Chapter 14

The mass of an electron is \(9.10938188 \times\) \(10^{-31} \mathrm{~kg} .\) To six significant figures, find (a) \(\gamma\) and (b) \(\beta\) for an electron with kinetic energy \(K=100.000 \mathrm{MeV}\)

What fraction of the volume of an iccberg (density \(\left.917 \mathrm{~kg} / \mathrm{m}^{3}\right)\) would be visible if the iceberg floats (a) in the ocean (salt water, density \(1024 \mathrm{~kg} / \mathrm{m}^{3}\) ) and (b) in a river (fresh water, density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) )? (When salt water freezes to form ice, the salt is excluded. So, an iceberg could provide fresh water to a community.)

What is the minimum energy that is required to break a nucleus of \({ }^{12} \mathrm{C}\) (of mass \(11.99671 \mathrm {u}\) ) into three nuclei of "He (of mass 4.00151 u each \() ?\)

A flotation device is in the shape of a right cylinder, with a height of \(0.500 \mathrm{~m}\) and a face area of \(4.00 \mathrm{~m}^{2}\) on top and bottom, and its density is 0.400 times that of fresh water. It is initially held fully submerged in fresh water, with its top face at the water surface. Then it is allowed to ascend gradually until it begins to float. How much work does the buoyant force do on the device during the ascent?

How much work must be done to increasc the speed of an electron (a) from \(0.18 c\) to \(0.19 \mathrm{c}\) and (b) from \(0.98 \mathrm{c}\) to \(0.99 \mathrm{c}\) ? Note that the speed increase is \(0.01 \mathrm{c}\) in both cascs.

When researchers find a reasonably complete fossil of a dinosaur, they can detcrmine the mass and weight of the living dinosaur with a scale model sculpted from plastic and based on the dimensions of the fossil bones. The scale of the model is \(1 / 20 ;\) that is, lengths are \(1 / 20\) actual length, areas are \((1 / 20)^{2}\) actual areas, and volumes are \((1 / 20)^{3}\) actual volumes. First, the model is suspended from one arm of a balance and weights are added to the other arm until equilibrium is reached. Then the model is fully submerged in water and enough weights are removed from the sccond arm to reestablish cquilibrium (Fig. \(14-42\) ). For a model of a particular \(T\), rex fossil, 637,76 g had to be removed to recstablish cquilibrium. What was the volume of (a) the model and (b) the actual \(T\). rex? (c) If the density of \(T\), rex was approximately the density of water, what was its mass?

In the reaction \(\mathrm{p}+{ }^{19} \mathrm{~F} \rightarrow a+{ }^{16} \mathrm {O},\) the masses are $$ \begin{array}{ll} m(p)=1.007825 u, & m(a)=4.002603 u \\ m(F)=18.998405 u, & m(O)=15.994915 u \end{array} $$ Calculate the \(Q\) of the reaction from these data.

A wood block (mass \(3.67 \mathrm{~kg}\), density \(\left.600 \mathrm{~kg} / \mathrm{m}^{\text {' }}\right)\) is fitted with lead (density \(1.14 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3}\) ) so that it floats in water with 0.900 of its volume submerged. Find the lead mass if the lead is fitted to the block's (a) top and (b) bottom.

In a high-energy collision between a cosmic-ray particle and a particle ncar the top of Earth's atmosphere, \(120 \mathrm {~km}\) above sca level, a pion is created. The pion has a total energy \(E\) of \(1.35 \times 10^{5} \mathrm{MeV}\) and is traveling vertically downward. In the pion's rest frame, the pion decays \(35.0 \mathrm{~ns}\) after its creation. At what altitude above sea level, as measured from Earth's reference frame. does the decay occur? The rest cnergy of a pion is \(139.6 \mathrm{MeV}\).

An iron casting containing a number of cavities weighs\(6000 \mathrm{~N}\) in air and \(4000 \mathrm{~N}\) in water. What is the total cavity volume in the casting? The density of solid iron is \(7.87 \mathrm{~g} / \mathrm{cm}^{3}\).

