A round cold-drawn 1045 steel rod has a mean strength Sy 5

Select a mechanical component from Part 3 of this book (roller bearings, springs, etc.), go to your universitys library or the appropriate internet website, and, using the Thomas Register of American Manufacturers (www.thomasnet.com), report on the information obtained on five manufacturers or suppliers

Select a mechanical component from Part 3 of this book (roller bearings, springs, etc.), go to the Internet, and, using a search engine, report on the information obtained on five manufacturers or suppliers.

Select an organization listed in Sec. 16, go to the Internet, and list what information is available on the organization.

Go to the Internet and connect to the NSPE website (www.nspe.org/ethics). Read the history of the Code of Ethics and briefly discuss your reading.

Go to the Internet and connect to the NSPE website (www.nspe.org/ethics). Read the complete NSPE Code of Ethics for Engineers and briefly discuss your reading.

Go to the Internet and connect to the NSPE website (www.nspe.org/ethics). Go to Ethics Resources and review one or more of the topics given. A sample of some of the topics may be: (a) Education Publications (b) Ethics Case Search (c) Ethics Exam (d) FAQ (e) Milton Lunch Contest ( f) Other Resources (g) You Be the Judge Briefly discuss your reading.

Estimate how many times more expensive it is to grind a steel part to a tolerance of 60.0005 in versus turning it to a tolerance of 60.003 in.

The costs to manufacture a part using methods A and B are estimated by CA 5 10 1 0.8 P and CB 5 60 1 0.8 P 2 0.005 P2 respectively, where the cost C is in dollars and P is the number of parts. Estimate the breakeven point.

A cylindrical part of diameter d is loaded by an axial force P. This causes a stress of PyA, where A 5 pd2 y4. If the load is known with an uncertainty of 610 percent, the diameter is known within 65 percent (tolerances), and the stress that causes failure (strength) is known within 615 percent, determine the minimum design factor that will guarantee that the part will not fail.

When one knows the true values x1 and x2 and has approximations X1 and X2 at hand, one can see where errors may arise. By viewing error as something to be added to an approximation to attain a true value, it follows that the error ei is related to Xi and xi as xi 5 Xi 1 ei (a) Show that the error in a sum X1 1 X2 is (x1 1 x2) 2 (X1 1 X2) 5 e1 1 e2 (b) Show that the error in a difference X1 2 X2 is (x1 2 x2) 2 (X1 2 X2) 5 e1 2 e2 (c) Show that the error in a product X1X2 is x1x2 2 X1X2 < X1X2 a e1 X1 1 e2 X2 b (d) Show that in a quotient X1yX2 the error is x1 x2 2 X1 X2 < X1 X2 a e1 X1 2 e2 X2 b

Use the true values x1 5 17 and x2 5 18 (a) Demonstrate the correctness of the error equation from Prob. 110 for addition if X1 and X2 are obtained by truncating x1 and x2 to three digits. (b) Demonstrate the correctness of the error equation for addition if X1 and X2 are obtained by rounding x1 and x2 to three significant figures.

A solid circular rod of diameter d undergoes a bending moment M 5 1000 lbf ? in inducing a stress s 5 32My(pd3 ). Using a material strength of 25 kpsi and a design factor of 2.5, determine the minimum diameter of the rod. Using Table A17 select a preferred fractional diameter and determine the resulting factor of safety.

A fatigue test is performed on rotating beam specimens where, for each rotation cycle, the specimens experience tensile and compressive stresses of equal magnitude. The cycles-to-failure experience with 69 specimens of 5160H steel from 1.25-in hexagonal bar stock was as follows: L 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 f 2 1 3 5 8 12 6 10 8 5 2 3 2 1 0 1 where L is the life in thousands of cycles, and f is the class frequency of failures. (a) Estimate the mean and standard deviation of the life for the population from which the sample was drawn. (b) Presuming the distribution is normal, how many specimens are predicted to fail at less than 115 kcycles?

