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Chapter: 11 -Problem: 2 >> Sketch a rough graph of the product cos (?x/L) sin (?x/L) from 0 to L and convince yourself that its integral from ?L to L vanishes.
Answer Preview: For purposes of the graph, we let u = x/L, so that we …

, Chapter: 16 -Problem: 6 >> Assume that the expected error in the logarithm of each concentration in Example 16.5 is equal to 0.010. Find the expected error in the rate constant, assuming the reaction to be first order.
Answer Preview: D = NS2 S = 9(7125) - (225) …

, Chapter: 1 -Problem: 2 >> Find the number of meters in 1.000 mile and the number of miles in 1.000 km, using the definition of the inch.
Answer Preview: (1 000 mi) = 1609 m (1 000 km) …

, Chapter: 3 -Problem: 6 >> Express the equation y = mx + b, where m and b are constants, in plane polar coordinates.
Answer Preview: p sin (6) = mp cos (p…

, Chapter: 8 -Problem: 3 >> Show that the reciprocal identity is satisfied by (?z/?x)y and (?x/?z)y if Transcribed Image Text: z = sin and x = y sin (z) = y arcsin (z).
Answer Preview: From the table of derivatives wh…

, Chapter: 13 -Problem: 17 >> Find the two matrix productsThe left factor in one product is equal to the right factor in the other product, and vice versa. Are the two products equal to each other? Transcribed Image Text: 13 2 2 3 3 2 1 -1 2 1 2 2 -1 -2 1 –1 1 3 2 2 3 3 2 1 2 2 -1 -2 –1 –1 1 -1 2
Answer Preview: The two products are not equal …

, Chapter: 13 -Problem: 21 >> Use Mathematica or another software package to verify the inverse found in the preceding example.
Answer Preview: 210 121 […

, Chapter: 2 -Problem: 9 >> If the two ends of a completely flexible chain (one that requires no force to bend it) are suspended at the same height near the surface of the earth, the curve representing the shape of the chain is called a catenary. It can be shown that the catenary is represented byy = a cosh (x/a),where a = T /g? and where ? is the mass per unit length, g is the acceleration due to gravity, and T is the tensi
Answer Preview: For this graph, we have plotted y 11 4…

, Chapter: 3 -Problem: 10 >> The solutions to the Schrödinger equation for the electron in a hydrogen atom have three quantum numbers associated with them, called n, l, and m, and these solutions are denoted by ?nlm. (a)?The ?210?function is given by where a0 = 0.529 × 10??10 m is called the Bohr radius. Write this function in terms of Cartesian coordinates. (b)?The ?211 function is given by Write an expression for the magn
Answer Preview: a. b. …

, Chapter: 8 -Problem: 7 >> Find (?2 f /?x2)y, (?2 f /?x?y), (?2 f /?y?x), and (?2 f /?y2), for each of the following functions,where a, b, and c are constants. a. f = e(ax2+by2), b. f = ln (bx2 + cy2).
Answer Preview: a. b. of ax a2 f 2 a2 f af a2f 2 ) a2 …

, Chapter: 12 -Problem: 2 >> A particle moves along the z axis. It is acted upon by a constant gravitational force equal to ˆ’kmg, where k is the unit vector in the z direction. It is also acted on by a frictional force given bywhere ζ is a constant called a €œfriction constant.€ Find the equation of motion and obtain a general solution. Find z as a function of time if z(0) = 0 and vx (0) = 0. Draw a graph of z as a functio
Answer Preview: The equation of motion is where This is a linear inhomogeneous equation. The complementary equation …

, Chapter: 13 -Problem: 18 >> Show that (AB)C = A(BC) for the matrices: Transcribed Image Text: 01 2 3 14 C=| A=! -2 0 1 3 21 3 1 B= -4 -4 2 3 1 23 3 1 -2
Answer Preview: AB = AB)C= = BC = (BC) = = = || 01 2 31-4 23 1 4 4 3 -…

, Chapter: 2 -Problem: 1 >> The following is a set of data for the vapor pressure of ethanol taken by a physical chemistry student. Plot these points by hand on graph paper, with the temperature on the horizontal axis (the abscissa) and the vapor pressure on the vertical axis (the ordinate). Decide if there are any bad data points. Draw a smooth curve nearly through the points. Use Excel to construct another graph and notice
Answer Preview: Here is a graph constructed with E…

, Chapter: 5 -Problem: 15 >> Solve the cubic equation by trial and error, factoring, or by using Mathematica or Excel:x3 + x2 ? 4x ? 4 = 0.
Answer Preview: This equation ca…

, Chapter: 6 -Problem: 13 >> Find the value of the limit:If a limiting expression appears to approach 0 × ?, it can be put into a form that appears to approach 0/0 or ?/? by using the reciprocal of one factor. In addition the rule must sometimes be applied more than once in order to find the value of the limit. Transcribed Image Text:
Answer Preview: We apply lHpitals rule. …

, Chapter: 10 -Problem: 10 >> Find the coefficients of the first few terms of the Maclaurin seriessinh (x) = a0 + a1x + a2x2 +· · ·What is the radius of convergence of the series?
Answer Preview: The function is analytic everywhere, …

, Chapter: 13 -Problem: 24 >> The BF3 molecule is planar with bond angles of 120?. (a) List the symmetry operators that belong to this molecule. We place the molecule in the x–y plane with the boron atom at the origin and one of the fluorine atoms on the x axis. Call the fluorine atoms a, b, and c starting at the x axis and proceeding counterclockwise around the x–y plane. The symmetry elements are: a threefold rotation axis,
Answer Preview: a. The operators are E, C 3(z) , a , b , c , h , C 2a , C 2b …

, Chapter: 16 -Problem: 7 >> Vaughan obtained the following data for the dimerization of butadiene at 326 ?C.Time (min)        Partial pressure of butadiene (atm) 0 to be deduced 3.25........................................................0.7961 8.02........................................................0.7457 12.18......................................................0.7057 17.30.............................................
Answer Preview: A linear fit of the logarithm of the partial pressure against time shows considerable …

, Chapter: 4 -Problem: 12 >> An object has a force on it given by F = (4.75 N)i + (7.00 N)j + (3.50 N)k. (a) Find the magnitude of the force. (b) Find the projection of the force in the x-y plane. That is, find the vector in the x-y plane whose head is reached from the head of the force vector by moving in a direction perpendicular to the x-y plane.
Answer Preview: a. F = (4 75 N) 2 + …

, Chapter: 6 -Problem: 33 >> Draw a rough graph of the function y = tan (x)/x in the interval ??/2 < x < ?/2. Use l’Hôpital’s rule to evaluate the function at x = 0.
Answer Preview: Here is an accurate graph. The …

, Chapter: 9 -Problem: 11 >> Find the Jacobian for the transformation from Cartesian to cylindrical polar coordinates.
Answer Preview: Without resorting to a determinant, w…

, Chapter: 13 -Problem: 15 >> The potential energy of two electric charges Q1and Q2in a vacuum iswhere r12 is the distance between the charges and ε0 is a constant called the permittivity of a vacuum, equal to 8.854187817 × 10ˆ’12 F mˆ’1 = 8.854187817 × 10ˆ’12 C2 Nˆ’1 mˆ’2. The potential energy of a hydrogen molecules is given bywhere A and B represent the nuclei and 1 and 2 represent the electrons, and where the two indexe
Answer Preview: We use the fact that the origin is midway between the nuclei. Unde…

