Questions and Answers of Mechanics

A particle falls along a cycloidal path from the origin to the final point \((x, y)=\) \((\pi a / 2, a)\); the time required is \(\pi \sqrt{a / 2 g}\). How long would it take the particle to slide
A unique transport system is built between two stations \(1 \mathrm{~km}\) apart on the surface of the moon. A tunnel in the shape of a full cycloid cycle is dug, and the tunnel is lined with a
A hollow glass tube is bent into the form of a slightly tilted rectangle, as shown in the figure. Two small ball bearings can be introduced into the tubes at one corner; one rolls clockwise and the
Assume earth's atmosphere is essentially flat, with index of refraction \(n=1\) at the top and \(n=n(y)\) below, with \(y\) measured from the top, and the positive \(y\) direction downward. Suppose
Consider earth's atmosphere to be spherically symmetric above the surface, with index of refraction \(n=n(r)\), where \(r\) is measured from the center of the earth. Using polar coordinates \(r,
(a) Show that the pressure difference between two points in an incompressible liquid of density \(ho\) in static equilibrium is \(\Delta P=ho g s\), where \(s\) is the vertical separation between the
The surface of a paraboloid of revolution is defined by \(z=a\left(x^{2}+y^{2}ight)\) where \(a\) is a constant. Find the differential equation for a geodesic originating at a point \((x, y)=\)
According to Einstein's general theory of relativity, light rays are deflected as they pass by a massive object like the sun. The trajectory of a ray influenced by a central, spherically symmetric
A clock is thrown straight upward on an airless planet with uniform gravity \(g\), and it falls back to the surface at a time \(t_{f}\) after it was thrown, according to clocks at rest on the
(a) An automobile driver, stopped at an intersection, ties a helium-filled balloon on a string attached to the floor of her car, so the balloon floats up. When the light turns green she accelerates
A skyscraper elevator comes equipped with two weighing scales: The first is a typical bathroom scale containing springs that compress when someone stands on it, and the second is the type often used
A laser is aimed horizontally near earth's surface, a distance \(y_{0}\) above the ground; a pulse of light is then emitted.(a) How far will the pulse fall by the time it has travelled a distance
In Example 4.3 we found the equation of motion of a block on an inclined plane, using the generalized coordinate \(X\), the distance of the block from the bottom of the incline. Solve the equation
Note that in the Hafele-Keating experiments the total error in the eastward and westward flights was comparable, \(\pm 23\) and \(\pm 21\) nanoseconds, respectively, but that the percentage error was
A hypothetical planet has an equatorial circumference of 40,000 km, a gravity \(g=10 \mathrm{~m} / \mathrm{s}^{2}\), and completes one revolution every 24 hours. Aircraft A circles eastward around
A particle of mass \(m\) slides inside a smooth hemispherical bowl of radius \(R\). Beginning with spherical coordinates \(r, \theta\) and \(\varphi\) to describe the dynamics, select generalized
A small block of mass \(m\) and a weight of mass \(M\) are connected by a string of length \(D\). The string has been threaded through a small hole in a tabletop, so the block can slide without
Two blocks of equal mass \(m\), connected by a Hooke's-law spring of unstretched length \(\ell\), are free to move in one dimension. Find the equations of motion of the system, using the relative and
In certain situations, it is possible to incorporate frictional effects in a simple way into a Lagrangian problem. As an example, consider the Lagrangian(a) Find the equation of motion for the
Consider a vertical circular hoop of radius \(R\) rotating about its vertical symmetry axis with constant angular velocity \(\Omega\). A bead of mass \(m\) is threaded onto the hoop, so is free to
Consider a particle moving in three dimensions with Lagrangian L = \((1 / 2) m\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}ight)+a \dot{x}+b\), where \(a\) and \(b\) are constants.(a) Find the equations
Consider a Lagrangian that depends on second derivatives of the coordinatesThrough the variational principle, find the resulting differential equations of motion. L = L(qk, ak, ak, t).