Determinations of the ultimate tensile strength Sut of stainless-steel sheet (17-7PH, condition TH 1050), in sizes from 0.016 to 0.062 in, in 197 tests combined into seven classes were Sut (kpsi) 174 182 190 198 206 214 222 f 6 9 44 67 53 12 6 where Sut is the class midpoint and f is the class frequency. Estimate the mean and standard deviation

The lives of parts are often expressed as the number of cycles of operation that a specified percentage of a population will exceed before experiencing failure. The symbol L is used to bud98209_ch01_002-040.indd Page 37 9/23/13 4:23 PM f-496 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles 38 designate this definition of life. Thus we can speak of L10 life as the number of cycles to failure exceeded by 90 percent of a population of parts. Given a normal distribution model, with a mean of L 5 122.9 kilocycles and standard deviation of sL 5 30.3 kilocycles, estimate the corresponding L10 life

The tensile 0.2 percent offset yield strength of AISI 1137 cold-drawn steel bars up to 1 inch in diameter from 2 mills and 25 heats is reported as follows: Sy 93 95 97 99 101 103 105 107 109 111 f 19 25 38 17 12 10 5 4 4 2 where Sy is the class midpoint in kpsi and f is the number in each class. Presuming the distribution is normal, what is the yield strength exceeded by 99 percent of the population?

A mechanical system comprises three subsystems in series with reliabilities of 98, 96, and 94 percent. What is the overall reliability of the system?

From Sec. 312, the maximum shear stress in a solid round bar of diameter, d, due to an applied torque, T, is given by tmax 5 16 Ty(pd3 ). A round, cold-drawn 1018 steel rod is subjected to a mean torsional load of T 5 1.5 kN ? m with a standard deviation of 145 N ? m. The rod material has a mean shear yield strength of Ssy 5 312 MPa with a standard deviation of 23.5 MPa. Assuming the strength and load have normal distributions, what value of the design factor nd corresponds to a reliability of 0.99 against yielding? Determine the corresponding diameter of the rod.

A round cold-drawn 1045 steel rod has a mean strength Sy 5 95.5 kpsi with a standard deviation of s Sy 5 6.59 kpsi. The rod is to be subjected to a mean static axial load of P 5 65 kip with a standard deviation of s P 5 5.0 kip. Assuming the strength and load have normal distributions, determine the reliabilities corresponding to the design factors of (a) 1.2, (b) 1.5. Also, determine the diameter corresponding to each case

A beam subjected to axial loading will experience an axial stress, sa. If, in addition, the beam is subjected to a bending load, a bending stress, sb, will also occur at the outer fibers of the beam. The maximum stress at the outer fibers of the beam will be smax 5 sa 1 sb. Assume that sa and sb are independent and sa 5 90 MPa, s sa 5 8.4 MPa, sb 5 383 MPa, s sb 5 22.3 MPa. The rod is made of a steel with Sy 5 553 MPa and s Sy 5 42.7 MPa. Assuming the strength and load have normal distributions, determine the design factor and the reliability guarding against yielding

Three blocks A, B, and C and a grooved block D have dimensions a, b, c, and d as follows: a 5 1.500 6 0.001 in b 5 2.000 6 0.003 in c 5 3.000 6 0.004 in d 5 6.520 6 0.010 in

The volume of a rectangular parallelepiped is given by V 5 xyz. If x 5 a 6 Da, y 5 b 6 Db, z 5 c 6 Dc, show that V V < a a 1 b b 1 c c Use this result to determine the bilateral tolerance on the volume of a rectangular parallelepiped with dimensions

A pivot in a linkage has a pin in the figure whose dimension a 6 ta is to be established. The thickness of the link clevis is 1.500 6 0.005 in. The designer has concluded that a gap of between 0.004 and 0.05 in will satisfactorily sustain the function of the linkage pivot. Determine the dimension a and its tolerance.

A circular cross section O ring has the dimensions shown in the figure. In particular, an AS 568A standard No. 240 O ring has an inside diameter Di and a cross-section diameter d of Di 5 3.734 6 0.028 in di 5 0.139 6 0.004 in Estimate the mean outside diameter Do and its bilateral tolerance.

For the table given, repeat Prob. 124 for the following O rings, given the AS 568A standard number. Solve Problems 125 and 126 using SI units. Solve Problems 127 and 128 using ips units. Note: The solutions require research.

For the table given, repeat Prob. 124 for the following O rings, given the AS 568A standard number. Solve Problems 125 and 126 using SI units. Solve Problems 127 and 128 using ips units. Note: The solutions require research.

