In this circuit, V = 10 volts, R = 3,000 ohms, and C = 50 x 10-6 farads. The circuit has a time constant t, which depends on the resistance, R, and the capacitance, C, as t = R x C = 0.15 second. 1. Use a for loop. 2. Use the math library function exp(x) to compute ex. You will need to include the system header file math.h. 3. On Unix you will need –lm in your command line to tell the Linker to search the math library. 4. Use macro definition for all the constants. 5. Format the output so the output looks like the following. The time and voltage should display two digits after the decimal point.

## In this circuit, V = 10 volts, R = 3,000 ohms, and C = 50 x 10-6 farads. The circuit has a time constant t, which depends on the resistance, R, and the capacitance, C, as t = R x C = 0.15 second. 1. Use a for loop. 2. Use the math library function exp(x) to compute ex. You will need to include the system header file math.h. 3. On Unix you will need –lm in your command line to tell the Linker to search the math library. 4. Use macro definition for all the constants. 5. Format the output so the output looks like the following. The time and voltage should display two digits after the decimal point.

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Question 1, chap 33, sect 3. part 1 of 2 10 points The compound eyes of bees and other insects are highly sensitive to light in the ultraviolet portion of the spectrum, particularly light with frequencies between 7.5 × 1014 Hz and 1.0 × 1015 Hz. The speed of light is 3 × 108 m/s. What is the largest wavelength to which these frequencies correspond? Question 3, chap 33, sect 3. part 1 of 3 10 points A plane electromagnetic sinusoidal wave of frequency 10.7 MHz travels in free space. The speed of light is 2.99792 × 108 m/s. Determine the wavelength of the wave. Question 4, chap 33, sect 3. part 2 of 3 10 points Find the period of the wave. Question 2, chap 33, sect 3. part 2 of 2 10 points What is the smallest wavelength? Question 5, chap 33, sect 3. part 3 of 3 10 points At some point and some instant, the electric field has has a value of 998 N/C. Calculate the magnitude of the magnetic field at this point and this instant. Question 6, chap 33, sect 3. part 1 of 2 10 points A plane electromagnetic sinusoidal wave of frequency 10.7 MHz travels in free space. The speed of light is 2.99792 × 108 m/s. Determine the wavelength of the wave. Question 8, chap 33, sect 3. part 1 of 1 10 points The magnetic field amplitude of an electromagnetic wave is 9.9 × 10−6 T. The speed of light is 2.99792 × 108 m/s . Calculate the amplitude of the electric field if the wave is traveling in free space. Question 7, chap 33, sect 3. part 2 of 2 10 points At some point and some instant, the electric field has has a value of 998 V/m. Calculate the magnitude of the magnetic field at this point and this instant. Question 9, chap 33, sect 5. part 1 of 1 10 points The cable is carrying the current I(t). at the surface of a long transmission cable of resistivity ρ, length ℓ and radius a, using the expression ~S = 1 μ0 ~E × ~B . Question 10, chap 33, sect 5. part 1 of 1 10 points In 1965 Penzias and Wilson discovered the cosmic microwave radiation left over from the Big Bang expansion of the universe. The energy density of this radiation is 7.64 × 10−14 J/m3. The speed of light 2.99792 × 108 m/s and the permeability of free space is 4π × 10−7 N/A2. Determine the corresponding electric field amplQuestion 11, chap 33, sect 5. part 1 of 5 10 points Consider a monochromatic electromagnetic plane wave propagating in the x direction. At a particular point in space, the magnitude of the electric field has an instantaneous value of 998 V/m in the positive y-direction. The wave is traveling in the positive x-direction. x y z E wave propagation The speed of light is 2.99792×108 m/s, the permeability of free space is 4π×10−7 T ・ N/A and the permittivity of free space 8.85419 × 10−12 C2/N ・ m2. Compute the instantaneous magnitude of the magnetic field at the same point and time.itude. Question 12, chap 33, sect 5. part 2 of 5 10 points What is the instantaneous magnitude of the Poynting vector at the same point and time? Question 13, chap 33, sect 5. part 3 of 5 10 points What are the directions of the instantaneous magnetic field and theQuestion 14, chap 33, sect 5. part 4 of 5 10 points What is the instantaneous value of the energy density of the electric field? Question 16, chap 33, sect 6. part 1 of 4 10 points Consider an electromagnetic plane wave with time average intensity 104 W/m2 . The speed of light is 2.99792 × 108 m/s and the permeability of free space is 4 π × 10−7 T・m/A. What is its maximum electric field? What is the instantaneous value of the energy density of the magnetic field? Question 17, chap 33, sect 6. part 2 of 4 10 points What is the the maximum magnetic field? Question 19, chap 33, sect 6. part 4 of 4 10 points Consider an electromagnetic wave pattern as shown in the figure below. Question 18, chap 33, sect 6. part 3 of 4 10 points What is the pressure on a surface which is perpendicular to the beam and is totally reflective? Question 20, chap 33, sect 8. part 1 of 1 10 points A coin is at the bottom of a beaker. The beaker is filled with 1.6 cm of water (n1 = 1.33) covered by 2.1 cm of liquid (n2 = 1.4) floating on the water. How deep does the coin appear to be from the upper surface of the liquid (near the top of the beaker)? An cylindrical opaque drinking glass has a diameter 3 cm and height h, as shown in the figure. An observer’s eye is placed as shown (the observer is just barely looking over the rim of the glass). When empty, the observer can just barely see the edge of the bottom of the glass. When filled to the brim with a transparent liquid, the observer can just barely see the center of the bottom of the glass. The liquid in the drinking glass has an index of refraction of 1.4 . θi h d θr eye Calculate the angle θr . Question 22, chap 33, sect 8. part 2 of 2 10 points Calculate the height h of the glass.

