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mechanics
Questions and Answers of
Mechanics
A bicycle wheel of diameter \(D=26\) inches \((1\) inch \(=2.54 \mathrm{~cm})\) is rotating with frequency \(f=1.0 \mathrm{~s}^{-1}\). Neglecting finite size and tire deformation, assuming the wheel
On a rotating platform with angular frequency \(\omega=1.5 \mathrm{rad} / \mathrm{s}\) a small spring cannon is fixed, at a distance \(R=2.0 \mathrm{~m}\) from the center. The small cannon, arranged
A constant force of magnitude \(10 \mathrm{~N}\) lies in the \(x y\) plane and forms an angle of \(30^{\circ}\) with the \(x\) axis. Calculate the work done when its point of application moves from
A simple pendulum, with mass \(m=60 \mathrm{~g}\), and inextensible wire \(L=120 \mathrm{~cm}\) long, starts from rest initially at an angle of \(60^{\circ}\) to the vertical. Determine the speed
A simple pendulum serves as a device for estimating wind speed. A small ball of mass \(m=100\) \(\mathrm{g}\) hangs from the end of the pendulum of length \(l=1 \mathrm{~m}\). Blow the wind in a
With reference to the figure and data from the Exercise 4.4, determine the work done by the frictional force when the bucket dropped \(h=42 \mathrm{~cm}\). Find the velocity required in step 3 )
An ideal spring that is initially compressed is released at the initial time to throw an object of mass \(m=60 \mathrm{~g}\) upward. The body reaches a height \(h=2.0 \mathrm{~m}\) from its initial
The following vector field is given: \(\mathbf{A}=\frac{1}{2} x^{2} \hat{\mathbf{i}}-x y \hat{\mathbf{j}}+x y z \hat{\mathbf{k}}\). Check whether the field is irrotational.
The vector field of the question 6 is given. Determine the value of its divergence at the point \(\mathrm{P}\) of Cartesian coordinates \((2,4,3)\).Question 6The following vector field is given:
Show that the operatoris identically null.
Assume the scalar field of question 8.Determine the values of the Cartesian components of the gradient of the scalar field \(V\) at the point \(\mathrm{P}\) of Cartesian coordinates
A force is described by the vector field \(\mathbf{F}=-\alpha\left(9 x^{2} \hat{\mathbf{i}}+2 z^{2} \hat{\mathbf{j}}+6 y z \hat{\mathbf{k}}\right)\). After obtaining the dimensions of \(\alpha\),
A force field in the plane is described by the relation \(F_{x}=F_{o} x^{k} ; F_{y}=F_{o} y^{k}\), with \(k\) and \(F_{o}\) real constants. For what values of \(k\) and \(F_{o}\) is the field
A force field in the plane is described by the relation \(F_{x}=a x+b y ; F_{y}=c x+d y\), with \(a,b, c, d\) real constants. Determine for what values of the constants the field is conservative and
It is given the force field of the question 13.Determine for what values of the constants the field is central and determine the potential.Question 13A force field in the plane is described by the
A device carries a mass \(m=1200 \mathrm{~kg}\), at a constant speed of \(80 \mathrm{~km} / \mathrm{h}\), has an acceleration equal to \(0.1 \mathrm{~m} / \mathrm{s}^{2}\), concordant with the speed.
Determine the angular frequency \(\omega_{k}\) for the Brillouin peaks in water for \(90^{\circ}\) laser scattering, using a He-Ne laser with \(\lambda=632.8 \mathrm{~nm}\). Determine the width of
Describe the dynamical structure factor for Raman scattering for a He-Ne laser with \(\lambda=632.8 \mathrm{~nm}\). The energy level responsible for this scattering has an energy of \(0.05
The spontaneous magnetization of a two-dimensional Ising model on a square lattice at \(T
Apply the theory of Section 13.4 to a two-dimensional lattice gas and show that, at \(T=T_{c}\), the quantity \(k T_{c} / P_{c} v_{c} \simeq 10.35\).