Consider the Lagrangian \(L^{\prime}=m \dot{x} \dot{y}-k x y\) for a particle free to move in two dimensions, where \(x\) and \(y\) are Cartesian coordinates, and \(m\) and \(k\) are constants.(a)
A pendulum consists of a plumb bob of mass \(m\) on the end of a string that swings back and forth in a plane. The upper end of the string passes through a small hole in the ceiling, and the angle
A spherical pendulum consists of a particle of mass \(m\) on the end of a string of length \(R\). The position of the particle can be described by a polar angle \(\theta\) and an azimuthal angle
The Hamiltonian of a bead on a parabolic wire turning with constant angular velocity ω iswhere \(H\) is a constant. Reduce the problem to quadrature: That is, find an equation for the time \(t\) is
One end of a wire is tied to a point A on the ceiling and the other end is tied to a point on a ring of radius \(R\) and negligible mass. The ring therefore hangs from the wire in a vertical plane
A particle moves in a cylindrically symmetric potential \(U(ho, z)\). Use cylindrical coordinates \(ho, \varphi\), and \(z\) to parameterize the space.(a) Write the Lagrangian for an unconstrained
A particle of mass \(m\) slides inside a smooth paraboloid of revolution whose axis of symmetry \(z\) is vertical. The surface is defined by the equation \(z=\alpha ho^{2}\), where \(z\) and \(ho\)
A spring pendulum features a pendulum bob of mass \(m\) attached to one end of a spring of force-constant \(k\) and unstretched length \(R\). The other end of the spring is attached to a fixed point
A pendulum is constructed from a bob of mass \(m\) on one end of a light string of length \(D\). The other end of string is attached to the top of a circular cylinder of radius \(R\) \((R
A plane pendulum is made with a plumb bob of mass \(m\) hanging on a Hooke'slaw spring of negligible mass, force constant \(k\), and unstretched length \(\ell_{0}\). The spring can stretch but is not
A particle of mass \(m\) and charge \(q\) moves within a parallel-plate capacitor whose charge \(Q\) decays exponentially with time, \(Q=Q_{0} e^{-t / \tau}\), where \(\tau\) is the time constant of
A particle of mass \(m\) travels between two points \(x=0\) and \(x=x_{1}\) on Earth's surface, leaving at time \(t=0\) and arriving at \(t=t_{1}\). The gravitational field \(g\) is uniform.(a)
Suppose the particle of the preceding problem moves instead at constant speed along an isoceles triangular path between the beginning point and the end point, with the high point at height \(z_{1}\)
A plane pendulum consists of a light rod of length R supporting a plumb bob of mass \(m\) in a uniform gravitational field \(g\). The point of support of the top end of the rod is forced to oscillate
Solve the preceding problem if instead of being forced to oscillate in the horizontal direction, the upper end of the rod is forced to oscillate in the vertical direction with \(y=A \cos \omega
A particle of mass \(m\) on a frictionless table top is attached to one end of a light string. The other end of the string is threaded through a small hole in the table top, and held by a person
A rod is bent in the middle by angle \(\alpha\). The bottom portion is kept vertical and the top portion is therefore oriented at angle \(\alpha\) to the vertical. A bead of mass \(m\) is slipped
Center of mass and relative coordinates. Show that for two particles moving in one dimension, with coordinates \(x_{1}\) and \(x_{2}\), with a potential that depends only upon their separation
Consider a Lagrangian \(L^{\prime}=L+d f / d t\), where the Lagrangian is \(L=\) \(L\left(q_{k}, \dot{q}_{k}, tight)\), and the function \(f=f\left(q_{k}, tight)\).(a) Show that