, Chapter: 3 -Problem: 8 >> Find the values of the Cartesian coordinates that correspond to r = 5.00, ? = 45?, ? = 135?.
Answer Preview: Spherical coordinate is (r, theeta, phi) We have to change this into Cartesian coordinate i e…

, Chapter: 3 -Problem: 13 >> Find the difference 3.00e?i ? 2.00e2i.
Answer Preview: 3 00e i …

, Chapter: 6 -Problem: 31 >> According to the Planck theory of black-body radiation, the radiant spectral emittance is given by the formulawhere ? is the wavelength of the radiation, h is Planck’s constant, kB is Boltzmann’s constant, c is the speed of light, and T is the temperature on the Kelvin scale. Treat T as a constant and find an equation that gives the wavelength of maximum emittance.
Answer Preview: At the maximum, this derivative vanishes. We place both terms in the square br…

, Chapter: 6 -Problem: 14 >> The volume of a cube is given byV = V(a) = a3,where a is the length of a side. Estimate the percent error in the volume if a 1.00% error is made in measuring the length, using the formulaCheck the accuracy of this estimate by comparing V(a) and V(1.0100a). Transcribed Image Text: dV AV = da (*) ??.
Answer Preview: V V V(1 0100) V(1 000) …

, Chapter: 13 -Problem: 8 >> Find the result of each operation on the given point (represented by Cartesian coordinates): (a) i? (2,4,6). (b) C?2(y)(1,1,1).
Answer Preview: a. i (2,4,6…

, Chapter: 15 -Problem: 8 >> Find the probability that x lies between ? ? 1.500? and ? + 1.500? for a Gaussian distribution.
Answer Preview: Let u = y/2 fraction …

, Chapter: 5 -Problem: 3 >> Carry out the algebraic manipulations to obtain the cubic equation in Eq. (5.9).
Answer Preview: K a = xy y = K a /x where we let y = [A ]/c . Since t…

, Chapter: 8 -Problem: 5 >> Find (?f/?x)y, and (?f /?y)x for each of the following functions, where a, b, and c are constants. (a) f = a cos2 (bxy), (b) f = a exp(?b(x2 + y2).
Answer Preview: a. b. af ax af …

, Chapter: 6 -Problem: 2 >> The natural logarithm of 1 + x is represented by the seriesUse the identityto find a series to represent 1/(1 + x). Transcribed Image Text: x2 In (1+ x) = x – x3 3 4 (valid for x? < 1 and x = 1).
Answer Preview: d In (1+x) dx …

, Chapter: 13 -Problem: 12 >> Find the 3 by 3 matrix that is equivalent in its action to each of the symmetry operators: (a) S?2(z). (b) C?2(x).
Answer Preview: a. x' = x y' …

, Chapter: 15 -Problem: 1 >> Assume the following discrete probability distribution:Find the mean and the standard deviation. Find the probability that x lies between Œ©xŒª ˆ’σx and Œ©xŒª ˆ’σx. Transcribed Image Text: 3 4 Pn 0.00193 0.1832 0.1054 0.3679 0.7788 1.0000 6. 10 0.7788 0.3679 0.1054 0.00193 0.18 Pn
Answer Preview: probability th…

, Chapter: 3 -Problem: 2 >> Find the value of the expression Transcribed Image Text: 3 (2+ 4) – 6 (7+ |–17)* + (/37=|-1)* (1+22)* – (1–71 + 6³)² + /T2+|-4| 3 2
Answer Preview: 3(2+4)-6(7+ | 171) + (37 | 11) (1 +2)4 (…

, Chapter: 7 -Problem: 19 >> Write Mathematica entries to obtain the following integrals:(a) ? cos3 (x)dx. (b) ?21 e5x2 dx.
Answer Preview: The correct v…

, Chapter: 11 -Problem: 13 >> Find the inverse Laplace transform of1/s(s2 + k2).
Answer Preview: We recognize k/(s 2 + k …

, Chapter: 13 -Problem: 1 >> Find the following commutators, where Dx = d/dx:(a) [D?x , sin (x)];(b) [D?3x ,x];
Answer Preview: a. b. d d [s…

, Chapter: 16 -Problem: 3 >> The van der Waals equation of state isFor carbon dioxide, a = 0.3640 Pa m6 molˆ’1 and b = 4.267 × 10ˆ’5 m3 molˆ’1. Find the pressure of 0.7500 mol of carbon dioxide if V = 0.0242 m3 and T = 298.0 K. Find the uncertainty in the pressure if the uncertainty in the volume is 0.00004 m3 and the uncertainty in the temperature is 0.5 K. Assume that the uncertainty in n is negligible. Find the pressure p
Answer Preview: From the ideal gas equation of state The difference between the value from the van der W…

, Chapter: 4 -Problem: 9 >> Find the angle between A and B if A = 1.00i + 2.00j + 1.00k and B = 1.00i ? 1.00k.
Answer Preview: A/B = 1 00 + 0 2 00 = 1 00 A = 1 00 + …

, Chapter: 8 -Problem: 13 >> (a) Find the local minimum of the functionf(x,y) = x2 + y2 + 2x.(b) Without using the method of Lagrange, find the constrained minimum subject to the constraint,x = ?y.(c) Find the constrained minimum using the method of Lagrange.
Answer Preview: a. At a relative extremum At the extremum 2x + 2 = 0 2y = 0 This corresponds to x …

, Chapter: 8 -Problem: 16 >> The average distance from the center of the sun to the center of the earth is 1.495 × 1011 m. The mass of the earth is 5.983 × 1024 kg, and the mass of the sun is greater than the mass of the earth by a factor of 332958. Find the magnitude of the force exerted on the earth by the sun and the magnitude of the force exerted on the sun by the earth.
Answer Preview: The magnitude of the force on the earth due to the …

, Chapter: 12 -Problem: 4 >> The frequency of vibration of the H2 molecule is 1.3194 × 1014 s?1. Find the value of the force constant.
Answer Preview: NAV = (1 0078 g mol-1) :-) 2(1 0078 g mol…

, Chapter: 13 -Problem: 18 >> Show that the properties of Eqs. (13.42) and (13.43) are obeyed by the particular matrices Transcribed Image Text: [10 1 123 -3 1 1 -2 -3 A=| 4 5 6 7 8 9 C=| 0 3 –2 B= 27 -7
Answer Preview: 123 0 2 2 -A-A 456 B = -3 12 789 1 -2-3 10 1 xC 03-2 27…

, Chapter: 4 -Problem: 7 >> (a) Find the Cartesian components of the position vector whose spherical polar coordinates are r = 2.00, ? = 90?, ? = 0?. Call this vector A. (b) Find the scalar product of the vector A from part a and the vector B whose Cartesian components are (1.00,2.00,3.00). (c) Find the angle between these two vectors.
Answer Preview: a. x = 2 00 y = 0 00 z = r cos () = 0 00 A = …