## Question 1, chap 33, sect 3. part 1 of 2 10 points The compound eyes of bees and other insects are highly sensitive to light in the ultraviolet portion of the spectrum, particularly light with frequencies between 7.5 × 1014 Hz and 1.0 × 1015 Hz. The speed of light is 3 × 108 m/s. What is the largest wavelength to which these frequencies correspond? Question 3, chap 33, sect 3. part 1 of 3 10 points A plane electromagnetic sinusoidal wave of frequency 10.7 MHz travels in free space. The speed of light is 2.99792 × 108 m/s. Determine the wavelength of the wave. Question 4, chap 33, sect 3. part 2 of 3 10 points Find the period of the wave. Question 2, chap 33, sect 3. part 2 of 2 10 points What is the smallest wavelength? Question 5, chap 33, sect 3. part 3 of 3 10 points At some point and some instant, the electric field has has a value of 998 N/C. Calculate the magnitude of the magnetic field at this point and this instant. Question 6, chap 33, sect 3. part 1 of 2 10 points A plane electromagnetic sinusoidal wave of frequency 10.7 MHz travels in free space. The speed of light is 2.99792 × 108 m/s. Determine the wavelength of the wave. Question 8, chap 33, sect 3. part 1 of 1 10 points The magnetic field amplitude of an electromagnetic wave is 9.9 × 10−6 T. The speed of light is 2.99792 × 108 m/s . Calculate the amplitude of the electric field if the wave is traveling in free space. Question 7, chap 33, sect 3. part 2 of 2 10 points At some point and some instant, the electric field has has a value of 998 V/m. Calculate the magnitude of the magnetic field at this point and this instant. Question 9, chap 33, sect 5. part 1 of 1 10 points The cable is carrying the current I(t). at the surface of a long transmission cable of resistivity ρ, length ℓ and radius a, using the expression ~S = 1 μ0 ~E × ~B . Question 10, chap 33, sect 5. part 1 of 1 10 points In 1965 Penzias and Wilson discovered the cosmic microwave radiation left over from the Big Bang expansion of the universe. The energy density of this radiation is 7.64 × 10−14 J/m3. The speed of light 2.99792 × 108 m/s and the permeability of free space is 4π × 10−7 N/A2. Determine the corresponding electric field amplQuestion 11, chap 33, sect 5. part 1 of 5 10 points Consider a monochromatic electromagnetic plane wave propagating in the x direction. At a particular point in space, the magnitude of the electric field has an instantaneous value of 998 V/m in the positive y-direction. The wave is traveling in the positive x-direction. x y z E wave propagation The speed of light is 2.99792×108 m/s, the permeability of free space is 4π×10−7 T ・ N/A and the permittivity of free space 8.85419 × 10−12 C2/N ・ m2. Compute the instantaneous magnitude of the magnetic field at the same point and time.itude. Question 12, chap 33, sect 5. part 2 of 5 10 points What is the instantaneous magnitude of the Poynting vector at the same point and time? Question 13, chap 33, sect 5. part 3 of 5 10 points What are the directions of the instantaneous magnetic