Show that for the spherical model in one dimension the free energy at constant \(\lambda\) is given by\[\frac{\beta A_{\lambda}}{N}=\frac{1}{2} \ln \left[\frac{\beta\left\{\lambda+\sqrt{
Starting with expression (13.3.8) for the partition function of a one-dimensional \(n\)-vector model, with \(J_{i}=n J^{\prime}\), show that\[\operatorname{Lim}_{n, N \rightarrow \infty} \frac{1}{n
Show that the low-field susceptibility, \(\chi_{0}\), of the spherical model at \(T
In view of the fact that only those fluctuations whose length scale is large play a dominant role in determining the nature of a phase transition, the quantity
Consider a spherical model whose spins interact through a long-range potential varying as \((a / r)^{d+\sigma}(\sigma>0), r\) being the distance between two spins. This replaces the quantity
Refer to Section 13.6 on the ideal Bose gas in \(d\) dimensions, and complete the steps leading to equations (13.6.9) through (13.6.15) and (13.6.23).
The derivations here proceed exactly as in Problem 7.7. The singularity of these quantities arises from the last term of the two expressions, and is qualitatively similar to the singularity of the
Show that for any given fluid\[C_{P}=V T(\partial P / \partial T)_{S}(\partial P / \partial T)_{V} \kappa_{T}\]and\[C_{V}=V T(\partial P / \partial T)_{S}(\partial P / \partial T)_{V
Show that for any given fluid\[\kappa_{T}=ho^{-2}(\partial ho / \partial \mu)_{T}\]where \(ho(=N / V)\) is the particle density and \(\mu\) the chemical potential of the fluid. For the ideal Bose gas
Consider an ideal relativistic Bose gas composed of \(N_{1}\) particles and \(N_{2}\) antiparticles, each of rest mass \(m_{0}\), with occupation numbers\[\frac{1}{\exp
Derive equation (13.1.9) for hard spheres in one dimension from equation (13.1.7). Plot the pair correlation function for \(n D=0.25,0.50,0.75\), and 0.90 . Determine the structure factor \(S(k)\)
Use the pair correlation function (13.1.8) and (13.1.9) to determine analytically the structure factor for hard spheres in one dimension. Show that \(S(k)\) is given by equation (13.1.21). Plot
Use the Takahashi method of Section 13.1 for a system of point masses and harmonic springs of length \(a\). Allow the particles to pass through each other, so that the partition function can be
Confirm the first few coefficients in the low-temperature series for the two-dimensional Ising model in equation (13.4.53). Write a program to calculate the energies of all \(2^{16}\) states for a
Calculate the exact zero field partition function of the one-dimensional Ising model on a periodic chain of \(n\) spins using equation (13.2.5) and write \(Q_{N}(0, T)\) in the form of equation
Use the code posted at www.elsevierdirect.com to evaluate equations (13.4.56) through (13.4.59) to determine the low-temperature series coefficients for the two-dimensional Ising model for an \(8
Use the data posted at www.elsevierdirect.com to evaluate equations (13.4.56) through (13.4.59) to plot the two-dimensional Ising model internal energy and specific heat as a function of temperature
Show that the decimation transformation of a one-dimensional Ising model, with \(l=2\), can be written in terms of the transfer matrix \(\boldsymbol{P}\)
Verify that expression (15) of Section 14.2 indeed satisfies the functional equation (14) for the field-free Ising model in one dimension. Next show (or at least verify) that, with the field present,
Verify that expression (32) of Section 14.2 indeed satisfies the functional equation (31) for the field-free spherical model in one dimension. Next show (or at least verify) that, with the field
Making the suggested substitution into eqn. (14.2.24), we get\[\begin{aligned}Q_{N}=\int & \cdots \int \exp \left[\sum_{j=1}^{N^{\prime}}\left\{K_{0}^{\prime}+K_{1}^{\prime} \cdot \frac{2
An approximate way of implementing an \(\mathrm{RG}\) transformation on a square lattice is provided by the so-called Migdal-Kadanoff transformation \({ }^{17}\) shown in Figure 14.8. It consists of
Consider the linearized RG transformation (14.3.12), with\[\mathfrak{A}_{l}^{*}=\left(\begin{array}{ll}a_{11} & a_{12} \tag{3}\\a_{21} & a_{22}\end{array}\right)\]such that \(\left(a_{11}
Check that the critical exponents (14.4.38) through (14.4.40), in the limit \(n \rightarrow \infty\), agree with the corresponding exponents for the spherical model of Section 13.5 with \(d \lesssim
Show, from equations (14.4.43) through (14.4.46), that for \(d \lesssim 4\)\[\eta \simeq \frac{1}{2 n} \varepsilon^{2}, \quad \gamma \simeq 1+\frac{1}{2}\left(1-\frac{6}{n}\right) \varepsilon, \quad
Using the various scaling relations, derive from equations (14.4.43) through (14.4.45) comparable expressions for the remaining exponents \(\beta, \delta\), and \(v\). Repeat for these exponents the
Making use of expressions (15.1.11) and (15.1.12) for \(\Delta S\) and \(\Delta P\), and expressions (15.1.14) for \(\overline{(\Delta T)^{2}}, \overline{(\Delta V)^{2}}\), and \(\overline{(\Delta T
Establish the probability distribution (15.1.15), which leads to the expressions in (15.1.16) for \(\overline{(\Delta S)^{2}}, \overline{(\Delta P)^{2}}\), and \(\overline{(\Delta S \Delta P)}\).