(a) If \(m\) is a particle's mass, \(p\) is its momentum magnitude, and \(K\) is its kinctic energy, show that $$ m=\frac{(p c)^{2}-K^{2}}{2 K c^{2}} $$ (b) For low particle speeds, show that the right side of the equation reduces to \(m\). (c) If a particle has \(K=55.0 \mathrm{MeV}\) when \(p=\) \(121 \mathrm{MeV} i c,\) what is the ratio \(\mathrm{m} / \mathrm{m}_{e}\) of its mass to the clectron mass?

Suppose that you release a small ball from rest at a depth of \(0.600 \mathrm{~m}\) below the surface in a pool of water. If the density of the ball is 0.300 that of water and if the drag force on the ball from the water is negligible, how high above the water surface will the ball shoot as it emerges from the water? (Neglect any transfer of energy to the splashing and waves produced by the emerging ball.)

A 5.00 -grain aspirin tablet has a mass of \(320 \mathrm{mg}\). Forhow many kilometers would the encrgy cquivalent of this mass power an automobile? Assume \(12.75 \mathrm{~km} / \mathrm{L}\) and a heat of combustion of \(3.65 \times 10^{7} \mathrm{~J} / \mathrm{L}\). for the gasoline used in the automobile.

The mass of a muon is 207 times the clectron mass; the average lifetime of muons at rest is \(2.20 \mu \mathrm{s}\). In a certain experiment, muons moving through a laboratory are measured to have an average lifetime of \(6.90 \mu \mathrm{s}\). For the moving muons, what are (a) \(\beta, (\) b) \(K,\) and \((c) p(\) in \(\mathrm{MeV} / c) ?\)

Canal effect. Figure \(14-45\) shows an anchored barge that cxtcnds across a canal by distance \(d=30 \mathrm{~m}\) and into the water by distance \(b=12 \mathrm{~m} .\) The canal has a width \(D=55 \mathrm{~m},\) a water depth \(H=14 \mathrm{~m},\) and a uniform water-flow speed \(v_{i}=1.5 \mathrm{~m} / \mathrm{s} .\) Assume that the flow around the barge is uniform. As the water passes the bow, the water level undergoes a dramatic dip known as the canal effect. If the dip has depth \(h=0.80 \mathrm{~m},\) what is the water speed alongside the boat through the vertical cross sections at (a) point \(a\) and (b) point \(b ?\) The erosion due to the speed increase is a common concern to hydraulic cngincers.

To four significant figures, find the following when the kinetic energy is \(10.00 \mathrm{MeV}:\) (a) \(\gamma\) and \((\mathrm{b}) \beta\) for an clectron \(\left (E_{\mathrm{i}}=\right.\) \(0.510998 \mathrm{MeV}\) ), (c) \(\gamma\) and (d) \( \beta\) for a proton \(\left(E_{0}=938.272 \mathrm{MeV}\right)\) and (c) \(\gamma\) and (f) \(\beta\) for an \(\alpha\) particle \(\left (E_{0}=3727.40 \mathrm{McV}\right)\)

What must be the momentum of a particle with mass \(m\) so that the total encrgy of the particle is 3.00 times its rest cnergy?

A garden hose with an internal diameter of \(1.9 \mathrm{~cm}\) is connected to a (stationary) lawn sprinkler that consists merely of a container with 24 holes, each \(0.13 \mathrm{~cm}\) in diameter. If the water in the hose has a speed of \(0.91 \mathrm{~m} / \mathrm{s},\) at what speed does it leave the sprinkler holes?

In Module \(28-4,\) we showed that a particle of charge \(q\) and mass \(m\) will move in a circle of radius \(r=m v /|q| B\) when its velocity \(\vec{v}\) is perpendicular to a uniform magnetic field \(\vec{B}\). We also found that the period \(T\) of the motion is independent of speed \(v\) These two results are approximately correct if \(v

Water is pumped steadily out of a flooded basement at \(5.0 \mathrm{~m} / \mathrm{s}\) through a hose of radius \(1.0 \mathrm{~cm},\) passing through a window \(3.0 \mathrm{~m}\) above the waterline. What is the pump's power?