For the table given, repeat Prob. 124 for the following O rings, given the AS 568A standard number. Solve Problems 125 and 126 using SI units. Solve Problems 127 and 128 using ips units. Note: The solutions require research.

For the table given, repeat Prob. 124 for the following O rings, given the AS 568A standard number. Solve Problems 125 and 126 using SI units. Solve Problems 127 and 128 using ips units. Note: The solutions require research.

Convert the following to appropriate ips units: (a) A stress, s 5 150 MPa. (b) A force, F 5 2 kN. (c) A moment, M 5 150 N ? m. (d) An area, A 5 1 500 mm2 . (e) A second moment of area, I 5 750 cm4 . (f) A modulus of elasticity, E 5 145 GPa. (g) A speed, v 5 75 km/h. (h) A volume, V 5 1 liter.

Convert the following to appropriate SI units: (a) A length, l 5 5 ft. (b) A stress, s 5 90 kpsi. (c) A pressure, p 5 25 psi. (d) A section modulus, Z 5 12 in3 . (e) A unit weight, w 5 0.208 lbf/in. ( f ) A deflection, d 5 0.001 89 in. (g) A velocity, v 5 1 200 ft/min. (h) A unit strain, e 5 0.002 15 in/in. (i) A volume, V 5 1830 in3 .

Generally, final design results are rounded to or fixed to three digits because the given data cannot justify a greater display. In addition, prefixes should be selected so as to limit number strings to no more than four digits to the left of the decimal point. Using these rules, as well as those for the choice of prefixes, solve the following relations: (a) s 5 MyZ, where M 5 1770 lbf ? in and Z 5 0.934 in3 . (b) s 5 FyA, where F 5 9440 lbf and A 5 23.8 in2 . (c) y 5 Fl3 y3EI, where F 5 270 lbf, l 5 31.5 in, E 5 30 Mpsi, and I 5 0.154 in4 . (d) u 5 TlyGJ, where T 5 9 740 lbf ? in, l 5 9.85 in, G 5 11.3 Mpsi, and J 5 pd4 y32 with d 5 1.00 in.

Repeat Prob. 131 for the following: (a) s 5 Fywt, where F 5 1 kN, w 5 25 mm, and t 5 5 mm. (b) I 5 bh3 y12, where b 5 10 mm and h 5 25 mm. (c) I 5 pd 4 y64, where d 5 25.4 mm. (d) t 5 16 Typd3 , where T 5 25 N ? m, and d 5 12.7 mm.

Repeat Prob. 131 for: (a) t 5 FyA, where A 5 pd2 y4, F 5 2 700 lbf, and d 5 0.750 in. (b) s 5 32 Faypd3 , where F 5 180 lbf, a 5 31.5 in, and d 5 1.25 in. (c) Z 5 p(do 4 2 di 4 )y(32 do) for do 5 1.50 in and di 5 1.00 in. (d) k 5 (d4 G)y(8 D3 N), where d 5 0.062 5 in, G 5 11.3 Mpsi, D 5 0.760 in, and N 5 32 (a dimensionless number).

Select a mechanical component from Part 3 of this book (roller bearings, springs, etc.), go to your university's library or the appropriate internet website, and, using the Thomas Register of American Manufacturers (www.thomasnet.com), report on the information obtained on five manufacturers or suppliers.

Select a mechanical component from Part 3 of this book (roller bearings, springs, etc.), go to the Internet, and, using a search engine, report on the information obtained on five manufacturers or suppliers.

Select an organization listed in Sec. 1-6, go to the Internet, and list what information is available on the organization.

Go to the Internet and connect to the NSPE website (www.nspe.org/ethics). Read the history of the Code of Ethics and briefly discuss your reading.

Go to the Internet and connect to the NSPE website (www.nspe.org/ethics). Read the complete NSPE Code of Ethics for Engineers and briefly discuss your reading.

Go to the Internet and connect to the NSPE website (www.nspe.org/ethics). Go to Ethics Resources and review one or more of the topics given. A sample of some of the topics may be: (a) Education Publications (b) Ethics Case Search (c) Ethics Exam (d) FAQ (e) Milton Lunch Contest (f) Other Resources (g) You Be the Judge Briefly discuss your reading.

Estimate how many times more expensive it is to grind a steel part to a tolerance of \(\pm 0.0005\) in versus turning it to a tolerance of \(\pm 0.003\) in.