, Chapter: 6 -Problem: 32 >> The thermodynamic energy of a collection of N harmonic oscillators (approximate representations of molecular vibrations) is given by(a) Drawa rough sketch of the thermodynamic energy as a function of T.(b) The heat capacity of this system is given by C = dU/dT.Show that the heat capacity is given by(c) Find the limit of the heat capacity as T ? 0 and as T ? ?. Note that the limit as T ? ? is the s
Answer Preview: a. Here is an accurate graph. For this graph, we omit the constant …

, Chapter: 12 -Problem: 16 >> Find the speed of propagation of a traveling wave in an infinite string with the same mass per unit length and the same tension force as the violin string in the previous exercise.
Answer Preview: C = 1/2 =) (0 )…

, Chapter: 14 -Problem: 3 >> Solve the set of equations, using Cramer’s rule: 3x1 + x2 + x3 = 19, x1 ? 2x2 + 3x3 = 13, x1 + 2x2 + 2x3 = 23.Verify your result using Mathematica.
Answer Preview: x2 = 19 1 1 13 -2 3 23 22 311 1-23 122 3 19 1 1 13 3 …

, Chapter: 4 -Problem: 7 >> Find A × B if A = (0.00,1.00,2.00) and B = (2.00,1.00,0.00).
Answer Preview: A B = A B = i(A y B z A z B y ) …

, Chapter: 7 -Problem: 9 >> Evaluate the definite integral: ??/20 x sin (x2) dx.
Answer Preview: x sin (x 2 ) dx = -1/2 cos x …

, Chapter: 13 -Problem: 14 >> Give the function that results if the given symmetry operator operates on the given function for each of the following: (a) C?4(z)x2. (b) ??h x cos (x/y).
Answer Preview: a. C 4(z) …

, Chapter: 15 -Problem: 9 >> Find the expression for (v2x)1/2, the rootmean-square value of vx, and the expression for the standard deviation of vx.
Answer Preview: 1/2 m m (0) = (2747) x 1 exp (-24627) du, 2 2KBT …

, Chapter: 5 -Problem: 13 >> Verify the real solutions in the preceding example by substituting them into the equation.
Answer Preview: The equation is f(x) = x 4 5x 3 + 4x 2 …

, Chapter: 7 -Problem: 13 >> Solve the simultaneous equations to obtain the result of the previous example.
Answer Preview: A 1 + A 2 = 6 A 1 + 2A 2 = 30 …

, Chapter: 12 -Problem: 4 >> A harmonic oscillator has a mass m = 0.300 kg and a force constant k = 155 N m?1: (a) Find the period and the frequency of oscillation. (b) Find the value of the friction constant ? necessary to produce critical damping with this oscillator. Find the value of the constant ?1. (c) Construct a graph of the position of the oscillator as a function of t for the initial conditions z(0) = 0, vz(0) = 0.1
Answer Preview: a. b. For critical damping For crit…

, Chapter: 15 -Problem: 1 >> List as many sources of error as you can for some of the following measurements. Classify each one as systematic or random and estimate the magnitude of each source of error. (a) The measurement of the diameter of a copper wire using a micrometer caliper. (b) The measurement of the mass of a silver chloride precipitate in a porcelain crucible using a digital balance. (c) The measurement of the res
Answer Preview: a. Systematic : faulty calibration of the caliper 0 1 mm Ran…

, Chapter: 5 -Problem: 20 >> In the theory of black-body radiation, the following equationx = 5(1 ˆ’ eˆ’x)needs to be solved to find the wavelength of maximum spectral radiant emittance. The variable x iswhere λmax is the wavelength of maximum spectral radiant emittance, h is Planck€™s constant, c is the speed of light, kB is Boltzmann€™s constant, and T is the absolute temperature. Solve the equation numerically for a value
Answer Preview: A graph indicates a root near x = 4 96. By trial and e…

, Chapter: 6 -Problem: 12 >> Find the following derivatives:(a) d(yz)/dx, where y = ax2, z = sin (bx).(b) dP/dV, where(c) d?/d?, where Transcribed Image Text: nRT an? P = (V – nb) V2 *
Answer Preview: a. b. c. d(y…

, Chapter: 13 -Problem: 13 >> List the symmetry elements for (a) H2O (bent). (b) CO2 (linear).
Answer Preview: a. E, C 2(z) …

, Chapter: 16 -Problem: 11 >> Change the data set of Table 16.1 by adding a value of the vapor pressure at 70—¦C of 421 torr ± 40 torr. Find the least-squares line using both the unweighted and weighted procedures.Table 16.1 Transcribed Image Text: I (K) t (°C) Vapor pressure Expected error (Torr) (Torr) 25.00 298.15 55.9 3.0 3.
Answer Preview: After the point was added, the results were as follows: For the unweighted procedur…

, Chapter: 3 -Problem: 12 >> Find the sum of 4e3i and 5e2i.
Answer Preview: 4 003 00i 5 00 2 00i (4 00) [cos (3 00) + i si…

, Chapter: 6 -Problem: 16 >> Draw a rough graph of the functiony = y(x) = sin (|x|).Is the function differentiable at x = 0? Draw a rough graph of the derivative of the function.
Answer Preview: Your graph should look like thi…

, Chapter: 11 -Problem: 4 >> Show that the an coefficients for the series representing the function in the previous example all vanish.
Answer Preview: Integrate by parts: Let u = x, …

, Chapter: 14 -Problem: 1 >> Solve the set of simultaneous equations: 3x + 4y + 5z = 1, 4x ? 3y + 6z = 3, 7x + 2y ? 6z = 2.
Answer Preview: The inverse matrix x = 2, y = …

, Chapter: 6 -Problem: 5 >> Assume that y = 3.00x2 ? 4.00x + 10.00. If x = 4.000 and ?x = 0.500, find the value of ?y using Eq. (6.10). Find the correct value of ?y.
Answer Preview: Now we compute the correct va…

, Chapter: 6 -Problem: 36 >> Carry out Newton’s method to find the smallest positive root of the equation5.000x ? ex = 0.Do the calculation by hand and verify your result by use of Excel.
Answer Preview: A graph indicates a root near x = 0 300. we take x 0 = 0 300. We discontinu…

, Chapter: 11 -Problem: 9 >> Repeat the calculation of the previous example with a = 0.500 s?1. Show that a narrower line width occurs.
Answer Preview: The Fourier transform is: Here is the gr…

, Chapter: 15 -Problem: 14 >> A sample of seven individuals has the following set of annual incomes: $40,000, $41,000, $41,000, $62,000, $65,000, $125,000, and $650,000. Find the mean income, the median income, and the mode of this sample.
Answer Preview: The presence of two high-income members of the set cause the mean …

, Chapter: 4 -Problem: 8 >> From the definition, show that A × B = ?(B × A)
Answer Preview: This result follows immediately fro…

, Chapter: 7 -Problem: 4 >> Find the area bounded by the curve representing y = x3, the positive x axis, and the line x = 3.
Answer Preview: The positive x axis, and …

, Chapter: 11 -Problem: 2 >> Find the Fourier series to represent the functionConstruct a graph showing the first three terms of the series and the third partial sums. Transcribed Image Text: -1 if – L
Answer Preview: This is an even function, so the Fourier series will be a cosine series. We use the fac…

, Chapter: 15 -Problem: 7 >> For the lowest-energy state of a particle in a box of length L, find the probability that the particle will be found between L/4 and 3L/4.
Answer Preview: (probability) 2 0 7500L L JO 2500L To sin (x/L)dx -2 …

