Questions and Answers of Mechanics

Write a Monte Carlo code for a fluid of \(N\) hard spheres in a two-dimensional \(L \times L\) square box with periodic boundary conditions in each direction. Calculate the pair correlation function
Write a code to test a uniform pseudorandom number generator. If you do not have a canned generator available, write a generator based on L'Ecuyer's recommended generator in Appendix I. Apply the
The energy levels of an imperfect Fermi gas in the presence of an external magnetic field \(\boldsymbol{B}\), to first order in \(a\), may be written
For the imperfect Bose gas studied in Section 11.2, calculate the specific heat \(C_{V}\) near absolute zero and show that, as \(T ightarrow 0\), the specific heat vanishes in a manner characteristic
Show that, in the case of a degenerate gas of fermions \(\left(T \ll T_{F}ight.\) ), the correlation function \(g(r)\), for \(r \gg \hbar / p_{F}\), reduces to the expression\[g(r)-1=-\frac{3(m k
Show that for a gas of noninteracting bosons, or fermions, the pair correlation function \(g(r)\) is given by the expression\[g(r)=1 \pm \frac{g_{s}}{n^{2} h^{6}}\left|\int_{-\infty}^{\infty}
Using the data and the number of stars in the Galaxy estimated in the question 7 , determine the average distance \(d\) (in pc) between star and star assuming for the Galaxy a spherical
Using the data and the number of stars in the Galaxy estimated in the question 7, determine the average distance \(d\) (in pc) between star to star assuming for the Galaxy a flat disk of height \(300
Discuss what changes, if any, should be made in the question 3 in the case of elliptical, rather than circular, orbits.Question 3:Show that in the case of a binary system in which the orbit is
The Sun has mass \(M_{\odot}=1.9910^{30} \mathrm{~kg}\), radius \(R_{\odot}=7.010^{8} \mathrm{~m}\) and rotates on itself (approximately) in 30 days. Using the data in Table 11.1, determine:1. the
Draw and determine the C.M. positions of the following molecules. Use a periodic table of the elements for the masses of the elements (it is enough to consider the mass number). Atomic distances are
A neutron star is an object with a mass of about \(1.4 M_{\odot}\), where the solar mass \(M_{\odot}=2 \times 10^{30}\) \(\mathrm{kg}\). The neutron star is aggregated to have density equal to that
There are about \(10^{11}\) stars in the Galaxy with mass comparable to the Sun. Determine total mass \(M_{S}\) in the stars and the fraction of the Galaxy mass in the form of interstellar medium
In the case of the previous question, if the effect of the finite speed of sound were neglected, would \(h\) be larger or smaller? What percentage error would be made?Previous question:One wants to
A pendulum in a local reference frame (see Fig. 5.7) is constrained in a point along the \(z^{\prime}\) axis so that it can swing freely in the \(x^{\prime} y^{\prime}\) plane. The pendulum is
The following scalar field is given: \(V(x, y, z)=2 x^{2}+x y z+y\). Verify that Schwarz's theorem is satisfied.Schwarz's theorem: Mention should now be made of an important theorem of Mathematical
Write a code to test a Gaussian pseudorandom number generator. If you do not have a canned generator available, write a generator based on the Box-Muller algorithm in Appendix I. Apply the following
Define a sequence of correlated random numbers\[s_{k}=\alpha s_{k-1}+(1-\alpha) r_{k}\]where \(r_{k}\) is a unit-variance, uncorrelated, Gaussian pseudorandom number while \(0
Write a Monte Carlo code for a system of \(N\) hard spheres of diameter \(D\) on a one-dimensional ring of length \(L\) with periodic boundary conditions. Calculate the pair correlation function and
Determine the intensity of the gravitational force exerted by a sphere of radius \(R=15 \mathrm{~cm}\) and \(M=158 \mathrm{~kg}\) and one of radius \(r=3 \mathrm{~cm}\) with mass \(m=0.73
Determine the intensities of the gravitational force of attraction between Earth and the Moon and between Earth and the Sun, and calculate their ratio \(R\).