Show that the function \(L^{\prime}\) given in the preceding problem must obey Lagrange's equations if \(L\) does, directly from the principle of stationary action. Lagrange's equations do not have
In Example 4.8 we analyzed the case of a bead on a rotating parabolic wire. The energy of the bead was not conserved, but the Hamiltonian was:There is an equilibrium point at \(r=0\) which is
One point on a horizontal circular wire \(\mathrm{C}\) of radius \(R\) is attached to a thin, vertical axle which turns at constant angular velocity \(\Omega\) about the vertical axis, causing
A frictionless slide is constructed in the shape of a cycloid. The horizontal coordinate x and vertical coordinate y of the slide are given in parametric form bywhere \(A\) is a constant. Here the
The wire described in the preceding problem is now forced to rotate about its vertical axis of symmetry with constant angular velocity \(\Omega\).(a) Find \(\Omega_{c}\), the critical value of
A wire bent in the shape of a hyperbolic cosine function y = a cosh(x/x0) is supported in a vertical plane, where \(x\) and \(y\) are the horizontal and vertical coordinates, respectively. and \(a\)
A wire is bent into the shape of a quartic function \(y=a x^{4}\) and oriented in a vertical plane, with \(x\) horizontal, \(y\) vertical, and \(a\) a positive constant. A bead of mass \(m\) is
A bead of mass \(m\) is placed on a vertically-oriented circular hoop of radius \(R\) that is forced to rotate with constant angular velocity \(\omega\) about a vertical axis through its center.(a)
A surface with \(N_{0}\) adsorption centers has \(N\left(\leq N_{0}ight)\) gas molecules adsorbed on it. Show that the chemical potential of the adsorbed molecules is given by\[\mu=k T \ln
Study the state of equilibrium between a gaseous phase and an adsorbed phase in a singlecomponent system. Show that the pressure in the gaseous phase is given by the Langmuir
Show that for a system in the grand canonical ensemble\[\{\overline{(N E)}-\bar{N} \bar{E}\}=\left(\frac{\partial U}{\partial N}ight)_{T, V} \overline{(\Delta N)^{2}}\]
Define a quantity \(J\) as\[J=E-N \mu=T S-P V\]Show that for a system in the grand canonical ensemble\[\overline{(\Delta J)^{2}}=k T^{2} C_{V}+\left\{\left(\frac{\partial U}{\partial N}ight)_{T,
Assuming that the latent heat of vaporization of water \(L_{\mathrm{V}}=2260 \mathrm{~kJ} / \mathrm{kg}\) is independent of temperature and the specific volume of the liquid phase is negligible
Assuming that the latent heat of sublimation of ice \(L_{\mathrm{S}}=2500 \mathrm{~kJ} / \mathrm{kg}\) is independent of temperature and the specific volume of the solid phase is negligible compared
Calculate the slope of the solid-liquid transition line for water near the triple point \(T=273.16 \mathrm{~K}\), given that the latent heat of melting is \(80 \mathrm{cal} / \mathrm{g}\), the
Show that the Clausius-Clapeyron equation (4.7.7) guarantees that each of the coexistence curves at the triple point of a material "points into" the third phase; for example, the slope of the
Sketch the \(P-V\) phase diagram for helium-4 using the sketch of the \(P-T\) phase diagram in Figure 4.3. Ps S P Pc+ superfluid! TA T V To
Derive the equivalent of the Clausius-Clapeyron equation (4.7.7) for the slope of the coexistence chemical potential as a function of temperature. Use the fact that the pressures \(P(\mu, T)\) in two
Sketch the \(P-T\) and \(P-V\) phase diagrams of water, taking into account the fact that the mass density of the liquid phase is larger than the mass density of the solid phase.