, Chapter: 3 -Problem: 4 >> The equation x2 + y2 + z2 = c2, where c is a constant, represents a surface in three dimensions. Express the equation in spherical polar coordinates. What is the shape of the surface?
Answer Preview: x 2 + y 2 + z …

, Chapter: 6 -Problem: 11 >> Find the inflection points for the function y = sin (x).
Answer Preview: The inflection points occur a…

, Chapter: 12 -Problem: 8 >> Substitute this trial solution into Eq. (12.43), using the condition of Eq. (12.54), and show that the equation is satisfied. Transcribed Image Text: ´d²z' dz (12.43) - 3 kz = m dt dt2
Answer Preview: The trial solution is z(t) = te t We substitute the trial so…

, Chapter: 14 -Problem: 4 >> Solve the set of equations, using Gauss–Jordan elimination: x1 + x2 = 6, 2x2 ? x3 = 1, x1 + 2x2 = 5. Use Mathematica to confirm your solution.
Answer Preview: The augmented matrix Multiply the first row by 2 Subtract t…

, Chapter: 3 -Problem: 19 >> Estimate the number of grains of sand on the beaches of the major continents of the earth. Exclude islands and inland bodies of water. You should come up with a number somewhere near Avogadro’s number.
Answer Preview: Assume that the earth has seven continents with an average radius of 2000 km…

, Chapter: 7 -Problem: 11 >> Evaluate the integralwithout using a table of integrals. Transcribed Image Text: esin (0) cos (0)d0, *IT/2 0,
Answer Preview: We let y = sin () dy = …

, Chapter: 11 -Problem: 6 >> Derive the orthogonality relation expressed above.
Answer Preview: We have looked up the integrals in the table of definite integral…

, Chapter: 14 -Problem: 4 >> Find the value of x1that satisfies the set of equations Transcribed Image Text: 10 X1 1-1 1 1 1 -1 1 [1 1 6. 1 ?2 4 ?? -1] [ x4 ||
Answer Preview: The complete solution is …

, Chapter: 5 -Problem: 1 >> Show by substitution that the quadratic formula provides the roots to a quadratic equation.
Answer Preview: For simplicity. we assume …

, Chapter: 8 -Problem: 1 >> A certain nonideal gas is described by the equation of statewhere T is the temperature on the Kelvin scale, Vm is the molar volume, P is the pressure, and R is the gas constant. For this gas, the second virial coefficient B2 is given as a function of T byFind (ˆ‚ P/ˆ‚Vm)T and (ˆ‚ P/ˆ‚T )Vm and an expression for dP. Transcr
Answer Preview: P = RT (SP), - - RZ av T V2 m dP RT + V…

, Chapter: 13 -Problem: 10 >> Find C?2(x)(1,2, ? 3).
Answer Preview: C 2(z) …

, Chapter: 15 -Problem: 5 >> Calculate the mean and standard deviation of the Gaussian distribution, showing that ? is the mean and that ? is the standard deviation.
Answer Preview: (x) = = = = = = = xe-(x-u/2 di 1-00 10 +u)e-w2/26d…

, Chapter: 3 -Problem: 7 >> Find the values of the plane polar coordinates that correspond to x = 3.00, y = 4.00.
Answer Preview: P p = 9 00+ …

, Chapter: 8 -Problem: 13 >> Complete the formula Transcribed Image Text: = ), as as +? ?,? ?V P,n
Answer Preview: (V). - (V) + +? n T, n d S s = ( ) …

, Chapter: 12 -Problem: 16 >> A pendulum of length L oscillates in a vertical plane. Assuming that the mass of the pendulum is all concentrated at the end of the pendulum, show that it obeys the differential equationwhere g is the acceleration due to gravity and Ï• is the angle between the pendulum and the vertical. This equation cannot be solved exactly. For small oscillations such thatsin (Ï•) ‰ˆ Ï•,find the solution to the
Answer Preview: The real solution to this e…

, Chapter: 16 -Problem: 5 >> The vibrational contribution to the molar heat capacity of a gas of nonlinear molecules is given in statistical mechanics by the formula i=1 " style="" class="fr-fic fr-dib">where ui = hνi/kBT . Here νi is the frequency of the ith normal mode of vibration, of which there are 3n ˆ’ 6 if n is the number of nuclei in the molecule, h is Planck€™s constant, kB is Boltzmann€™s constant, R is the ideal
Answer Preview: u = 4 = 43 uz = 4 588 hv2 KBT = = hvi kB T || (6 6260755 10-34 J s)(4 78 1013 s-1) (1 3806568 x 10-2…

, Chapter: 1 -Problem: 19 >> The volume of a sphere is equal to 4/3? r3 where r is the radius of the sphere. Assume that the earth is spherical with a radius of 3958.89 miles. (This is the radius of a sphere with the same volume as the earth, which is flattened at the poles by about 30 miles.) Find the volume of the earth in cubic miles and in cubic meters. Using a value of ? with at least six digits give the correct number o
Answer Preview: 3 V = = 4 - 3 4 3 = 2 59508 …

, Chapter: 6 -Problem: 6 >> Find the first and second derivatives of the following functions: (a) y = y(x) = 3x3 ln (x). (b) y = y(x) = 1/(c ? x2), where c is a constant. (c) y = y(x) = ce?a cos (bx), where a, b, and c are constants.
Answer Preview: a. b. c. d…

, Chapter: 6 -Problem: 10 >> Find the following derivatives and evaluate them at the points indicated: (a) (dy/dx)x=0 if y = sin (bx), where b is a constant. (b) (d f /dt)t=0 if f = Ae?kt, where A and k are constants.
Answer Preview: a. b. dy dx …

, Chapter: 11 -Problem: 4 >> Find the one-sided Fourier cosine transform of the function f(x) = xe?ax.
Answer Preview: where have used …

, Chapter: 16 -Problem: 3 >> In the cryoscopic determination of molar mass, the molar mass in kg molˆ’1is given bywhere W is the mass of the solvent, w is the mass of the unknown solute, ΔTf is the amount by which the freezing point of the solution is less than that of the pure solvent, and Kf and Kf are constants characteristic of the solvent. A sample of an unknown substance was dissolved in benzene, for which Kf = 5.12 K
Answer Preview: We assume that errors in K f and k f are negligible. The principal sou…

, Chapter: 1 -Problem: 4 >> Find the speed of light in miles per hour.
Answer Preview: 1 in 0 0254 m 1 …

, Chapter: 4 -Problem: 6 >> Find the magnitude of the vector A = (? 3.00,4.00,? 5.00).
Answer Preview: A = 9 0…

, Chapter: 8 -Problem: 8 >> Find (?2 f /?x2)y, (?2 f /?x?y), (?2 f /?y?x), and (?2 f /? y2), for each of the following functions,where a, b, and c are constants (a) f = (x2 + y2)?1, (b) f = sin (xy).
Answer Preview: a. b. af (7), - a2 f 2 (31) - - (x + y)3 (…

, Chapter: 11 -Problem: 7 >> Find the one-sided Fourier sine transform of the function ae?bx.
Answer Preview: where we have us…

, Chapter: 15 -Problem: 8 >> Find the expectation values for px and p2x for the lowest-energy state of our particle in a box of length L. Find the standard deviation.
Answer Preview: This vanishing value of the momentum corresponds to the fact that the …