If we choose the quantities \(E\) and \(V\) as "independent" variables, then the probability distribution function (15.1.8) does not reduce to a form as simple as (15.1.13) or (15.1.15); it is marked
A string of length \(l\) is stretched, under a constant tension \(F\), between two fixed points \(A\) and \(B\). Show that the mean square (fluctuational) displacement \(y(x)\) at point \(P\),
A string of length \(l\) is stretched, under a constant tension \(F\), between two fixed points \(A\) and \(B\). Show that the mean square (fluctuational) displacement \(y(x)\) at point \(P\),
Pospišil (1927) observed the Brownian motion of soot particles, of radii \(0.4 \times 10^{-4} \mathrm{~cm}\), immersed in a water-glycerine solution, of viscosity 0.0278 poise at a temperature of
In the notation of Section 15.3, show that for a Brownian particle\[\langle\boldsymbol{v}(t) \cdot \boldsymbol{F}(t)angle=3 k T / \tau, \quad \text { while }\langle\boldsymbol{v}(t) \cdot
Integrate equation (15.3.14) to obtain\[\boldsymbol{r}(t)=\boldsymbol{v}(0) \tau\left(1-e^{-t / \tau}\right)+\tau \int_{0}^{t}\left\{1-e^{(u-t) / \tau}\right\} \boldsymbol{A}(u) d u\]so that
While detecting a very feeble current with the help of a moving-coil galvanometer, one must ensure that an observed deflection is not just a stray kick arising from the Brownian motion of the
(a) Integrate Langevin's equation (15.3.5) for the velocity component \(v_{x}\) over a small interval of time \(\delta t\), and show thatEquation (15.3.5).\[\frac{\left\langle\delta
Generalize the analysis of the Langevin theory of a harmonic oscillator, as given by equation (15.3.33), to the case of an oscillator starting at time \(t=0\) with the initial position \(x(0)\) and
Generalize the Fokker-Planck equation to the case of a particle executing Brownian motion in three dimensions. Determine the general solution of this equation and study its important features.
The autocorrelation function \(K(s)\) of a certain statistically stationary variable \(y(t)\) is given by(a) \(K(s)=K(0) e^{-\alpha s^{2}} \cos \left(2 \pi f^{*} s\right)\)or by(b) \(K(s)=K(0)
(a) From the defining equation of the variable \(Y(t)\), we get\[\begin{equation*}\left\langle Y^{2}(t)\rightangle=\int_{u}^{u+t} \int_{u}^{u+t}\left\langle y\left(u_{1}\right)
Show that if the autocorrelation function \(K(s)\) of a certain statistically stationary variable \(y(t)\) is given by\[K(s)=K(0) \frac{\sin (a s)}{a s} \frac{\sin (b s)}{b s} \quad(a>b>0)\]then the
Show that the power spectra \(w_{\boldsymbol{v}}(f)\) and \(w_{\boldsymbol{A}}(f)\) of the fluctuating variables \(\boldsymbol{v}(t)\) and \(\boldsymbol{A}(t)\) that appear in the Langevin equation
(a) Verify equations (15.6.7) through (15.6.9).(b) Substituting expression (15.6.9) for \(\boldsymbol{K}_{v}(s)\) into equation (15.6.6), derive formula (15.3.31) for \(\left\langle
Determine \(\hat{\chi}_{v x}^{\prime \prime}(\omega)\) and \(S_{v x}(\omega)\) for a Brownian particle in a harmonic oscillator potential. Show that the response function and the power spectrum for
Derive the linear response density matrix (15.6.29) from the equation of motion (15.6.28).