Evaluate the density matrix \(ho_{m n}\) of an electron spin in the representation that makes \(\hat{\sigma}_{x}\) diagonal.Next, show that the value of \(\left\langle\sigma_{z}ightangle\), resulting
Prove that\[\left\langle q\left|e^{-\beta \hat{H}}ight| q^{\prime}ightangle=\exp \left[-\beta \hat{H}\left(-i \hbar \frac{\partial}{\partial q}, qight)ight] \delta\left(q-q^{\prime}ight)\]where
(a) Solve the integral∫3N0≤∑3Ni=1|xi|≤R(dx1…dx3N)∫3N0≤∑i=13N|xi|≤R(dx1…dx3N)and use it to determine the "volume" of the relevant region of the phase space of an extreme
(a) Derive formula (3.2.36) from equations (3.2.14) and (3.2.35).Data From Equations (3.2.14)Data From Equations (3.2.35)(b) Derive formulae (3.2.39) and (3.2.40) from equations (3.2.37) and
Prove that the quantity \(g^{\prime \prime}\left(x_{0}ight)\), see equations (3.2.25), is equal to \(\left\langle(E-U)^{2}ightangle \exp (2 \beta)\). Thus show that equation (3.2.28) is physically
Using the fact that \((1 / n !)\) is the coefficient of \(x^{n}\) in the power expansion of the function \(\exp (x)\), derive an asymptotic formula for this coefficient by the method of saddle-point
Verify that the quantity \((k / \mathcal{N}) \ln \Gamma\), where\[\Gamma(\mathcal{N}, U)=\sum_{\left\{n_{r}ight\}}^{\prime} W\left\{n_{r}ight\}\]is equal to the (mean) entropy of the given system.
Making use of the fact that the Helmholtz free energy \(A(N, V, T)\) of a thermodynamic system is an extensive property of the system, show that\[N\left(\frac{\partial A}{\partial N}ight)_{V,
Let's go to part (c) right away. Our problem here is to maximize the expression \(S / k=-\sum_{r, s} P_{r, s} \ln P_{r, s}\), subject to the constraints \(\sum_{r, s} P_{r, s}=\) 1, \(\sum_{r, s}
Prove that, quite generally,\[C_{P}-C_{V}=-k \frac{\left[\frac{\partial}{\partial T}\left\{T\left(\frac{\partial \ln Q}{\partial V}ight)_{T}ight\}ight]_{V}^{2}}{\left(\frac{\partial^{2} \ln
Show that, for a classical ideal gas,\[\frac{S}{N k}=\ln \left(\frac{Q_{1}}{N}ight)+T\left(\frac{\partial \ln Q_{1}}{\partial T}ight)_{P}\]
If an ideal monatomic gas is expanded adiabatically to twice its initial volume, what will the ratio of the final pressure to the initial pressure be? If during the process some heat is added to the
(a) The volume of a sample of helium gas is increased by withdrawing the piston of the containing cylinder. The final pressure \(P_{f}\) is found to be equal to the initial pressure \(P_{i}\) times
Determine the work done on a gas and the amount of heat absorbed by it during a compression from volume \(V_{1}\) to volume \(V_{2}\), following the law \(P V^{n}=\) const.
If the "free volume" \(\bar{V}\) of a classical system is defined by the equation\[\bar{V}^{N}=\int e^{\left\{\bar{U}-U\left(\boldsymbol{q}_{i}ight)ight\} / k T} \prod_{i=1}^{N} d^{3} q_{i}\]where
(a) Evaluate the partition function and the major thermodynamic properties of an ideal gas consisting of \(N_{1}\) molecules of mass \(m_{1}\) and \(N_{2}\) molecules of mass \(m_{2}\), confined to a
Consider a system of \(N\) classical particles with mass \(m\) moving in a cubic box with volume \(V=L^{3}\). The particles interact via a short-ranged pair potential \(u\left(r_{i j}ight)\) and each
Show that the partition function \(Q_{N}(V, T)\) of an extreme relativistic gas consisting of \(N\) monatomic molecules with energy-momentum relationship \(\varepsilon=p c, c\) being the speed of
Consider a system similar to the one in the preceding problem but consisting of \(3 N\) particles moving in one dimension. Show that the partition function in this case is given by\[Q_{3 N}(L,
If we take the function \(f(q, p)\) in equation (3.5.3) to be \(U-H(q, p)\), then clearly \(\langle fangle=0\); formally, this would meanData From Equation (3.5.3)\[\int[U-H(q, p)] e^{-\beta H(q, p)}
Show that for a system in the canonical ensemble\[\left\langle(\Delta E)^{3}ightangle=k^{2}\left\{T^{4}\left(\frac{\partial C_{V}}{\partial T}ight)_{V}+2 T^{3} C_{V}ight\}\]Verify that for an ideal
Consider the long-time averaged behavior of the quantity \(d G / d t\), where\[G=\sum_{i} q_{i} p_{i}\]and show that the validity of equation (3.7.5) implies the validity of equation (3.7.6), and
Show that, for a statistical system in which the interparticle potential energy \(u(\boldsymbol{r})\) is a homogeneous function (of degree \(n\) ) of the particle coordinates, the virial
(a) Calculate the time-averaged kinetic energy and potential energy of a one-dimensional harmonic oscillator, both classically and quantum-mechanically, and show that the results obtained are
The restoring force of an anharmonic oscillator is proportional to the cube of the displacement. Show that the mean kinetic energy of the oscillator is twice its mean potential energy.
