Chapter 14

The mean lifetime of stationary muons is measured to be \(2.2000 \mu \mathrm{s}\). The mean lifetime of high-speed muns in a burst of cosmic rays observed from Earth is measured to be \(16.000 \mu \mathrm{s}\). To five significant figures, what is the speed parameter \(\beta\) of these cosmic-ray muons relative to Farth?

\(\cdot 1\) ILW A fish maintains its depth in fresh water by adjusting the air content of porous bone or air sacs to make its average density the same as that of the water. Suppose that with its air sacs collapsed. a fish has a density of \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\). To what fraction of its expanded body volume must the fish inflate the air sacs to reduce its density to that of water?

To eight significant figures, what is speed parameter \(\beta\) if the Lorentz factor \(y\) is (a) 1.0100000 , (b) 10.000000 (c) 100.00000 . and (d) \(1000.0000 ?\)

\(\cdot 2\) A partially cvacuated airtight container has a tight-fitting lid of surface area \(77 \mathrm{~m}^{2}\) and negligible mass. If the force required to remove the lid is \(480 \mathrm{~N}\) and the atmospheric pressure is \(1.0 \times 10^{5} \mathrm{~Pa}\), what is the internal air pressure?

You wish to make a round trip from Earth in a spaceship. traveling at constant speed in a straight line for exactly 6 months (as you measure the time interval) and then returning at the same constant speed. You wish further, on your return, to find Earth as it will be exactly 1000 years in the future. (a) To eight significant figurcs, at what speed parameter \(\beta\) must you travel? (b) Does it matter whether you travel in a straight line on your journey?

\(\cdot 3\) ssm Find the pressure increase in the fluid in a syringe when a nurse applies a force of \(42 \mathrm{~N}\) to the syringe's circular piston, which has a radius of \(1.1 \mathrm{~cm}\).

(Come) back to the future. Suppose that a father is \(20.00 \mathrm{y}\) older than his daughter. He wants to travel outward from Earth for \(2.000 \mathrm{y}\) and then back for another \(2.000 \mathrm{y}\) (both intervals as he measures them ) such that he is then \(20.00 \mathrm{y}\) younger than his daughter. What constant speed parameter \(\beta\) (relative to Earth) is required?

\(\cdot 4\) Three liquids that will not mix are poured into a cylindrical container. The volumes and densities of the liquids are \(0.50 \mathrm{~L}, 2.6 \mathrm{~g} / \mathrm{cm}^{3}\); \(0.25 \mathrm{~L}, 1.0 \mathrm{~g} / \mathrm{cm}^{3} ;\) and \(0.40 \mathrm{~L}, 0.80 \mathrm{~g} / \mathrm{cm}^{3} .\) What is the force on the bot- tom of the container duc to these liquids? One liter \(=1 \mathrm{~L}=1000 \mathrm{~cm}^{3}\). (Ignore the contribution due to the atmosphere.)

An unstable high-energy particle enters a detector and leaves a track of length \(1.05 \mathrm{~mm}\) before it decays. Its speed relative to the detector was \(0.992 c\). What is its proper lifetime? That is, how long would the particle have lasted before decay had it been at rest with respect to the detector?

\(\cdot 5\) ssin An office window has dimensions \(3.4 \mathrm{~m}\) by \(2.1 \mathrm{~m} .\) As a result of the passage of a storm, the outside air pressure drops to 0.96 atm, but inside the pressure is held at 1.0 atm. What net force pushes out on the window?

The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into Earth's future, to a time when human civilization has been replaced by an ape civilization. Considering only special relativity, determine how far into Earth's future the astronauts would travel if they slept for \(120 \mathrm{y}\) whilc traveling relative to Earth with a speed of \(0.9990 \mathrm{c},\) first outward from Earth and then back again.

An clectron of \(\beta=0.999987\) moves along the axis of an cvacuated tube that has a length of \(3.00 \mathrm{~m}\) as measured by a laboratory observer \(S\) at rest relative to the tube. An observer \(S^{\prime \prime}\) who is at rest relative to the electron, however, would see this tube moving with speed \(v(=\beta c) .\) What length would observer \(S^{\prime}\) measure for the tube?