The water flowing through a \(1.9 \mathrm{~cm}\) (inside diameter) pipe flows out through three \(1.3 \mathrm{~cm}\) pipes. (a) If the flow rates in the three smaller pipes are \(26,19,\) and 11 L. \(/\) min, what is the flow rate in the \(1.9 \mathrm{~cm}\) pipe? (b) What is the ratio of the speed in the \(1.9 \mathrm{~cm}\) pipe to that in the pipe carrying \(26 \mathrm{~L} / \mathrm{min} ?\)

A certain particle of mass \(m\) has momentum of magnitude \(m c .\) What are (a) \(\beta,(b) \gamma,\) and \((c)\) the ratio \(K / E_{0} ?\)

How much work is done by pressure in forcing \(1.4 \mathrm{~m}^{3}\) of water through a pipe having an internal diameter of \(13 \mathrm{~mm}\) if the difference in pressure at the two ends of the pipe is \(1.0 \mathrm{~atm} ?\)

The energy released in the explosion of \(1.00 \mathrm{~mol}\) of \(\mathrm{TNT}\) is \(3.40 \mathrm{MJ}\). The molar mass of TNT is \(0.227 \mathrm{~kg} / \mathrm {mol}\). What weight of TNT is needed for an explosive release of \(1.80 \times 10^{14} \mathrm{~J} ?\) (b) Can you carry that weight in a backpack, or is a truck or train required? (c) Suppose that in an cxplosion of a fission bomb, \(0.080 \%\) of the fissionable mass is converted to released energy. What weight of fissionable material is needed for an explosive release of \(1.80 \times 10^{14} \mathrm{~J} ?\) (d) Can you carry that weight in a backpack, or is a truck or train required?

Suppose that two tanks, 1 and \(2,\) each with a large opening at the top, contain different liquids. A small hole is made in the side of each tank at the same depth \(h\) below the liquid surface, but the hole in tank 1 has half the cross-sectional area of the hole in tank 2 . (a) What is the ratio \(\rho_{i} / \rho_{2}\) of the densities of the liquids if the mass flow rate is the same for the two holes? (b) What is the ratio \(R_{n} / R_{V 2}\) of the volume flow rates from the two tanks? (c) At one instant, the liquid in tank 1 is \(12.0 \mathrm{~cm}\) above the hole. If the tanks are to have equal volume flow rates, what height above the hole must the liquid in tank 2 be just then?

The mass of an electron is \(9.10938188 \times 10^{31} \mathrm{~kg} .\) To eight significant figures, find the following for the given electron kinetic energy: (a) \(\gamma\) and (b) \(\beta\) for \(K=1.0000000 \mathrm {keV},\) (c) \(\gamma\) and \((\mathrm{d}) \beta\) for \(K=\) \(1.0000000 \mathrm{MeV},\) and then \((\mathrm{e}) \gamma\) and \((\mathrm{f}) \beta\) for \(K=1.0000000 \mathrm{GeV}\)

An alpha particle with kinetic energy \(7.70 \mathrm{MeV}\) collides with an \({ }^{14} \mathrm{~N}\) nucleus at rest, and the two transform into an \({ }^{17} \mathrm{O}\) nucleus and a proton. The proton is emitted at \(90^{n}\) to the direction of the incident alpha particle and has a kinetic energy of \(4.44 \mathrm{MeV}\). The masses of the various particles are alpha particle, \(4.00260 \mathrm{u} ;{ }^{14} \mathrm{~N}\), \(14.00307 \mathrm{u} ;\) proton, \(1.007825 \mathrm{u} ;\) and \({ }^{17} \mathrm{O}, 16.99914 \mathrm{u} .\) In \(\mathrm{MeV},\) what are (a) the kinetic energy of the oxygen nucleus and (b) the \(Q\) of the reaction? (Hint: The speeds of the particles are much less than \(c\) )

Water is moving with a speed of \(5.0 \mathrm{~m} / \mathrm{s}\) through a pipe with a cross-sectional area of \(4.0 \mathrm{~cm}^{2}\). The watcr gradually descends \(10 \mathrm{~m}\) as the pipe cross-sectional area increases to \(8.0 \mathrm{~cm}^{2}\). (a) What is the spced at the lower level? (b) If the pressure at the upper level is \(1.5 \times 10^{5} \mathrm{~Pa}\), what is the pressure at the lower level?