The costs to manufacture a part using methods A and B are estimated by \(C_A=10+0.8 P\) and \(C_B=60+0.8 P-0.005 P^2\) respectively, where the cost C is in dollars and P is the number of parts. Estimate the breakeven point.

A cylindrical part of diameter d is loaded by an axial force P. This causes a stress of P / A, where \(A=\pi d^2 / 4\). If the load is known with an uncertainty of \(\pm 10\) percent, the diameter is known within \(\pm 5\) percent (tolerances), and the stress that causes failure (strength) is known within \(\pm 15\) percent, determine the minimum design factor that will guarantee that the part will not fail.

When one knows the true values \(x_1\) and \(x_2\) and has approximations \(X_1\) and \(X_2\) at hand, one can see where errors may arise. By viewing error as something to be added to an approximation to attain a true value, it follows that the error \(e_i\) is related to \(X_i\) and \(x_i\) as \(x_i=X_i+e_i\) (a) Show that the error in a sum \(X_1+X_2\) is \(\left(x_1+x_2\right)-\left(X_1+X_2\right)=e_1+e_2\) (b) Show that the error in a difference \(X_1-X_2\) is \(\left(x_1-x_2\right)-\left(X_1-X_2\right)=e_1-e_2\) (c) Show that the error in a product \(X_1 X_2\) is \(x_1 x_2-X_1 X_2 \approx X_1 X_2\left(\frac{e_1}{X_1}+\frac{e_2}{X_2}\right)\) (d) Show that in a quotient \(X_1 / X_2\) the error is \(\frac{x_1}{x_2}-\frac{X_1}{X_2} \approx \frac{X_1}{X_2}\left(\frac{e_1}{X_1}-\frac{e_2}{X_2}\right)\)

Use the true values \(x_1=\sqrt{7}\) and \(x_2=\sqrt{8}\) (a) Demonstrate the correctness of the error equation from Prob. 1-10 for addition if \(X_1\) and \(X_2\) are obtained by truncating \(x_1\) and \(x_2\) to three digits. (b) Demonstrate the correctness of the error equation for addition if \(X_1\) and \(X_2\) are obtained by rounding \(x_1\) and \(x_2\) to three significant figures.

A solid circular rod of diameter d undergoes a bending moment \(M=1000 \mathrm{lbf} \cdot\) in inducing a stress \(\sigma=32 M /\left(\pi d^3\right)\). Using a material strength of \(25 \mathrm{kpsi}\) and a design factor of 2.5, determine the minimum diameter of the rod. Using Table A-17 select a preferred fractional diameter and determine the resulting factor of safety.

A fatigue test is performed on rotating beam specimens where, for each rotation cycle, the specimens experience tensile and compressive stresses of equal magnitude. The cycles-to-failure experience with 69 specimens of 5160H steel from 1.25-in hexagonal bar stock was as follows: where L is the life in thousands of cycles, and f is the class frequency of failures. (a) Estimate the mean and standard deviation of the life for the population from which the sample was drawn. (b) Presuming the distribution is normal, how many specimens are predicted to fail at less than 115 kcycles?

Determinations of the ultimate tensile strength \(S_{u t}\) of stainless-steel sheet (17-7PH, condition TH 1050), in sizes from 0.016 to 0.062 in, in 197 tests combined into seven classes were where \(S_{u t}\) is the class midpoint and f is the class frequency. Estimate the mean and standard deviation.

The lives of parts are often expressed as the number of cycles of operation that a specified percentage of a population will exceed before experiencing failure. The symbol L is used to designate this definition of life. Thus we can speak of \(L_{10}\) life as the number of cycles to failure exceeded by 90 percent of a population of parts. Given a normal distribution model, with a mean of \(\bar{L}=122.9\) kilocycles and standard deviation of \(s_L=30.3\) kilocycles, estimate the corresponding \(L_{10}\) life.

The tensile 0.2 percent offset yield strength of AISI 1137 cold-drawn steel bars up to 1 inch in diameter from 2 mills and 25 heats is reported as follows: where \(S_y\) is the class midpoint in kpsi and f is the number in each class. Presuming the distribution is normal, what is the yield strength exceeded by 99 percent of the population?