\(\cdot 8 \Rightarrow 6\) The bends during flight. Anyone who scuba dives is adviscd not to fly within the next 24 h because the air mixture for diving can introduce nitrogen to the bloodstream. Without allowing the nitrogen to come out of solution slowly, any sudden air-pressure reduction (such as during airplane ascent) can result in the nitrogen forming bubbles in the blood, creating the bends, which can be painful and even fatal. Military special operation forces are especially at risk. What is the change in pressure on such a special-op soldier who must scuba dive at a depth of \(20 \mathrm{~m}\) in seawater one day and parachute at an altitude of \(7.6 \mathrm{~km}\) the next day? Assume that the average air density within the altitude range is \(0.87 \mathrm{~kg} / \mathrm{m}^{3}\).

A spaceship of rest length \(130 \mathrm {~m}\) races past a timing station at a speed of \(0.740 \mathrm{c}\). (a) What is the length of the spaceship as measured by the timing station? (b) What time interval will the station clock record between the passage of the front and back cnds of the ship?

\(\cdot 9 \Rightarrow\) Blood pressure in Argentinosaurus. (a) If this longnecked, gigantic sauropod had a head height of \(21 \mathrm{~m}\) and a heart height of \(9.0 \mathrm{~m},\) what (hydrostatic) gauge pressure in its blood was required at the heart such that the blood pressure at the brain was 80 torr (just enough to perfuse the brain with blood)? Assume the blood had a density of \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} .\) (b) What was the blood pressure (in torr or \(\mathrm{mm} \mathrm{Hg}\) ) at the feet?

A meter stick in frame \(S^{\prime}\) makes an angle of \(30^{\circ}\) with the \(x^ {\prime}\) axis. If that frame moves parallel to the \(x\) axis of frame \(S\) with speed \(0.90 \mathrm{c}\) relative to frame \(S\), what is the length of the stick as measured from \(S ?\)

A rod lics parallel to the \(x\) axis of reference frame \(S,\) moving along this axis at a speed of \(0.630 \mathrm{c}\). Its rest length is \(1.70 \mathrm{~m}\). What will be its measured length in frame \(S ?\)

\(\cdot 11\) As Giruffe bending to drink. In a giraffe with its head \(2.0 \mathrm{~m}\) above its heart, and its heart \(2.0 \mathrm{~m}\) above its feet, the (hydrostatic) gauge pressure in the blood at its heart is 250 torr. Assume that the giraffe stands upright and the blood density is \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). In torr (or \(\mathrm{mm} \mathrm{Hg}\) ), find the (gauge) blood pressure (a) at the brain (the pressure is enough to perfuse the brain with blood, to keep the giraffe from fainting) and (b) at the feet (the pressure must be countered by tight-fitting skin acting like a pressure stocking). (c) If the giraffe were to lower its head to drink from a pond without splaying its legs and moving slowly, what would be the increase in the blood pressure in the brain? (Such action would probably be lethal.)

The length of a spaceship is measured to be exactly half its rest length. (a) To three significant figures, what is the speed parameter \(\beta\) of the spaceship relative to the observer's frame? (b) By what factor do the spaceship's clocks run slow relative to clocks in the observer's frame?

\(\cdot 12 \Rightarrow 3=\) The maximum depth \(d_{\max }\) that a diver can snorkel is sct by the density of the water and the fact that human lungs can function against a maximum pressure difference (between inside and outside the chest cavity) of 0.050 atm. What is the difference in \(d_{\max }\) for fresh water and the water of the Dead Sea (the saltiest natural water in the world, with a density of \(1.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) )?

A space traveler takes off from Earth and moves at speed \(0.9900 c\) toward the star Vega, which is 26.00 ly distant. How much time will have elapsed by Earth clocks (a) when the traveler reaches Vega and (b) when Earth observers receive word from the traveler that she has arrived? (c) How much older will Earth observers calculate the traveler to be (measured from her frame) when she reaches Vega than she was when she started the trip?