A quantity of mass \(M\) in space splits into two fractions, of mass \(\alpha M\) and \((1-\alpha) M\). Assuming fixed separation \(D\) between the two fragments, for what value of \(\alpha\) is the
Determine to what altitude you need to rise above the Earth's surface for the acceleration of gravity to change by \(1 \%\). Repeat the operation, imagining that you can descend by digging into the
The writer has a mass (measured by the scale) of \(73 \mathrm{~kg}\). Assuming the Earth's radius doubles, determine (i) my weight if \(M_{T}\) remained constant, and (ii) if the Earth's density
Determine the value of the Earth's mass using the value of \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\) and that of \(G\) given in (10.30). Earth's radius is \(R_{T}=6370 \mathrm{~km}\). G = 6.67430(15) x
If two planets have the same mass, but \(A\) has twice the radius of \(B\), determine what the ratio of the accelerations of gravity is worth \(g_{A} / g_{B}\).
If two planets have the same density, but \(A\) has twice the radius of \(B\), determine how much the ratio of the accelerations of gravity is worth \(g_{A} / g_{B}\).
The mass of the proton is \(m_{p}=1.6710^{-27} \mathrm{~kg}\), that of the electron 1840 less than \(m_{p}\). The Coulomb potential energy for the electromagnetic interaction between electron and
Determine the height \(h\) relative to the Earth's surface of a geostationary satellite traveling in a circular orbit in the equatorial plane. Also determine its speed \(v\).
Neglecting friction with the atmosphere, determine how fast a projectile must be launched in order to travel in a circular orbit around the Earth so that it can return to its starting point.
The ISS (International Space Station) is on an orbit \(420 \mathrm{~km}\) above the Earth's surface, and is moving with a speed of about \(7.66 \mathrm{~km} / \mathrm{s}\). What is the value of the
Determine the work required to transport \(10 \mathrm{~kg}\) of material from Earth to the ISS, the International Space Station, in orbit \(420 \mathrm{~km}\) above the Earth's surface.
An artificial satellite of mass \(m=3.4\) ton is in a circular orbit at the altitude \(h_{1}=5000 \mathrm{~km}\). Due to various causes, it gradually loses energy and reaches an altitude \(h_{2}=600
A meteorite is at a distance from the center of the Earth equal to 5 times the Earth's radius \(R_{T}\) with negligible speed and direction of impact toward the ground. Determine the speed with which
An artificial satellite of mass \(m=2.5\) ton is in a circular orbit around the Earth at a distance of \(1600 \mathrm{~km}\) from the surface. Determine the magnitude of the satellite's angular
We normally think that dissipative forces tend to decrease the velocity of an object. This is correct for isolated systems. Consider the case of an artificial satellite of mass \(m\) in a circular
Show that the force field expressed in spherical coordinates \(\mathbf{F}=f(r, \theta) \mathbf{r}\) is conservative if and only if the function \(f\) is independent of \(\theta\).
Calculate the radius \(R_{S}\) of the orbit of a geostationary satellite using the lunar period, \(T=27.3\)d, and the Earth-Moon distance, \(D_{T L}=3.810^{5} \mathrm{~km}\), as data.
A dwarf planet moves around the Sun on an elliptical orbit with semi-major axis \(a\), semi-minor axis \(b\) and period of revolution \(T\). Determine the speed when its direction is perpendicular to
Show that in the case of a binary system in which the orbit is circular of radius \(r\), the kinetic energy corresponds to half the absolute value of the potential energy. In other words, the total
In a gravitationally bound binary system on a circular orbit of radius \(r\), there is energy dissipation by some mechanism (radiation emission, gravitational wave emission). Determine whether, as a
In the case of a gravitationally bound two-body system on an elliptical orbit in which there is energy dissipation as in the case of the question 4, explain why the system first tends to have a
As the Earth revolves around the Sun, the Sun revolves around the center of our Galaxy (where the black hole described in Fig. 11.12 resides). The Sun's distance from the Galactic center is \(D=8.5
Using the expression of the orbit of an ellipse of small eccentricity \(e\) in polar coordinates, determine the ratio between the velocities of a planet at apogee, \(v_{A}\), and perigee, \(v_{P}\),
Using the expression of the orbit of an ellipse of small eccentricity \(e\) in polar coordinates, show that the variation of the planet's angular velocity as a function of eccentricity is equal to
Halley's comet is visible when it is close to perihelion, which is \(0.586 \mathrm{AU}\) from the Sun. The last perihelion passage was in 1986 and the next one will be in 2061. Determine the distance
Neutron stars have masses equal to \(1.5 M_{\odot}\). Knowing that quantum mechanics sets their volume density equal to \(\sim 1.4 \times 10^{14} \mathrm{~g} / \mathrm{cm}^{3}\), estimate the radius.