A mechanical system comprises three subsystems in series with reliabilities of 98, 96, and 94 percent. What is the overall reliability of the system?

From Sec. 3-12, the maximum shear stress in a solid round bar of diameter, d, due to an applied torque, T, is given by \(\tau_{\max }=16 T /\left(\pi d^3\right)\). A round, cold-drawn 1018 steel rod is subjected to a mean torsional load of \(\bar{T}=1.5 \mathrm{kN} \cdot \mathrm{m}\) with a standard deviation of \(145 \mathrm{~N} \cdot \mathrm{m}\). The rod material has a mean shear yield strength of \(\bar{S}_{\mathrm{xy}}=312 \mathrm{MPa}\) with a standard deviation of 23.5 MPa. Assuming the strength and load have normal distributions, what value of the design factor \(\bar{n}_d\) corresponds to a reliability of 0.99 against yielding? Determine the corresponding diameter of the rod.

A round cold-drawn 1045 steel rod has a mean strength \(\bar{S}_y=95.5 \mathrm{kpsi}\) with a standard deviation of \(\hat{\sigma}_{S,}=6.59 \mathrm{kpsi}\). The rod is to be subjected to a mean static axial load of \(\bar{P}=65 \mathrm{kip}\) with a standard deviation of \(\hat{\sigma}_P=5.0\) kip. Assuming the strength and load have normal distributions, determine the reliabilities corresponding to the design factors of (a) 1.2, (b) 1.5. Also, determine the diameter corresponding to each case.

A beam subjected to axial loading will experience an axial stress, \(\sigma_a\). If, in addition, the beam is subjected to a bending load, a bending stress, \(\sigma_b\), will also occur at the outer fibers of the beam. The maximum stress at the outer fibers of the beam will be \(\sigma_{\max }=\sigma_a+\sigma_b\). Assume that \(\sigma_a\) and \(\sigma_b\) are independent and \(\bar{\sigma}_a=90 \mathrm{MPa}, \hat{\sigma}_{\sigma_s}=8.4 \mathrm{MPa}, \bar{\sigma}_b=383 \mathrm{MPa}, \hat{\sigma}_{\sigma_s}=22.3 \mathrm{MPa}\). The rod is made of a steel with \(\bar{S}_y=553 \mathrm{MPa}\) and \(\hat{\sigma}_{\mathrm{S},}=42.7 \mathrm{MPa}\). Assuming the strength and load have normal distributions, determine the design factor and the reliability guarding against yielding.

Three blocks A, B, and C and a grooved block D have dimensions a, b, c, and d as follows: \(a=1.500 \pm 0.001\) in \(b=2.000 \pm 0.003\) in \(c=3.000 \pm 0.004\) in \(d=6.520 \pm 0.010\) in (a) Determine the mean gap \(\bar{w}\) and its tolerance. (b) Determine the mean size of d that will assure that \(w \geq 0.010\) in.

The volume of a rectangular parallelepiped is given by V = xyz. If \(x=a \pm \Delta a, y=b \pm \Delta b\), \(z=c \pm \Delta c\), show that \(\frac{\Delta V}{\bar{V}} \approx \frac{\Delta a}{\bar{a}}+\frac{\Delta b}{\bar{b}}+\frac{\Delta c}{\bar{c}}\) Use this result to determine the bilateral tolerance on the volume of a rectangular parallelepiped with dimensions \(a=1.500 \pm 0.002\) in \(b=1.875 \pm 0.003\) in \(c=3.000 \pm 0.004\) in

A pivot in a linkage has a pin in the figure whose dimension \(a \pm t_a\) is to be established. The thickness of the link clevis is \(1.500 \pm 0.005\) in. The designer has concluded that a gap of between 0.004 and 0.05 in will satisfactorily sustain the function of the linkage pivot. Determine the dimension a and its tolerance.

A circular cross section O ring has the dimensions shown in the figure. In particular, an AS 568A standard No. 240 O ring has an inside diameter \(D_i\) and a cross-section diameter d of \(D_i=3.734 \pm 0.028\) in \(d_i=0.139 \pm 0.004\) in Estimate the mean outside diameter \(\bar{D}_o\) and its bilateral tolerance.