Temporal separation between two events. Events \(A\) and \(B\) occur with the following spacctime coordinates in the reference frames of Fig. 37 - 25 : according to the unprimed frame, \(\left(x_{A}, t_{A}\right)\) and \(\left(x_{B}, t_{B}\right) ;\) according to the primed frame, \(\left(x_{A}^{\prime}, t_{A} ^{\prime}\right)\) and \(\left(x_{B}^{\prime}, r_ {B}^{\prime}\right) .\) In the unprimed frame, \(\Delta t=t_{n}-t_{A}=1.00 \mu \mathrm{s}\) and \(\Delta x=x_{B}-x_{A}=240 \mathrm {~m}\) (a) Find an expression for \(\Delta t^{\prime}\) in terms of the speed parameter \(\beta\) and the given data. Graph \(\Delta r^{\prime}\) versus \(\beta\) for the following two ranges of \(\beta\) (b) 0 to 0.01 and (c) 0.1 to 1 . (d) At what value of \(\beta\) is \(\Delta t^{\prime}\) minimum and (c) what is that minimum? (f) Can one of these events cause the other? Explain.

ILW A water pipe having a \(2.5 \mathrm{~cm}\) inside diameter carries water into the bascment of a house at a speed of \(0.90 \mathrm{~m} / \mathrm{s}\) and a pressure of \(170 \mathrm{kPa}\). If the pipe tapers to \(1.2 \mathrm{~cm}\) and rises to the second floor \(7.6 \mathrm{~m}\) above the input point, what are the (a) speed and (b) water pressure at the second floor?

A pitot tube (Fig. \(14-48\) ) is used to determine the airspeed of an airplane. It consists of an outer tube with a number of small holes \(B\) (four are shown) that allow air into the tube; that tube is connected to one arm of a U-tube. The other arm of the U-tube is connected to hole \(A\) at the front end of the device, which points in the direction the plane is headed. At \(A\) the air becomes stagnant so that \(v_{A}=0 .\) At \(B,\) however, the speed of the air presumably equals the airspecd \(v\) of the plane. (a) Use Bernoulli's equation to show that $$ v=\sqrt{\frac{2 \rho g h}{\rho_{\text {ar }}}} $$ where \(\rho\) is the density of the liquid in the \(U\) -tube and \(h\) is the difference in the liquid levels in that tube. (b) Suppose that the tube contains alcohol and the level difference \(h\) is \(26.0 \mathrm{~cm}\). What is the plane's speed relative to the air? The density of the air is \(1.03 \mathrm{~kg} / \mathrm{m}^{3}\) and that of alcohol is \(810 \mathrm{~kg} / \mathrm{m}^{3}\)

A venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipc (Fig. \(14-50) ;\) the cross-scetional arca \(A\) of the entrance and exit of the meter matches the pipe's cross-sectional arca. Between the cntrance and exit, the fluid flows from the pipe with speed \(V\) and then through a narrow "throat" of cross- scetional area \(a\) with speed \(v\). A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's specd is accompanied by a change \(\Delta p\) in the fluid's pressure, which causes a height difference \(h\) of the liquid in the two arms of the manomcter. (Here \(\Delta p\) means pressure in the throat minus pressurc in the pipe.) (a) By applying Bernoulli's equation and the equation of continuity to points 1 and 2 in Fig. \(14-50,\) show that $$ V=\sqrt{\frac{2 a^{2} \Delta p}{\rho\left(a^{2}-A^{2}\right)}} $$ where \(\rho\) is the density of the fluid. (b) Suppose that the fluid is fresh water, that the cross-scctional arcas are \(64 \mathrm{~cm}^{2}\) in the pipe and \(32 \mathrm{~cm}^{2}\) in the throat, and that the pressure is \(55 \mathrm{kPa}\) in the pipe and \(41 \mathrm{kPa}\) in the throat. What is the rate of water flow in cubic meters per second?