At a depth of \(10.9 \mathrm{~km},\) the Challenger Deep in the Marianas Trench of the Pacific Ocean is the decpest site in any occan. Yet, in \(1960,\) Donald Walsh and Jacques Piccard reached the Challenger Deep in the bathyscaph Trieste. Assuming that scawater has a uniform density of \(1024 \mathrm{~kg} / \mathrm{m}^{3}\), approximate the hydrostatic pressure (in atmospheres) that the Trieste had to withstand. (Even a slight defect in the Trieste structure would have been disastrous.)

Calculate the hydrostatic difference in blood pressure between the brain and the foot in a person of height \(1.83 \mathrm{~m}\). The density of blood is \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\).

The center of our Milky Way galaxy is about 23000 ly away. (a) To cight significant figures, at what constant speed parameter would you need to travel exactly 23000 ly (measured in the Galaxy frame) in cxactly \(30 \mathrm{y}\) (measured in your frame)? (b) Measured in your frame and in lightyears, what length of the Galaxy would pass by you during the trip?

What gaugc pressure must a machine produce in order to suck mud of density \(1800 \mathrm{~kg} / \mathrm{m}^{3}\) up a tube by a height of \(1.5 \mathrm{~m} ?\)

Observer \(S\) reports that an event occurred on the \(x\) axis of his reference frame at \(x=3.00 \times 10^{8} \mathrm {~m}\) at time \(t=2.50 \mathrm{~s}\). Observer \(S^{\prime}\) and her frame are moving in the positive dircction of the \(x\) axis at a speed of \(0.400 \mathrm{c}\). Further, \(x=x^{\prime}=0\) at \(t=t^{\prime} =0\), What are the (a) spatial and (b) temporal coordinate of the cvent according to \(S^{\prime \prime}\) If \(S^{\prime}\) were, instead, moving in the negative direction of the \(x\) axis, what would be the (c) spatial and (d) temporal coordinate of the cvent according to \(S^ {\prime \prime}\) ?

Crew members attempt to escape from a dam-aged submarine \(100 \mathrm{~m}\) below the surface. What force must be applied to a pop-out hatch, which is \(1.2 \mathrm{~m}\) by \(0.60 \mathrm{~m},\) to push it out at that depth? Assume that the density of the ocean water is \(1024 \mathrm{~kg} / \mathrm{m}^{3}\) and the internal air pressure is at \(1.00 \mathrm{~atm}\)

Crew members attempt to escape from a dam-aged submarine \(100 \mathrm{~m}\) below the surface. What force must be applied to a pop-out hatch, which is \(1.2 \mathrm{~m}\) by \(0.60 \mathrm{~m},\) to push it out at that depth? Assume that the density of the ocean water is \(1024 \mathrm{~kg} / \mathrm{m}^{3}\) and the internal air pressure is at \(1.00 \mathrm{~atm}\)

Inertial frame \(S^{\prime}\) moves at a speed of \(0.60 \mathrm{c}\) with respect to frame \(S\) (Fig. 37-9). Further, \(x=x^ {\prime}=0\) at \(t=t^{\prime}=0 .\) Two cvents are recorded. In frame \(S\), event 1 occurs at the origin at \(t=0\) and event 2 occurs on the \(x\) axis at \(x=3,0 \mathrm{~km}\) at \(t=4.0 \mathrm{prs}\). According to observer \(S^{\prime},\) what is the time of (a) event 1 and (b) event 27 (c) Do the two observers see the same sequence or the reverse?

A large aquarium of height \(5.00 \mathrm{~m}\) is filled with fresh water to a depth of \(2.00 \mathrm{~m}\). One wall of the aquarium consists of thick plastic \(8.00 \mathrm{~m}\) wide. By how much does the total force on that wall increase if the aquarium is next filled to a depth of \(4.00 \mathrm{~m} ?\)

Two identical cylindrical ves-scls with their bascs at the same level cach contain a liquid of density \(1.30 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). The arca of cach base is \(4.00 \mathrm{~cm}^{2}\), but in one vessel the liquid height is \(0.854 \mathrm{~m}\) and in the other it is \(1.560 \mathrm{~m}\). Find the work done by the gravitational force in equaliz.ing the levels when the two vessels are connected.