For the table given, repeat Prob. 1-24 for the following O rings, given the AS 568A standard number. Solve Problem 1-25 using SI units. Note: The solutions require research. Problem number 1-25 AS 568A No. 110

For the table given, repeat Prob. 1-24 for the following O rings, given the AS 568A standard number. Solve Problem 1-26 using SI units. Note: The solutions require research. Problem number 1-26 AS 568A No. 220

For the table given, repeat Prob. 1-24 for the following O rings, given the AS 568A standard number. Solve Problem 1-27 using ips units. Note: The solutions require research. Problem number 1-27 AS 568A No. 160

For the table given, repeat Prob. 1-24 for the following O rings, given the AS 568A standard number. Solve Problem 1-28 using ips units. Note: The solutions require research. Problem number 1-28 AS 568A No. 320

Convert the following to appropriate ips units: (a) A stress, \(\sigma=150 \mathrm{MPa}\). (b) A force, F = 2 kN. (c) A moment, \(M=150 \mathrm{~N} \cdot \mathrm{m}\). (d) An area, \(A=1500 \mathrm{~mm}^2\). (e) A second moment of area, \(I=750 \mathrm{~cm}^4\). (f) A modulus of elasticity, E = 145 GPa. (g) A speed, v = 75 km/h. (h) A volume, V = 1 liter.

Convert the following to appropriate SI units: (a) A length, l = 5 ft. (b) A stress, \(\sigma=90 \mathrm{kpsi}\). (c) A pressure, p = 25 psi. (d) A section modulus, Z = 12 \(in^3\). (e) A unit weight, w = 0.208 lbf/in. (f) A deflection, \(\delta=0.00189\) in. (g) A velocity, v = 1200 ft/min. (h) A unit strain, \(\epsilon=0.00215 \mathrm{in} / \mathrm{in}\). (i) A volume, V = 1830 \(in^3\).

Generally, final design results are rounded to or fixed to three digits because the given data cannot justify a greater display. In addition, prefixes should be selected so as to limit number strings to no more than four digits to the left of the decimal point. Using these rules, as well as those for the choice of prefixes, solve the following relations: (a) \(\sigma=M / Z\), where \(M=1770 \mathrm{lbf} \cdot\) in and \(Z=0.934 \mathrm{in}^3\). (b) \(\sigma=F / A\), where \(F=9440 \mathrm{lbf}\) and \(A=23.8 \mathrm{in}^2\). (c) \(y=F l^3 / 3 E I\), where \(F=270 \mathrm{lbf}, l=31.5 \mathrm{in}, E=30 \mathrm{Mpsi}\), and \(I=0.154 \mathrm{in}^4\). (d) \(\theta=T l / G J\), where \(T=9740 \mathrm{lbf} \cdot\) in, \(l=9.85 \mathrm{in}, G=11.3 \mathrm{Mpsi}\), and \(J=\pi d^4 / 32\) with \(d=1.00 \mathrm{in}\).

Repeat Prob. 1-31 for the following: (a) \(\sigma=F / w t\), where \(F=1 \mathrm{kN}, w=25 \mathrm{~mm}\), and \(t=5 \mathrm{~mm}\). (b) \(I=b h^3 / 12\), where \(b=10 \mathrm{~mm}\) and \(h=25 \mathrm{~mm}\). (c) \(I=\pi d^4 / 64\), where \(d=25.4 \mathrm{~mm}\). (d) \(\tau=16 T / \pi d^3\), where \(T=25 \mathrm{~N} \cdot \mathrm{m}\), and \(d=12.7 \mathrm{~mm}\).

Repeat Prob. 1-31 for: (a) \(\tau=F / A\), where \(A=\pi d^2 / 4, F=2700 \mathrm{lbf}\), and \(d=0.750\) in. (b) \(\sigma=32 \mathrm{Fa} / \pi d^3\), where \(F=180 \mathrm{lbf}, a=31.5 \mathrm{in}\), and \(d=1.25 \mathrm{in}\). (c) \(Z=\pi\left(d_o^4-d_i^4\right) /\left(32 d_o\right)\) for \(d_o=1.50\) in and \(d_i=1.00\) in. (d) \(k=\left(d^4 G\right) /\left(8 D^3 N\right)\), where \(d=0.0625 \mathrm{in}, G=11.3 \mathrm{Mpsi}, D=0.760 \mathrm{in}\), and N = 32 (a dimensionless number).