The car-in-the-garage problem. Carman has just purchased the world's longest stretch limo, which has a proper length of \(L_{c}=30.5 \mathrm{~m} .\) In Fig. \(37-32 a,\) it is shown parked in front of a garage with a proper length of \(L_{g}=6.00 \mathrm{~m}\). The garage has a front door (shown open) and a back door (shown closed). The limo is obviously longer than the garage. Still, Garageman, who owns the garage and knows something about relativistic length contraction, makes a bet with Carman that the limo can fit in the garage with both doors closed. Carman, who dropped his physics course before reaching special relativity, says such a thing, even in principle, is impossible. To analyze Garageman's scheme, an \(x_{c}\) axis is attached to the limo, with \(x_{c}=0\) at the rear bumper, and an \(x_{k}\) axis is attached to the garage, with \(x_{g}=0\) at the (now open) front door. Then Carman is to drive the limo directly toward the front door at a velocity of \(0.9980 c\) (which is, of course, both technically and financially impossible). Carman is stationary in the \(x_{c}\) reference frame; Garageman is stationary in the \(x_{z}\) reference frame. There are two events to consider. Event 1 : When the rear bumper clears the front door, the front door is closed. Let the time of this event be zero to both Carman and Garageman: \(t_{\mathrm{s} 1}=t_{c 1}=0\). The event occurs at \(x_{c}=x_{g}=0 .\) Figure \(37-32 b\) shows event 1 according to the \(x_{g}\) reference frame. Event 2 : When the front bumper reaches the back door, that door opens. Figure \(37-32 c\) shows event 2 according to the \(x_{g}\) reference frame. According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) \(x_{R 2}\) and (c) \(t_{g 2}\) of event \(2 ?\) (d) For how long is the limo temporarily "trapped" inside the garage with both doors shut? Now consider the situation from the \(x_{c}\) reference frame, in which the garage comes racing past the limo at a velocity of \(-0.9980 \mathrm{c}\). According to Carman, (e) what is the length of the passing garage, what are the spacetime coordinates (f) \(x_{c 2}\) and \((g) t_{c 2}\) of event \(2,(h)\) is the limo ever in the garage with both doors shut, and (i) which event occurs first? (j) Sketch events 1 and 2 as seen by Carman. (k) Are the events causally related; that is, does one of them cause the other? (I) Finally, who wins the bet?

A liquid of density \(900 \mathrm{~kg} / \mathrm{m}^{3}\) flows through a horizontal pipe that has a cross-sectional area of \(1.90 \times 10^{-2} \mathrm{~m}^{2}\) in region \(A\) and a cross-sectional area of \(9.50 \times 10^{-2} \mathrm{~m}^{2}\) in region \(B\). The pressure difference between the two regions is \(7.20 \times 10^{3} \mathrm{~Pa}\). What are (a) the volume flow rate and (b) the mass flow rate?

An airplane has rest length \(40.0 \mathrm{~m}\) and speed \(630 \mathrm{~m} / \mathrm{s} .\) To a ground observer, (a) by what fraction is its length contracted and (b) how long is needed for its clocks to be \(1.00 \mu \mathrm{s}\) slow?

To circle Earth in low orbit, a satellite must have a speed of about \(2.7 \times 10^{4} \mathrm{~km} / \mathrm{h}\). Suppose that two such satellites orbit Earth in opposite directions. (a) What is their relative speed as they pass, according to the classical Galilean velocity transformation equation? (b) What fractional error do you make in (a) by not using the (correct) relativistic transformation equation?