When a pilot takes a tight turn at high speed in a modern fighter airplane, the blood pressure at the brain level decreases, blood no longer perfuses the brain, and the blood in the brain drains. If the heart maintains the (hydrostatic) gauge pressure in the aorta at 120 torr (or \(\mathrm{mm} \mathrm{Hg}\) ) when the pilot undergoes a horizontal centripetal acceleration of \(4 g,\) what is the blood pressure (in torr) at the brain, \(30 \mathrm{~cm}\) radially inward from the heart? The perfusion in the brain is small cnough that the vision switches to black and white and narrows to "tunnel vision" and the pilot can undergo \(g\) -LOC ("g-induced loss of consciousness"). Blood density is \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\)

A clock moves along an \(x\) axis at a speed of \(0.600 \mathrm{c}\) and reads zero as it passes the origin of the axis. (a) Calculate the clock's Lorentz factor. (b) What time does the clock read as it passcs \(x=180 \mathrm{~m} ?\)

In analyzing certain geo-logical features, it is often appropriate to assume that the pressure at some horizontal level of compensation, deep inside Earth, is the same over a large region and is equal to the pressure due to the gravitational force on the overlying material. Thus, the pressure on the level of compensation is given by the fluid pressure formula. This model requires, for one thing, that mountains have roots of continental rock extending into the denser mantle (Fig. 14-34). Consider a mountain of height \(H=6.0 \mathrm{~km}\) on a continent of thickness \(T=32 \mathrm{~km}\). The continental rock has a density of \(2.9 \mathrm{~g} / \mathrm{cm}^{3}\), and bencath this rock the mantle has a density of \(3.3 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate the depth \(D\) of the root. (Hint: Set the pressure at points \(a\) and \(b\) equal; the depth \(y\) of the level of compensation will cancel out.)

Bullwinkle in reference frame \(S^ {\prime}\) passes you in reference frame \(S\) along the common direction of the \(x^ {\prime}\) and \(x\) axes, as in Fig. \(37-9 .\) He carrics three meter sticks: meter stick 1 is parallel to the \(x^{\prime}\) axis, meter stick 2 is parallel to the \(y^{\prime \prime}\) axis, and meter stick 3 is parallel to the \(z^ {\prime}\) axis. On his wristwatch he counts off \(15.0 \mathrm{~s},\) which takes \(30.0 \mathrm{~s}\) according to you. Two events occur during his passage. According to you, event 1 occurs at \(x_{1}=33.0 \mathrm {~m}\) and \(t_{1}=22.0 \mathrm{~ns},\) and event 2 occurs at \(x_{2}=53.0 \mathrm {~m}\) and \(t_{2}=62.0 \mathrm{~ns}\) According to your measurements, what is the length of (a) meter stick \(1,\) (b) meter stick \(2,\) and (c) meter stick 3 ? According to Bullwinkle, what are (d) the spatial separation and (e) the temporal separation between events 1 and \(2,\) and \((f)\) which event occurs first?

In one observation, the column in a mercury barometer (as is shown in Fig. \(14-5 a\) ) has a mcasured height \(h\) of \(740.35 \mathrm{~mm}\). The temperature is \(-5.0^{\circ} \mathrm{C},\) at which temperature the density of mercury \(\rho\) is \(1.3608 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3} .\) The free-fall acceleration \(\mathrm{g}\) at the site of the baromcter is \(9.7835 \mathrm{~m} / \mathrm{s}^{2}\). What is the atmospheric pressure at that site in pascals and in torr (which is the common unit for barometer readings)?

To suck lemonade of density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) up a straw to a maximum height of \(4.0 \mathrm{~cm}\), what minimum gauge pressure (in atmospheres) must you produce in your lungs?