, Chapter: 2 -Problem: 14 >> Determine the number of significant digits in sin (95.5?).
Answer Preview: We calculate sin (95 45 ) and sin (95 45 ). Using a calculator that displays 8 dig…

, Chapter: 5 -Problem: 16 >> Find the real root of the equationx2 ? e?x = 0.
Answer Preview: The s…

, Chapter: 6 -Problem: 14 >> Investigate the limitfor any finite value of n. Transcribed Image Text: lim (x"e*) X00
Answer Preview: Additional applications of lHpitals rule give decreas…

, Chapter: 12 -Problem: 8 >> A tank contains a solution that is rapidly stirred, so that it remains uniform at all times. A solution of the same solute is flowing into the tank at a fixed rate of flow, and an overflow pipe allows solution from the tank to flow out at the same rate. If the solution flowing in has a fixed concentration that is different from the initial concentration in the tank, write and solve the differentia
Answer Preview: dn/dt = A Bn This is an inhomogeneous equation. The comp…

, Chapter: 14 -Problem: 9 >> Find the eigenvalues and eigenvectors of the matrix Transcribed Image Text: 111 1 1
Answer Preview: The eigenvalues are 0,3. The eigenvect…

, Chapter: 3 -Problem: 14 >> Find the square roots of x = 4.00 + 4.00i. Sketch an Argand diagram and locate the roots on it.
Answer Preview: z = re i r = 32 00 = 5 657 = arctan (1 00) = 0 785398 rad = 45 00 To sketch the Argand dia…

, Chapter: 5 -Problem: 16 >> Solve the set of equations 3x + 2y = 40, 2x ? y = 10.
Answer Preview: We multiply the second equation by 2 and add it …

, Chapter: 7 -Problem: 8 >> (a) By drawing rough graphs, satisfy yourself that ?1 is even about the center of the box. That is, ?1(x) = ?1(a ? x). Satisfy yourself that ?2 is odd about the center of box. (b) Draw a rough graph of the product ?1?2 and satisfy yourself that the integral of this product from x = 0 to x = a vanishes.
Answer Preview: a. For the purpose of the graphs, we let a = …

, Chapter: 13 -Problem: 20 >> Show that AA?1 = E and that A?1A = E for the matrices of the preceding example.
Answer Preview: Mathematica and other software packages can find a …

, Chapter: 15 -Problem: 7 >> Assume that a random variable, x, is governed by the probability distribution (a version of the Lorentzian function)where x ranges from ˆ’10.000 to 10.000. (a) Find the mean value of x and its variance and standard deviation. (b) Find the probability that x lies between Œ©xŒª ˆ’σx and Œ©xŒª +σx. Distribution The word "distribution" has several meanings in the financial w
Answer Preview: Here is a graph of the unnormalized function: a. We firs…

, Chapter: 1 -Problem: 8 >> In the USA, footraces were once measured in yards and at one time, a time of 10.00 s for this distance was thought to be unattainable. The best runners now run 100 m in 10 s or less. Express 100 m in yards, assuming three significant digits. If a runner runs 100.0 m in 10.00 s, find his time for 100 yd, assuming a constant speed.
Answer Preview: (100 0 m) (10 00 …

, Chapter: 5 -Problem: 23 >> The Dieterici equation of state isPea/Vm RT (Vm ? b) = RT,where P is the pressure, T is the temperature, Vm is the molar volume, and R is the ideal gas constant. The constant parameters a and b have different values for different gases. For carbon dioxide, a = 0.468 Pa m6 mol?2, b = 4.63 × 10?5 m3 mol?1. Without linearization, find the molar volume of carbon dioxide if T = 298.15 K and P = 10.000
Answer Preview: Divide this equation by (1 01325 10 6 Pa) and ignore the units Using tria…

, Chapter: 13 -Problem: 9 >> Find the result of each operation on the given point (represented by Cartesian coordinates): (a) C?3(z)(1,1,1). (b) S?4(z)(1,1,1).
Answer Preview: a. b. S 4(z…

, Chapter: 1 -Problem: 1 >> Find the number of inches in 1.000 m.
Answer Preview: (1 000 …

, Chapter: 4 -Problem: 11 >> A boy is swinging a weight around his head on a rope. Assume that the weight has a mass of 0.650 kg, that the rope plus the effective length of the boy’s arm has a length of 1.45 m, and that the weight makes a complete circuit in 1.34 s. Find the magnitude of the angular momentum, excluding the mass of the rope and the boy’s arm. If the mass is moving counterclockwise in a horizontal circle, what
Answer Preview: 2 (1 45 m) 1 34 s …

, Chapter: 12 -Problem: 9 >> An nth-order chemical reaction with one reactant obeys the differential equationwhere c is the concentration of the reactant and k is a constant. Solve this differential equation by separation of variables. If the initial concentration is c0 moles per liter, find an expression for the time required for half of the reactant to react.
Answer Preview: For half of the original amount to …

, Chapter: 1 -Problem: 11 >> The Rankine temperature scale is defined so that the Rankine degree is the same size as the Fahrenheit degree, and absolute zero is 0 ?R, the same as 0 K: (a) Find the Rankine temperature at 0.00 ?C. (b) Find the Rankine temperature at 0.00 ?F.
Answer Preview: a. b. 0 …

, Chapter: 5 -Problem: 24 >> Determine which, if any, of the following sets o equations is inconsistent or linearly dependent. Draw a graph for each set of equations, showing both equations. Find the solution for any set that has a unique solution.(a) x + 3y = 4, 2x + 6y = 8.(b) 2x + 4y = 24, x + 2y = 8.(c) 3x1 + 4x2 = 10, 4x1 ? 2x2 = 6.
Answer Preview: a. These equations are linearly dependent, since the second equation is equal to twice …

, Chapter: 13 -Problem: 2 >> Find the following commutators, where Dx = d/dx: (a) [D?3x, x2]; (b) [D?2x, f(x)].
Answer Preview: a. b. [22x] 5 = 232 1 1 - 24/1 …

, Chapter: 2 -Problem: 4 >> Find the value of the hyperbolic sine, cosine, and tangent for x = 0 and x = ?/2. Compare these values with the values of the ordinary (circular) trigonometric functions for the same values of the independent variable.
Answer Preview: Here are two table of value…

, Chapter: 5 -Problem: 14 >> Use the NSolve statement in Mathematica to find the numerical values of the roots of the equationx3 + 5.000x ? 42.00 = 0.Use the FindRoot statement to find the real root of the same equation.
Answer Preview: The result is Use the Find Ro…

, Chapter: 12 -Problem: 15 >> Radioactive nuclei decay according to the same differential equation which governs first-order chemical reactions, Eq. (12.71). In living matter, the isotope14C is continually replaced as it decays, but it decays without replacement beginning with the death of the organism. The half-life of the isotope (the time required for half of an initial sample to decay) is 5730 years. If a sample of charcoa
Answer Preview: The rate of disintegrations is prop…

, Chapter: 2 -Problem: 2 >> Use Excel or Mathematica to construct a graph representing the functiony(x) = x3 ? 2x2 + 3x + 4.
Answer Preview: Use Excel or Mathematica or s…