Find the speed parameter of a particle that takes \(2.0 \mathrm{y}\) longer than light to travel a distance of 6.0 ly.

How much work is needed to accelerate a proton from a speed of \(0.9850 c\) to a speed of \(0.9860 c ?\)

About one-third of the body of a person floating in the Dead Sca will be above the waterline. Assuming that the human body density is \(0.98 \mathrm{~g} / \mathrm{cm}^{3}\), find the density of the water in the Dead Sea. (Why is it so much greater than \(1.0 \mathrm{~g} / \mathrm{cm}^{3}\) ?)

A pion is created in the higher reaches of Earth's atmosphere when an incoming high-energy cosmic-ray particle collides with an atomic nucleus. A pion so formed descends toward Earth with a speed of \(0.99 c .\) In a reference frame in which they are at rest, pionsdecay with an average life of 26 ns. As measured in a frame fixed with respect to Earth, how far (on the average) will such a pion move through the atmosphere before it decays?

A simple open U-tube contains mercury. When \(11.2 \mathrm{~cm}\) of water is poured into the right arm of the tube, how high above its initial level does the mercury rise in the left arm?

Suppose that your body has a uniform density of 0.95 times that of water. (a) If you float in a swimming pool, what fraction of your body's volume is above the water surface? Quicksand is a fluid produced when water is forced up into sand, moving the sand grains away from one another so they are no longer locked together by friction. Pools of quicksand can form when water drains underground from hills into valleys where there are sand pockets. (b) If you float in a deep pool of quicksand that has a density 1.6 times that of water, what fraction of your body's volume is above the quicksand surface? (c) Are you unable to breathe?

A spaceship at rest in a certain reference frame \(S\) is given a speed increment of \(0.50 \mathrm{c}\). Relative to its new rest frame, it is then given a further \(0.50 \mathrm{c}\) increment. This process is continued until its speed with respect to its original frame \(S\) exceeds \(0.999 \mathrm{c}\). How many increments does this process require?

A glass ball of radius \(2.00 \mathrm{~cm}\) sits at the bottom of a container of milk that has a density of \(1.03 \mathrm{~g} / \mathrm{cm}^{3}\). The normal force on the ball from the container's lower surface has magnitude \(9.48 \times 10^{-2} \mathrm{~N}\). What is the mass of the ball?

In the red shift of radiation from a distant galaxy, a certain radiation, known to have a wavelength of \(434 \mathrm{nm}\) when observed in the laboratory, has a wavelength of \(462 \mathrm{nm}\). (a) What is the radial speed of the galaxy relative to Earth? (b) Is the galaxy approaching or receding from Earth?

Caught in an avalanche, a skier is fully submerged in flowing snow of density \(96 \mathrm{~kg} / \mathrm{m}^{3}\). Assume that the average density of the skier, clothing, and skiing equipment is \(1020 \mathrm{~kg} / \mathrm{m}^{3}\). What percentage of the gravitational force on the skier is offset by the buoyant force from the snow?

What is the momentum in MeV \(/ c\) of an electron with a kinetic energy of \(2.00 \mathrm{MeV} ?\)

An object hangs from a spring balance. The balance registers \(30 \mathrm{~N}\) in air, \(20 \mathrm{~N}\) when this object is immersed in water, and \(24 \mathrm{~N}\) when the object is immersed in another liquid of unknown density. What is the density of that other liquid?

The radius of Earth is \(6370 \mathrm{~km},\) and its orbital speed about the Sun is \(30 \mathrm{~km} / \mathrm{s}\). Suppose Earth moves past an observer at this speed. To the observer, by how much does Earth's diameter contract along the direction of motion?

In an experiment, a rectangular block with height \(h\) is allowed to float in four separate liquids. In the first liquid, which is water, it floats fully submergcd. In liquids \(A, B,\) and \(C\). it floats with heights \(h / 2,2 h / 3,\) and \(h / 4\) above the liquid surface, respectively. What are the relative densities (the densitics relative to that of water) of (a) \(A,(b) B,\) and \((c) C ?\)