Show that \(G_{A B}(t)=G_{B A}(t-i \beta \hbar)\) and use the cyclic property of the traces to derive the fluctuation-dissipation theorem \(\hat{\chi}_{A B}^{\prime \prime}(\omega)=\frac{1}{2
Show that \(G_{A B}(t)=G_{B A}(t-i \beta \hbar)\). Use this result to show that, in the classical limit, \(\hat{\chi}_{A B}^{\prime \prime}(t)\) becomes \(\left\langle\frac{d A(t)}{d t}
Determine the self-diffusion term in the dynamical structure factor \(S_{\text {self }}(\boldsymbol{k}, \omega)\) in equation (15.6.41b) for the case of a single particle that diffuses according to
Making use of expressions (12.3.17) through (12.3.19), (13.2.12), and (13.2.13), show that the expectation values of the numbers \(N_{+}, N_{-}, N_{++}, N_{--}\), and \(N_{+-}\)in the case of an
(a) Show that the partition function of an Ising lattice can be written as\[Q_{N}(B, T)=\sum_{N_{+}, N_{+-}}^{\prime} g_{N}\left(N_{+}, N_{+-}\right) \exp \left\{-\beta H_{N}\left(N_{+},
Using the approximate expression, see Fowler and Guggenheim (1940),\[g_{N}\left(N_{1}, N_{12}\right) \simeq \frac{\left(\frac{1}{2} q N\right) !}{N_{11} ! N_{22} !\left[\left(\frac{1}{2}
Making use of relation (13.2.37), along with expressions (13.2.8) for the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) of the transfer matrix \(\boldsymbol{P}\), determine the correlation length
Consider a one-dimensional Ising system in a fluctuating magnetic field \(B\), so that\[Q_{N}(s, T) \sim \int_{-\infty}^{\infty} d B \sum_{\left\{\sigma_{i}\right\}} \exp \left\{-\frac{\beta N
Solve exactly the problem of a field-free Ising chain with nearest-neighbor and next-nearest-neighbor interactions, so that\[H\left\{\sigma_{i}\right\}=-J_{1} \sum_{i} \sigma_{i} \sigma_{i+1}-J_{2}
Consider a double Ising chain such that the nearest-neighbor coupling constant along either chain is \(J_{1}\) while the one linking adjacent spins in the two chains is \(J_{2}\). Then, in the
Write down the transfer matrix \(\boldsymbol{P}\) for a one-dimensional spin-1 Ising model in zero field, described by the Hamiltonian\[H_{N}\left\{\sigma_{i}\right\}=-J \sum_{i} \sigma_{i}
(a) Apply the theory of Section 13.2 to a one-dimensional lattice gas and show that the pressure \(P\) and the volume per particle \(v\) are given by\[\frac{P}{k T}=\ln
For a one-dimensional system, such as the ones discussed in Sections 13.2 and 13.3, the correlation function \(g(r)\) at all temperatures is of the form \(\exp (-r a / \xi)\), where \(a\) is the
Show that for a one-dimensional, field-free Ising model\[\overline{\sigma_{k} \sigma_{l} \sigma_{m} \sigma_{n}}=\{\tanh \beta J\}^{n-m+l-k},\]where \(k \leq l \leq m \leq n\).