Ganymede is a satellite of the planet Jupiter, like the Moon for Earth. Using the data below, determine:1. the acceleration of gravity \(g_{G}\) on Jupiter (for comparison, the acceleration of
A body of mass \(m\) gravitationally attracted to a much larger body of mass \(M\) describes an elliptical trajectory if the total mechanical energy \(E_{T}\) is strictly negative, while it has a
An artificial satellite is initially located at a point A at a distance \(r_{0}=4.0 \times 10^{4} \mathrm{~km}\) from the center of the Earth, as in Fig. 11.17. Neglecting any friction, calculate
A neutron star (NS) is composed of neutrons (particles with no electric charge, mass \(m_{N}=1.67 \times 10^{-27} \mathrm{~kg}\) ). The distance between neutrons is \(d=2 \times 10^{-15}
A homogeneous rigid rod \(\mathrm{AB}\), of mass \(\mathrm{m}=4 \mathrm{~kg}\) and length \(L=500 \mathrm{~cm}\) rotates about an axis passing through endpoint \(\mathrm{A}\) and forming an angle
On a thin bar of negligible mass \(1 \mathrm{~m}\) long are arranged 5 bodies each of mass \(0.10 \mathrm{~kg}\) at distances of \(25 \mathrm{~cm}\) from each other starting from one end. Determine
Determine the moment of inertia of a homogeneous circular corona, of surface density \(\sigma=1.25\) \(\mathrm{kg} \mathrm{m}^{-2}\), with inner radius \(r_{1}=0.30 \mathrm{~m}\) and outer radius
A rectangular plate of density \(\sigma=1.2510^{2} \mathrm{~kg} \mathrm{~m}^{-2}\) is arranged with the longer side \(\ell=75\) \(\mathrm{cm}\) along the \(x\) axis of a Cartesian reference frame,
A rectangular slab of the same size and arranged in the same way as that in the question 4 has a surface density that varies with position according to the function \(\sigma(x, y)=a_{0}+a_{1} x y\)
A thin, homogeneous disk of radius \(\mathrm{R}=50 \mathrm{~cm}\) and mass \(\mathrm{m}=200 \mathrm{~g}\) lies in a plane. Calculate the moment of inertia with respect to any of its diameters.