Derive the virial equation of state equation (3.7.15) from the classical canonical partition function (3.5.5). Show that in the thermodynamic limit the interparticle terms dominate the ones that come
Show that in the relativistic case the equipartition theorem takes the form\[\left\langle m_{0} u^{2}\left(1-u^{2} / c^{2}ight)^{-1 / 2}ightangle=3 k T \text {, }\]where \(m_{0}\) is the rest mass of
Develop a kinetic argument to show that in a noninteracting system the average value of the quantity \(\sum_{i} p_{i} \dot{q}_{i}\) is precisely equal to \(3 P V\). Hence show that, regardless of
The energy eigenvalues of an \(s\)-dimensional harmonic oscillator can be written as\[\varepsilon_{j}=(j+s / 2) \hbar \omega ; \quad j=0,1,2, \ldots\]Show that the \(j\) th energy level has a
Obtain an asymptotic expression for the quantity \(\ln g(E)\) for a system of \(N\) quantum-mechanical harmonic oscillators by using the inversion formula (3.4.7) and the partition function (3.8.15).
(a) When a system of \(N\) oscillators with total energy \(E\) is in thermal equilibrium, what is the probability \(p_{n}\) that a particular oscillator among them is in the quantum state \(n\) ?
The potential energy of a one-dimensional, anharmonic oscillator may be written as\[V(q)=c q^{2}-g q^{3}-f q^{4}\]where \(c, g\), and \(f\) are positive constants; quite generally, \(g\) and \(f\)
The energy levels of a quantum-mechanical, one-dimensional, anharmonic oscillator may be approximated as\[\varepsilon_{n}=\left(n+\frac{1}{2}ight) \hbar \omega-x\left(n+\frac{1}{2}ight)^{2} \hbar
Study, along the lines of Section 3.8, the statistical mechanics of a system of \(N\) "Fermi oscillators," which are characterized by only two eigenvalues, namely 0 and \(\varepsilon\).
The quantum states available to a given physical system are (i) a group of \(g_{1}\) equally likely states, with a common energy \(\varepsilon_{1}\) and (ii) a group of \(g_{2}\) equally likely
Gadolinium sulphate obeys Langevin's theory of paramagnetism down to a few degrees Kelvin. Its molecular magnetic moment is \(7.2 \times 10^{-23} \mathrm{amp}-\mathrm{m}^{2}\). Determine the degree
Oxygen is a paramagnetic gas obeying Langevin's theory of paramagnetism. Its susceptibility per unit volume, at \(293 \mathrm{~K}\) and at atmospheric pressure, is \(1.80 \times 10^{-6}
(a) Consider a gaseous system of \(N\) noninteracting, diatomic molecules, each having an electric dipole moment \(\mu\), placed in an external electric field of strength \(E\). The energy of such a
Consider a pair of electric dipoles \(\boldsymbol{\mu}\) and \(\boldsymbol{\mu}^{\prime}\), oriented in the directions \((\theta, \phi)\) and \(\left(\theta^{\prime}, \phi^{\prime}ight)\),
Evaluate the high-temperature approximation of the partition function of a system of magnetic dipoles to show that the Curie constant \(C_{J}\) is given by\[C_{J}=\frac{N_{0} g^{2} \mu_{B}^{2}}{k}

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