Recall the symbol \(n(r)\), of equation (13.4.5), which denotes the number of closed graphs that can be drawn on a given lattice using exactly \(r\) bonds. Show that for a square lattice wrapped on a
According to Onsager, the field-free partition function of a rectangular lattice (with interaction parameters \(J\) and \(J^{\prime}\) in the two perpendicular directions) is given by\[\frac{1}{N}
Write the elliptic integral \(K_{1}(\kappa)\) in the form\[K_{1}(\kappa)=\int_{0}^{\pi / 2} \frac{1-\kappa \sin \phi}{\sqrt{ }\left(1-\kappa^{2} \sin ^{2} \phi\right)} d \phi+\int_{0}^{\pi / 2}
Using equations (13.4.22) and (13.4.28) at \(T=T_{c}\), show that the entropy of the two-dimensional Ising model on a square lattice at its critical point is given by\[\frac{S_{c}}{N k}=\frac{2
Using the Friedmann equation (9.1.1)\[\frac{d a}{d t}=\sqrt{\frac{8 \pi G u}{3 c^{2}}} a\]and the connection between scale factor \(a\) and blackbody temperature \(T\), \(T a=T_{0} a_{0}\), along
Determine the average energy per particle and average entropy per particle for the photons, electrons, positrons and neutrinos during the first second of the universe.
The average kinetic energy per relativistic electron/positron is of the order of \(u_{e} / n_{e} \sim k T\). The Coulomb energy per electron/positron is of the order of \(u_{c} \approx e^{2} /\left(4
Correction to the first printing of third edition: The exponent in the result should be \(-3 / 2\). For \(\beta m c^{2} \gg 1\) but before the time when the electron density approaches the protron
Correction to the first printing of third edition: The exponent in the result should be \(-3 / 2\). After the density of electrons levels off at the nearly the proton density, you can use equation
Correction to the first printing of third edition: the energy density in the statement of the problem should read\[u_{\text {total }}=\left(1+(21 / 8)(4 / 11)^{4 / 3} \right) u_{\gamma} .\]After the
If the current CMB temperature was \(27 \mathrm{~K}\) rather than \(2.7 \mathrm{~K}\), the baryonto-photon ratio would be \(10^{3}\) times smaller. Equation (9.7.8) implies that the nucleosynthesis
The strong interaction exhibits asymptotic freedom at high energies justifying treating the quarks an gluons as noninteracting. The effective number of species in equilibrium in these tiny
The strong interaction exhibits asymptotic freedom at high energies justifying treating the quarks an gluons as noninteracting. The effective number of species is much larger than during the time
By eqn. (10.2.3), the second virial coefficient of the gas with the given interparticle interaction would be\[\begin{aligned}a_{2} & =-\frac{2 \pi}{\lambda^{3}}\left[\int_{0}^{D}-1 \cdot r^{2} d
For this problem, we integrate (10.2.3) by parts and write\[a_{2} \lambda^{3}=-\frac{2 \pi}{3 k T} \int_{0}^{\infty} e^{-u(r) / k T} \frac{\partial u(r)}{\partial r} r^{3} d r\]cf. eqn. (3.7.17) and
(a) Using the thermodynamic relation\[C_{P}-C_{\mathrm{V}}=T(\partial P / \partial T)_{\mathrm{V}}(\partial V / \partial T)_{P}=-T(\partial P / \partial T)_{\mathrm{V}}^{2} /(\partial P / \partial
Since, by definition,\[\alpha=\mathrm{v}^{-1}(\partial \mathrm{v} / \partial T)_{P} \text { and } B^{-1} \equiv \kappa_{T}=-\mathrm{v}^{-1}(\partial \mathrm{v} / \partial P)_{T}\]we must
To the desired approximation,\[\begin{equation*}\frac{P}{k T} \equiv \frac{1}{V} \ln \mathscr{Q}=\frac{1}{\lambda^{3}}\left(z-a_{2} z^{2}\right), \quad n=\frac{N}{V}=\frac{1}{\lambda^{3}}\left(z-2
We consider a volume element \(d x_{1} d y_{1} d z_{1}\) around the point \(P\left(x_{1}, 0,0 \right)\) in solid 1 and a volume element \(d x_{2} d y_{2} d z_{2}\) around the point \(Q\left(x_{2},
Referring to equation (10.5.31) for the phase shifts \(\eta_{l}(k)\) of a hard-sphere gas, show that for \(k D \ll 1\)\[\eta_{l}(k) \simeq-\frac{(k D)^{2 l+1}}{(2 l+1)\{1 \cdot 3 \cdots(2
The symmetrized wave functions for a pair of non-interacting bosons/fermions are given by\[\Psi_{\alpha}\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right)=\frac{1}{\sqrt{2} V}\left(e^{i \mathbf{k}_{1} \cdot
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