A yo-yo consists of a homogeneous slotted cylinder, radius \(R=7.0 \mathrm{~cm}\) and mass \(\mathrm{m}=100 \mathrm{~g}\). The inner slot is of negligible width, and does not significantly affect the
The homogeneous circular corona of the question 3 rotates about its axis of symmetry by making one revolution every 90 s90 s around the axis of symmetry. Determine magnitude of the angular
Show that the moment of inertia for an axis perpendicular to the conjunction of a system of two bodies \(m_{1}, m_{2}\) spaced \(r\) apart and passing through C.M. is \(J=\mu r^{2}\), where \(\mu\)
Find the moment of inertia of the \(\mathrm{CO}_{2}\) molecule (data given in the question 3) with respect to an axis perpendicular to the molecule and passing through the C.M.Question 3Determine the
Find the moment of inertia of the \(\mathrm{H}_{2} \mathrm{O}\) molecule (data given in the question 3) with respect to the axis perpendicular to the molecule and passing through the C.M.Question
Find the moment of inertia of the \(\mathrm{NH}_{3}\) molecule (data given in the question 3) with respect to the axis perpendicular to the plane containing the three \(\mathrm{H}\) atoms and passing
Using a spherical coordinate system, show that the moment of inertia of a solid sphere of radius \(R\) and mass \(M\) with respect to any diameter passing through the center is \(\frac{2}{5} M
In geometry, an ellipsoid constitutes the three-dimensional analogue of the ellipse in the plane. The equation of an ellipsoid in a Cartesian coordinate system is given bywhere \(a, b\) and \(c\) are
A solid spherical ball of radius \(R=5.0 \mathrm{~cm}\) and mass \(M=100.0 \mathrm{~g}\) descends along an inclined plane of length \(L=1.5 \mathrm{~m}\) and with an angle of inclination with respect
A disk of mass \(M=470 \mathrm{~g}\) and radius \(R=15 \mathrm{~cm}\) rolls without crawling on a horizontal plane. A point mass \(m=240 \mathrm{~g}\) impacts on it and remains attached, Fig. 12.13.
A homogeneous sheet of mass \(M=8.0 \mathrm{~kg}\), side \(D=50 \mathrm{~cm}\) and negligible thickness can rotate without friction about the horizontal axis passing through point \(A\) and
A homogeneous disk of mass \(M=300 \mathrm{~g}\) and radius \(R=23.0 \mathrm{~cm}\) rotates with constant angular speed \(\omega_{0}=25.0 \mathrm{rad} / \mathrm{s}\) around a fixed vertical axis,
As shown in Fig.12.14, an object that can be considered point-like with mass \(m=0.50 \mathrm{~kg}\) is resting on a rough horizontal surface with dynamic friction coefficient \(\mu=0.30\). At the
A cylinder of mass \(M=2.0 \mathrm{~kg}\) and radius \(R=10 \mathrm{~cm}\) is resting on a horizontal plane, and its center \(\mathrm{O}\) is connected to a point \(\mathrm{P}\) on the plane by a
Consider two rods of equal length \(L=17.2 \mathrm{~cm}\) and masses \(M\) and \(3 M\) (with \(M=153 \mathrm{~g}\) ) glued to each other as in Fig. 12.16. The system of the two rods is constrained to
The roller in the Fig. 12.16 consists of two mutually integral (i.e., welded) disks of the same material and density. The roller can rotate without friction around a fixed pivot coincident with
The rigid system in Fig.12.17 consists of a homogeneous bent bar (thick line), of negligible cross section, total mass \(m=5.00 \mathrm{~kg}\) and total length \(L=1.10 \mathrm{~m}\). The two
Consider the system in Fig.12.17. A device with cylindrical symmetry, which rolls without crawling on a horizontal plane, has mass \(M=10.0 \mathrm{~kg}\) and radius \(R=8.0 \mathrm{~cm}\). The
A block of mass \(m=100 \mathrm{~g}\) is constrained to move in a smooth radial groove of a uniform disk of mass \(M=1.00 \mathrm{~kg}\) and radius \(R=10.0 \mathrm{~cm}\). The block is connected to
A homogeneous sphere of mass \(M\) and radius \(R\) initially stationary on top of the inclined plane in A with its center at an altitude \(h_{A}=5.0 \mathrm{~m}\) rolls along the plane AB inclined
A projectile of mass \(m=2.5 \mathrm{~kg}\) is shot tangentially (see Fig.12.19) at the edge of a ring having radius \(R=50 \mathrm{~cm}\) and whose mass is equal (within measurement errors) to that
Consider the homogeneous bar, of negligible cross section, mass \(M=1.333 \mathrm{~kg}\) and length \(L=60.0 \mathrm{~cm}\) in Fig.12.19. It is constrained to rotate in a vertical plane about its
A homogeneous disk, having mass \(M=2.50 \mathrm{~kg}\) and radius \(R=15.0 \mathrm{~cm}\), can rotate, without friction, about a fixed horizontal axis, passing through the point \(\mathrm{O}\) of
The system in Fig.12.21 consists of a homogeneous disk of mass \(M=300 \mathrm{~g}\) and radius \(R=40.0 \mathrm{~cm}\). At the disk, a slit has been produced along the entire length \(R\) of the
Determine the energy flux on Earth from the Sun, which is \(150 \times 10^{6} \mathrm{~km}\) away, using (13.12). Taking into account that a photovoltaic panel can have an efficiency of \(20 \%\),
Using the energy scale used in the microscopic world, chemical processes release about \(10 \mathrm{eV}\) per elementary reaction, while a nuclear reaction releases about \(10 \mathrm{MeV}\).
