (3) decrease by a factor of \(2,\) or (4) decrease by a factor of \(4 ?\) Why? (b) If the radius widens to three times its original size, what is the ratio of the flow speed in the wider section to that in the narrow section?

Water flows through a horizontal tube similar to that in Fig. 9.20. However in this case, the constricted part of the tube is half the diameter of the larger part. If the water speed is \(1.5 \mathrm{~m} / \mathrm{s}\) in the larger parts of the tube, by how much does the pressure drop in the constricted part? Express the final answer in atmospheres.

The speed of blood in a major artery of diameter \(1.0 \mathrm{~cm}\) is \(4.5 \mathrm{~cm} / \mathrm{s}\). (a) What is the flow rate in the artery? (b) If the capillary system has a total cross-sectional area of \(2500 \mathrm{~cm}^{2}\), the average speed of blood through the capillaries is what percentage of that through the major artery? (c) Why must blood flow at low speed through the capillaries?

The blood flow speed through an aorta with a radius of \(1.00 \mathrm{~cm}\) is \(0.265 \mathrm{~m} / \mathrm{s}\). If hardening of the arteries causes the aorta to be constricted to a radius of \(0.800 \mathrm{~cm},\) by how much would the blood flow speed increase?

In a dramatic lecture demonstration, a physics professor blows hard across the top of a copper penny that is at rest on a level desk. By doing this at the right speed, he can get the penny to accelerate vertically, into the airstream, and then deflect it into a tray, as shown in Fig. 9.42. Assuming the diameter of a penny is \(1.90 \mathrm{~cm}\) and its mass is \(2.50 \mathrm{~g}\), what is the minimum airspeed needed to lift the penny off the tabletop? Assume the air under the penny remains at rest.

Water flows at a rate of \(25 \mathrm{~L} / \mathrm{min}\) through a horizontal 7.0-cm-diameter pipe under a pressure of \(6.0 \mathrm{~Pa}\). At one point, calcium deposits reduce the cross-sectional area of the pipe to \(30 \mathrm{~cm}^{2}\). What is the pressure at this point? (Consider the water to be an ideal fluid.)

A hospital patient receives a quick 500 -cc blood transfusion through a needle with a length of \(5.0 \mathrm{~cm}\) and an inner diameter of \(1.0 \mathrm{~mm}\). If the blood bag is suspended \(0.85 \mathrm{~m}\) above the needle, how long does the transfusion take? (Neglect the viscosity of the blood flowing in the plastic tube between the bag and the needle.)

A rock is suspended from a string in air. The tension in the string is \(2.94 \mathrm{~N}\). When the rock is then dunked into a liquid and the string is allowed to go slack, it sinks and comes to rest on a spring with a spring constant of \(200 \mathrm{~N} / \mathrm{m} .\) The spring's final compression is \(1.00 \mathrm{~cm} .\) If the density of the rock is \(2500 \mathrm{~kg} / \mathrm{m}^{3},\) what is the density of the liquid?

In preparation for its tire rotation, a car weighing 2.25 tons is placed on a hydraulic garage lift. The mechanic then raises the car \(30.0 \mathrm{~cm} .\) (a) Calculate the work done on the car when it is lifted. (b) Assuming no frictional losses in the hydraulic fluid, how much work was done by the lift on the input side? (c) What was the force on the input side if its piston moved \(52.5 \mathrm{~cm} ?\) (d) Determine the ratio of the input side area to that of the lifting side (output) area?

A spherical object has an outside diameter of \(48.0 \mathrm{~cm} .\) Its outer shell is composed of aluminum and is \(2.00 \mathrm{~cm}\) thick. The remainder is uniform plastic with a density of \(800 \mathrm{~kg} / \mathrm{m}^{3}\). (a) Determine the object's average density. (b) Will this object float by itself in fresh water? Explain your reasoning. (c) If it does float, how much of it is above the water surface? If it doesn't float, determine the force required to keep it from sinking if it is entirely submerged.