, Chapter: 3 -Problem: 17 >> If wefind R(z),I (z), r, and ?. Transcribed Image Text: 3+2i 4 + 5i 2.
Answer Preview: The principal value of the arctangent is in the fourth quadrant, equal to 0 …

, Chapter: 13 -Problem: 20 >> Test the following matrices for singularity. Find the inverses of any that are nonsingular. Multiply the original matrix by its inverse to check your work. Use Mathematica to check your work.a.b. Transcribed Image Text: 0 1 2 3 4 5 678 6 8 1 7 3 2 4 6 –9
Answer Preview: a. b. Sin…

, Chapter: 1 -Problem: 3 >> Convert the following numbers to scientific notation: (a) 0.00000234. (b) 32.150.
Answer Preview: a. 0 00000…

, Chapter: 3 -Problem: 15 >> Find the four fourth roots of 3.000i.
Answer Preview: 3 000i 3 000i = 3 000i/2 3 000…

, Chapter: 12 -Problem: 13 >> Solve the equation (4x + y)dx + x dy = 0.
Answer Preview: Check for exactness The Pfaffian form is the differential of a function f …

, Chapter: 3 -Problem: 9 >> Express the following complex numbers in the form r ei?: (a) 4.00 + 4.00i. (b) ?1.00.
Answer Preview: a. b. z = 1 = e i …

, Chapter: 6 -Problem: 7 >> Find the first and second derivatives of the following functions: (a) y = (1/x)(1/(1 + x)). (b) f = f(v) = ce?mv2/(2kT) where m, c, k, and T are constants.
Answer Preview: a. b. dy dx = ( 72 ) ( + x) ( 7 …

, Chapter: 11 -Problem: 6 >> Find the Fourier transform of the function exp (?(x ? x0)2).
Answer Preview: Let u = x x 0 , x = u + x 0 The integrals with sin (u) vanish since the integrands are odd fun…

, Chapter: 3 -Problem: 10 >> Express the following complex numbers in the form x + iy: (a) z = e3?i/2. (b) z = 3e?i/2.
Answer Preview: a. x = r cos () = 3 cos (/2) = …

, Chapter: 6 -Problem: 7 >> Carry out Newton’s method by hand to find the smallest positive root of the equation1.000x2 ? 5.000x + 1.000 = 0.
Answer Preview: A graph indicates a root near x = 0 200. we take x 0 = 0 2000…

, Chapter: 11 -Problem: 5 >> Find the Fourier cosine series for the even functionf(x) = |x| for ? L < x < L.Sketch a graph of the periodic function that is represented by the series.
Answer Preview: This is an even function, …

, Chapter: 1 -Problem: 6 >> Calculate the following to the proper numbers of significant digits: (a) 17.13 + 14.6751 + 3.123 + 7.654 ? 8.123. (b) ln (0.000123).
Answer Preview: a. 17 13 + 14 6751 + 3 123 + 7 654 8 …

, Chapter: 13 -Problem: 4 >> If A? = x + d dx, find A?3.
Answer Preview: + ardu d x + dx dx2 -x + d …

, Chapter: 1 -Problem: 12 >> The volume of a sphere is given bywhere V is the volume and r is the radius. If a certain sphere has a radius given as 0.005250 m, find its volume, specifying it with the correct number of digits. Calculate the smallest and largest volumes that the sphere might have with the given information and check your first answer for the volume.
Answer Preview: The rule of thumb gives four significant digits, but the calcul…

, Chapter: 12 -Problem: 10 >> Find the solution to the differential equation Transcribed Image Text: d²y - 2y = –xe*. dx dy dx2
Answer Preview: This is an inhomogeneous equation. The complementary equation is With t…

, Chapter: 2 -Problem: 13 >> Verify the trigonometric identitycos (2x) = 1 ? 2 sin2 (x)for x = 0.50000 rad. Use as many digits as your calculator will display and check for round-off error.
Answer Preview: cos (1 00000) = 0 54030230587 1 2 sin …

, Chapter: 13 -Problem: 5 >> Find an expression for B?2 if B? = x(d/dx) and find B?2 f if f = bx4.
Answer Preview: d B f = = (x 2 ) 1 = x + ( ) …

, Chapter: 1 -Problem: 2 >> Express the following in terms of SI base units. The electron volt (eV), a unit of energy, equals 1.6022 × 10?19 J:(a) 13.6 eV. (b) 24.17 mi. (c) 55 mi h?1. (d) 7.53 nm ps?1.
Answer Preview: a. b. c. d. (13 …

, Chapter: 11 -Problem: 3 >> Show that Eq. (11.15) is correct. Transcribed Image Text: NT X f (x) sin bn (11.15) dx -L
Answer Preview: By orthogonality, all of the integrals vanish e…

, Chapter: 2 -Problem: 5 >> Without using a calculator or a table of logarithms, find the following: (a) ln (100.000), (b) ln (0.0010000), (c) log10 (e)
Answer Preview: Without using a calculator or a table of logarithms, find …

, Chapter: 13 -Problem: 7 >> Find the eigenfunction of the Hamiltonian operator for motion in the x direction if V(x) = E0 = constant.
Answer Preview: where we let If E E 0 the real solution is real = c 1 sin (x) + c 2 …

, Chapter: 3 -Problem: 12 >> Write a complex number in the form x + iy and show that the product of the number with its complex conjugate is real and nonnegative.
Answer Preview: (x + iy)(x iy) = x 2 + i …

, Chapter: 12 -Problem: 7 >> If z(0) = z0 and if vz(0) = 0, express the constants b1 and b2 in terms of z0.
Answer Preview: z(t) = [b cos (wt) + b2 sin (wt)] et/…

, Chapter: 2 -Problem: 3 >> Generate the negative logarithms in the short table of common logarithms.
Answer Preview: X 1 10 100 1000 y = log10 (x) 0 1 2 3 X 0 1 0 01 0 …

, Chapter: 13 -Problem: 11 >> Find the result of each operation on the given point (represented by Cartesian coordinates):(a) C?2(z) i? (1,1,1). (b)  i? C?2(z)(1,1,1).
Answer Preview: a. C 2(z) i (1,1,1) …

, Chapter: 2 -Problem: 12 >> Sketch graphs of the arcsine function, the arccosine function, and the arctangent function. Include only the principal values.
Answer Preview: Here are acc…

, Chapter: 12 -Problem: 15 >> A certain violin string has a mass per unit length of 20.00 mg cm?1 and a length of 55.0 cm. Find the tension force necessary to make it produce a fundamental tone of A above middle C (440 oscillations per second = 440 s?1 = 440 Hz).
Answer Preview: T = P V= = nc 2Lv -p (40) …

, Chapter: 2 -Problem: 2 >> Using the data from the previous problem, construct a graph of the natural logarithm of the vapor pressure as a function of the reciprocal of the Kelvin temperature. Why might this graph be more useful than the graph in the previous problem? Previous problemTemperature (?C) .......................Vapor pressure (torr) 25.00 ..........................................................55.9 30.00......
Answer Preview: This graphmight be more useful than the first graph because the functio…

, Chapter: 11 -Problem: 8 >> Find the one-sided Fourier cosine transform of the function a/(b2 + t2).
Answer Preview: where we have used E…