Show that if you have a mass \(M\) distributed of on a spherical shell, that is, on a sphere of radius \(R\) hollow inside, and a point mass \(m\) at a distance \(h\) from the sphere, the
Show that if you have a mass \(M\) distributed of on a spherical shell, and a point mass \(m\) lies inside from the sphere, the gravitational potential energy is \(V=-G \frac{m M}{R}\), that is,
Show that in the case described by the question 4 the point mass \(m\) inside the sphere feels no gravitational force from the spherical shell.Question 4Show that if you have a mass \(M\) distributed
Determine the gravitational potential \(\mathcal{V}_{G}\) in the case of a homogeneous solid sphere of radius \(R\) as a function of \(r>R\).
Making use of the result obtained in the question 3 , determine the gravitational potential \(\mathcal{V}_{G}\) in the case of a homogeneous solid sphere of radius \(R\) as a function of \(r \leq
In the stellar gravitational collapse of a star of mass \(12 M_{\odot}\) all mass (minus that which remains concentrated in the neutron star) is ejected from the explosion with velocity \(v \sim
Show that a necessary condition for a system to be in equilibrium under the conditions of the potential shown in Fig. 13.4, it must be at the position corresponding to the coordinates of point
Verify that if \(a=p_{a}+i q_{a}\) and \(b=p_{b}+i q_{b}\) are two complex numbers, then the product of the two is the complex number. Get real and imaginary part.
Determine the values of amplitude \(C\) and phase \(\phi\) in the relation (13.48) of harmonic oscillator given by (13.36). The proper frequency of the system is \(\omega_{o}=\sqrt{\frac{k}{m}}\) and
Determine the values of amplitude \(C\) and phase \(\phi\) in the relation (13.48) of harmonic oscillator given by (13.36). The proper frequency of the system is \(\omega_{o}=\sqrt{\frac{k}{m}}\),
Show that the elastic force exhibited by a bouncing ball of radius \(R\), assumed perfectly elastic, can be written as \(\mathbf{F}=-k(r-R) b \hat{f} r\) and that it admits potential energy
A rod of length \(D=150 \mathrm{~cm}\), mass \(M=150 \mathrm{~g}\) is constrained to rotate freely in the \(x y\) plane about one end, Fig. 13.8. At a distance of \(D / 5\) from the constraint, the
The CO (carbon monoxide) molecule is composed of one carbon atom \((\mathrm{A}=12)\) and one oxygen atom \((\mathrm{A}=16)\). The mass number \(\mathrm{A}\) represents the sum of the number of
In the device shown in the Fig. 8.15 the bumper of mass \(M=26.2 \mathrm{~g}\) is subject, in addition to the action of the spring of spring constant \(k=3.40 \mathrm{~N} / \mathrm{m}\), also to a
Consider the system shown in Fig. 6.8.. The spring has spring constant \(k=500\) \(\mathrm{N} / \mathrm{m}\) and rest length \(L_{0}=25.0 \mathrm{~cm}\). Initially a mass \(m=300 \mathrm{~g}\)
A body of mass \(m=11.0 \mathrm{~g}\) can move by resting on a rail, which is frictionless, that runs in the vertical plane with the shape shown in the Fig.6.8: a horizontal section to A, a quarter
An object of negligible size slides on a horizontal plane and its initial speed is \(v_{0}=4 \mathrm{~m} / \mathrm{s}\). The surface of the plane has increasing roughness and the corresponding

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