A particle moves along the \(x\) " axis of frame \(S^{\prime}\) with velocity \(0.40 \mathrm{c}\). Frame \(S^{\prime}\) moves with velocity \(0.60 \mathrm{c}\) with respect to frame \(S .\) What is the velocity of the particle with respect to frame \(S ?\)

A piston of cross-scctional area \(a\) is used in a hydraulic press to exert a small force of magnitude \(f\) on the enclosed liquid. A connecting pipe leads to a larger piston of cross-sectional area \(A\) (lig. 14.36 ). (a) What force magnitude \(F\) will the larger piston sustain without moving? (b) If the piston diameters are \(3.80 \mathrm{~cm}\) and \(53.0 \mathrm{~cm},\) what force magnitude on the small piston will balance a \(20.0 \mathrm{kN}\) force on the large piston?

A \(5.00 \mathrm{~kg}\) object is released from rest while fully submerged in a liquid. The liquid displaced by the submerged object has a mass of 3.00 kg. How far and in what direction does the object move in \(0.200 \mathrm{~s},\) assuming that it moves freely and that the drag force on it from the liquid is negligible?

A block of wood floats in fresh water with two-thirds of its volume \(V\) submerged and in oil with \(0.90 \mathrm{~V}\) submerged. Find the density of (a) the wood and (b) the oil.

An armada of spaceships that is 1.00 ly long (as measured in its rest frame) moves with speed \(0.800 \mathrm{c}\) relative to a ground station in frame \(S\). A messenger travels from the rear of the armada to the front with a speed of \(0.950 \mathrm{c}\) relative to \(S .\) How long docs the trip take as measured (a) in the rest frame of the messenger, (b) in the rest frame of the armada, and (c) by an observer in the ground frame \(S ?\)

An iron anchor of density \(7870 \mathrm{~kg} / \mathrm{m}^{3}\) appears \(200 \mathrm{~N}\) lighter in water than in air. (a) What is the volume of the anchor? (b) How much does it weigh in air?

A sodium light source moves in a horizontal circle at a constant speed of \(0.100 \mathrm{c}\) while emitting light at the proper wavelength of \(\lambda_{0} =589.00 \mathrm{nm}\). Wavelcngth \( \lambda\) is measured for that light by a detector fixed at the center of the circle. What is the wavelength shift \(\lambda- \lambda_{0} ?\)

A boat floating in fresh water displaces water weighing \(35.6 \mathrm{kN}\). (a) What is the weight of the water this boat displaces when floating in salt water of density \(1.10 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) ? (b) What is the difference between the volume of fresh water displaced and the volume of salt water displaced?

A spaceship, moving away from Earth at a speed of \(0.900 c,\) reports back by transmitting at a frequency (measured in the spaceship frame) of \(100 \mathrm{MHz}\). To what frequency must Earth receivers be tuned to receive the report?

Three children, each of weight \(356 \mathrm{~N},\) make a log raft by lashing together logs of diameter \(0.30 \mathrm{~m}\) and length \(1.80 \mathrm{~m}\). How many logs will be needed to keep them afloat in fresh water? Take the density of the logs to be \(800 \mathrm{~kg} / \mathrm{m}^{3}\).

Certain wavelengths in the light from a galaxy in the constellation Virgo are observed to be \(0.4 \%\) longer than the corresponding light from Earth sources. (a) What is the radial speed of this galaxy with respect to Earth? (b) Is the galaxy approaching or receding from Earth?

iLW A hollow spherical iron shell floats almost completely submerged in water. The outer diameter is \(60.0 \mathrm{~cm}\), and the density of iron is \(7.87 \mathrm{~g} / \mathrm{cm}^{3}\). Find the inner diameter.

A hollow sphere of inner radius \(8.0 \mathrm{~cm}\) and outer radius \(9.0 \mathrm{~cm}\) floats half-submerged in a liquid of density \(800 \mathrm{~kg} / \mathrm{m}^{3} .\) (a) What is the mass of the sphere? (b) Calculate the density of the material of which the sphere is made.