, Chapter: 1 -Problem: 9 >> Find the average length of a century in seconds and in minutes. Use the rule that a year ending in 00 is not a leap year unless the year is divisible by 400, in which case it is a leap year. Therefore, in four centuries there will by 97 leap years. Find the number of minutes in a microcentury.
Answer Preview: Number of days in 400 years = (365 d)(400 y) + 97 d = 146097 d …

, Chapter: 12 -Problem: 6 >> Show that e?1t does satisfy the differential equation.
Answer Preview: Divide by e 1t and sub…

, Chapter: 3 -Problem: 13 >> If z = (3.00 + 2.00i)2, find R(z),I (z),r, and ?.
Answer Preview: z = 9 00 + 6 00i 4 00 = 5 …

, Chapter: 5 -Problem: 11 >> Write an Excel worksheet that will convert a list of distance measurements in meters to miles, feet, and inches. If the length in meters is typed into a cell in column A, let the corresponding length in miles appear on the same line in column B, the length in feet in column C, and the length in inches in column C.
Answer Preview: The length in inches in column C. Here is the res…

, Chapter: 12 -Problem: 3 >> Show that the function of Eq. (12.25) satisfies Eq. (12.12).z = b1 cos(ωt) + b2 sin(ωt). (12.25) Transcribed Image Text: d²z (12.12) -kz. ? dt2
Answer Preview: m z = b cos (wt) + b sin (wt…

, Chapter: 15 -Problem: 12 >> Evaluate the most probable speed for nitrogen molecules at 298.15 K.
Answer Preview: Ump 2RT /2 M 2(…

, Chapter: 1 -Problem: 15 >> Some elementary chemistry textbooks give the value of R, the ideal gas constant, as 0.0821l atm K?1mol?1. (a) Obtain the value of R in l atm K?1 mol?1 to five significant digits. (b) Calculate the pressure in atmospheres and in (N m?2 Pa) of a sample of an ideal gas with n = 0.13678 mol, V = 10.000 l, T = 298.15 K.
Answer Preview: a. b. 1 Pa m -) () …

, Chapter: 4 -Problem: 15 >> According to the Bohr theory of the hydrogen atom, the electron in the atom moves around the nucleus in one of various circular orbits with radius r = a0n2, where a0 is a distance equal to 5.29 × 10?11 m, called the Bohr radius, and n is a positive integer. The mass of the electron is 9.109 × 10?31 kg. According to the theory, the magnitude of the angular momentum, L, is equal to nh/2?, where h is
Answer Preview: Since the orbit is circular, the position vector and t…

, Chapter: 12 -Problem: 17 >> Obtain the solution of Eq. (12.112) in the case of critical damping, using Laplace transforms. Transcribed Image Text: d²z' ? dz +"z = z2) +z") + z = 0, m dt k k (1) dt2 ? ? ? (12.112)
Answer Preview: The equation is with the condition. From the example in Chapter 11 we …

, Chapter: 15 -Problem: 2 >> Assume that a certain biased coin has a 51.0% probability of coming up “heads” when thrown. (a) Find the probability that in 10 throws five “heads” will occur. (b) Find the probability that in 10 throws seven “heads” will occur.
Answer Preview: a. b. probabili…

, Chapter: 1 -Problem: 1 >> Take a few fractions, such as 2/3, 4/9, or 3/7, and represent them as decimal numbers, finding either all of the nonzero digits or the repeating pattern of digits.
Answer Preview: 213 19317 …

, Chapter: 5 -Problem: 10 >> The following data were taken for the thermal decomposition of N2O3at a constant temperature:Using Excel, make three graphs: one with ln ([N2O3]) as a function of t, one with 1/[N2O3] as a function of t, and one with 1/[N2O3]2 as a function of t. Determine which graph is most nearly linear. If the first graph is most nearly linear, the reaction is first order; if the second graph is most nearly li
Answer Preview: Because of experimental error, it is a little …

, Chapter: 12 -Problem: 20 >> An object of mass m is subjected to a gradually increasing force given by F0(1?e?bt) where a and b are constants. Solve the equation of motion of the particle. Find the particular solution for the case that x(0) = 0 and dx/dt = 0 at t = 0.
Answer Preview: In the case that x(0) = 0 and dx/dt = 0 …

, Chapter: 3 -Problem: 1 >> Write the following expression in a simpler form: Transcribed Image Text: (x² + 2x)? – x²(x – 2)² + 12x4 6x3 + 12x4 B =
Answer Preview: B = || B = (x + 2x) = x(x - 2) + 12…

, Chapter: 5 -Problem: 7 >> Find approximately the smallest positive root of the equationtan (x) ? x = 0.
Answer Preview: Since tan (x) is larger than x in the entire range from x = 0 to …

, Chapter: 13 -Problem: 6 >> Show that the solution in the previous example satisfies the original equation.
Answer Preview: dy dx2 -3dy dx +2y=0 …

, Chapter: 3 -Problem: 3 >> (a) Find x and y if ? = 6.00 and ? = ?/6 rad. (b) Find ? and ? if x = 5.00 and y = 10.00.
Answer Preview: a. x = (6 00) cos (/6) = (6 00)(0 …

, Chapter: 5 -Problem: 2 >> For hydrocyanic acid (HCN), Ka = 4.9 × 10?10 at 25 ?C. Find [H+] if 0.1000 mol of hydrocyanic acid is dissolved in enough water to make 1.000 l. Assume that activity coefficients are equal to unity and neglect hydrogen ions from water.
Answer Preview: The neglect of hydrogen ions from water is acceptable, since n…

, Chapter: 13 -Problem: 11 >> Find S?2(y)(3,4,5).
Answer Preview: S 2(y…

, Chapter: 2 -Problem: 6 >> For a positive value of b find an expression in terms of b for the change in x required for the function ebx to double in size.
Answer Preview: f(x + x) f(x…

, Chapter: 6 -Problem: 1 >> Using graph paper plot the curve representing y = sin (x) for values of x lying between 0 and ?/2 radians. Using a ruler, draw the tangent line at x = ?/4. By drawing a right triangle on your graph and measuring its sides, find the slope of the tangent line.
Answer Preview: Your graph should look like …

, Chapter: 13 -Problem: 3 >> The components of the angular momentum correspond to the quantum-mechanical operators:These operators do not commute with each other. Find the commutator [LÌ‚x, LÌ‚y]. Transcribed Image Text: Îx Îy ?? y- ? ?? i ?z az ? ?? ?? (,
Answer Preview: We can now apply Eulers reciprocity relation to cancel all of the terms b…

, Chapter: 1 -Problem: 5 >> A furlong is exactly one-eighth of a mile and a fortnight is exactly 2 weeks. Find the speed of light in furlongs per fortnight, using the correct number of significant digits.
Answer Preview: (299792458 m s) ) ( X 1 in X 1 …

, Chapter: 5 -Problem: 8 >> Using a graphical procedure, find the most positive real root of the quartic equation:x4 ? 4.500x3 ? 3.800x2 ? 17.100x + 20.000 = 0.
Answer Preview: The curve representing this function crosses the x axis in only two places. This indicates th…

, Chapter: 12 -Problem: 2 >> Find the general solution to the differential equation Transcribed Image Text: d²y dy + 2y = 0. 3- dx2 dx
Answer Preview: Substitution of the trial solution y = e x gives the equation 2 e …

, Chapter: 1 -Problem: 14 >> The value of an angle is given as 31?. Find the measure of the angle in radians. Find the smallest and largest values that its sine and cosine might have and specify the sine and cosine to the appropriate number of digits.
Answer Preview: (31) 2 rad 360 = 0 54 rad s…

, Chapter: 3 -Problem: 11 >> Find the complex conjugate of the quantity e2.00i + 3ei?.
Answer Preview: e2 00i + 3eir (e2 00i + …

, Chapter: 12 -Problem: 14 >> Show that 1/y2 is an integrating factor for the equation in the previous example and show that it leads to the same solution.
Answer Preview: After multiplication by 1/y 2 the Pfaffian form is This …

, Chapter: 1 -Problem: 13 >> The volume of a right circular cylinder is given byV = ?r2h,where r is the radius and h is the height. If a right circular cylinder has a radius given as 0.134 m and a height given as 0.318 m, find its volume, specifying it with the correct number of digits. Calculate the smallest and largest volumes that the cylinder might have with the given information and check your first answer for the volume
Answer Preview: V = (0 134 m) 2 (0 318 m) …

, Chapter: 5 -Problem: 21 >> Solve the simultaneous equations by hand, using the method of substitution: x2 + x + 3y = 15, 3x + 4y = 18.Use Mathematica to check your result. Since the first equation is a quadratic equation, there will be two solution sets.
Answer Preview: Substitute this into …

, Chapter: 12 -Problem: 18 >> The differential equation for a secondorder chemical reaction without back reaction iswhere c is the concentration of the single reactant and k is the rate constant. Set up an Excel spreadsheet to carry out Euler€™s method for this differential equation. Carry out the calculation for the initial concentration 1.000 mol lˆ’1, k = 1.000 l molˆ’1 sˆ’1 for a time of 2.000 s and for Δt = 0.100 s. Comp
Answer Preview: Here are the numbers from the spreadsheet The resu…

, Chapter: 2 -Problem: 1 >> Enter a formula into cell D2 that will compute the mean of the numbers in cells A2, B2, and C2.
Answer Preview: = (…

, Chapter: 5 -Problem: 22 >> Stirling's approximation for ln (N!) isDetermine the validity of this approximation and of the less accurate versionln (N!) ? N ln (N) ? Nfor several values of N up to N = 100. Use a calculator, Excel, or Mathematica. Transcribed Image Text: In (N!) 1 In (27 N) + N In (N) – N. =
Answer Preview: Here are a few values N In (N!) In (27N) +N …

, Chapter: 13 -Problem: 1 >> Find the eigenfunctions and eigenvalues of the operator i d/dx, where i = ? ?1.
Answer Preview: i df/dx = a f Separate the variables: The single eigenfun…

, Chapter: 1 -Problem: 6 >> The distance by road from Memphis, Tennessee to Nashville, Tennessee is 206 mi. Express this distance in meters and in kilometers.
Answer Preview: 12 in F) (THT) …

, Chapter: 13 -Problem: 32 >> Show that the 1 by 1 matrices (scalars) in Eq. (13.74) obey the same multiplication table as does the group of symmetry operators. Transcribed Image Text: Ê + 1 ?z + 1 ? ? 1, ?a + –1 ?p ? -1 ?c +? -1. (13.74)
Answer Preview: Since the elements E 1 C 3 1 C 2 3 1, the product of any two of …

, Chapter: 3 -Problem: 2 >> Manipulate the van der Waals equation into an equation giving P as a function of T and Vm.
Answer Preview: Manipulate the van der Waals equation into an expression for P in …

, Chapter: 11 -Problem: 5 >> Find the one-sided Fourier sine transform of the function f(x) = e?ax/x.
Answer Preview: where have used Eq…

, Chapter: 3 -Problem: 5 >> Find the Cartesian coordinates of the point whose cylindrical polar coordinates are ? = 25.00, ? = 60.0?, z = 17.50
Answer Preview: x = cos () = 25 00 cos (6…

, Chapter: 9 -Problem: 8 >> A flywheel of radius R has a distribution of mass given by m(?) = a? + b, where ?, is the distance from the center, a and b are constants, and m(?) is the mass per unit area as a function of ?. The flywheel has a circular hole in the center with radius r. Find an expression for the moment of inertia, defined by I = ?? (?)?2 dA = ?Rr ?2?0 m(?)?2? d? d?. Distribution The wor
Answer Preview: R 2 I = [[ m(p)pdA = f* f** m(p)pp &$ …

, Chapter: 2 -Problem: 11 >> For an angle that is nearly as large as ?/2, find an approximate equality similar to Eq. (2.38) involving (?/2) ? ?, cos (?), and cot (?).
Answer Preview: Construct a right triangle with angle with the angle (/2) , where is small. The triangle is tall, …

, Chapter: 12 -Problem: 10 >> If zc(t) is a general solution to the complementary equation and zp(t) is a particular solution to the inhomogeneous equation, show that zc + zpis a solution to the inhomogeneous equation of Eq. (12.59). Transcribed Image Text: d z + f2(t)- dt3 d?z dz + fi(t) (12.59) = g(t), f3(t)- dt2 dt
Answer Preview: Since z c satisfies the complemen…

, Chapter: 2 -Problem: 12 >> Verify the trigonometric identity sin (x + y) = sin (x) cos (y) + cos (x) sin (y)for the angles x = 1.0000 rad, y = 2.00000 rad. Use as many digits as your calculator will display and check for round-off error.
Answer Preview: sin (3 00000) = 0 14112000806 sin …

, Chapter: 2 -Problem: 3 >> A reactant in a first-order chemical reaction without back reaction has a concentration governed by the same formula as radioactive decay,where [A]0 is the concentration at time t = 0,[A]t is the concentration at time t, and k is a function of temperature called the rate constant. If k = 0.123 sˆ’1 find the time required for the concentration to drop to 21.0% of its initial value.
Answer Preview: In ([A]o/…

, Chapter: 2 -Problem: 13 >> Make a graph of tanh(x) and coth(x) on the same graph for values of x ranging from 0.1 to 3.0.
Answer Preview: 11 10 9 …

, Chapter: 1 -Problem: 10 >> A light year is the distance traveled by light in one year: (a) Express this distance in meters and in kilometers. Use the average length of a year as described in the previous problem. How many significant digits can be given? (b) Express a light year in miles. Previous ProblemFind the average length of a century in seconds and in minutes. Use the rule that a year ending in 00 is not a leap year
Answer Preview: a. Since the number of significant digits in the number …

, Chapter: 2 -Problem: 5 >> Express the following with the correct number of significant digits. Use the arguments in radians: tan (0.600) sin (0.100) (a) tan(0.600). (b) sin(0.100). (c) cosh(12.0). (d) sinh(10.0).
Answer Preview: a. tan (0 600) = 0 684137 tan (0 5995) = 0 683403 tan …

, Chapter: 2 -Problem: 9 >> Using a calculator and displaying as many digits as possible, find the values of the sine and cosine of 49.500?. Square the two values and add the results. See if there is any round-off error in your calculator.
Answer Preview: sin (49 500) = 0 7604…

, Chapter: 2 -Problem: 15 >> Sketch rough graphs of the following functions. Verify your graphs using Excel or Mathematica.(a) e?x/5 sin (x). (b)

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