field and theQuestion 14, chap 33, sect 5. part 4 of 5 10 points What is the instantaneous value of the energy density of the electric field? Question 16, chap 33, sect 6. part 1 of 4 10 points Consider an electromagnetic plane wave with time average intensity 104 W/m2 . The speed of light is 2.99792 × 108 m/s and the permeability of free space is 4 π × 10−7 T・m/A. What is its maximum electric field? What is the instantaneous value of the energy density of the magnetic field? Question 17, chap 33, sect 6. part 2 of 4 10 points What is the the maximum magnetic field? Question 19, chap 33, sect 6. part 4 of 4 10 points Consider an electromagnetic wave pattern as shown in the figure below. Question 18, chap 33, sect 6. part 3 of 4 10 points What is the pressure on a surface which is perpendicular to the beam and is totally reflective? Question 20, chap 33, sect 8. part 1 of 1 10 points A coin is at the bottom of a beaker. The beaker is filled with 1.6 cm of water (n1 = 1.33) covered by 2.1 cm of liquid (n2 = 1.4) floating on the water. How deep does the coin appear to be from the upper surface of the liquid (near the top of the beaker)? An cylindrical opaque drinking glass has a diameter 3 cm and height h, as shown in the figure. An observer’s eye is placed as shown (the observer is just barely looking over the rim of the glass). When empty, the observer can just barely see the edge of the bottom of the glass. When filled to the brim with a transparent liquid, the observer can just barely see the center of the bottom of the glass. The liquid in the drinking glass has an index of refraction of 1.4 . θi h d θr eye Calculate the angle θr . Question 22, chap 33, sect 8. part 2 of 2 10 points Calculate the height h of the glass.

Consider the cubic equation ax3 + bx2 + cx + d = 0, (1) where a, b, c, and d are real input coefficients. Develop a matlab program to find all roots of equation (1) using the methods discussed in the Numerical Analysis class. Your program can not use the matlab built-in functions fzero and roots. You should turn in a .m file cubic24903674.m which contains a matlab function of the form function [rts,info] = cubic24903674(C) where, C = (a, b, c, d) is the input vector of coefficients, rts is the vector of roots and info is your output message. Your program will be stress-tested against cubic equations that may have: 1. (40 points) random roots; or 2. (20 points) very large or very small roots; or 3. (20 points) multiple roots or nearly multiple roots; or 4. (20 points) less than 3 roots or more than 3 roots. You will receive credit for a test polynomial only if your program gets the number of roots correctly, and only then will each correct root (accurate to within a relative error of at most 10^(−6) , as compared to the roots function in matlab) receive additional credit. Your program will receive 0 points if the strings fzero or roots (both in lower case letters) show up anywhere in your .m file.

## Consider the cubic equation ax3 + bx2 + cx + d = 0, (1) where a, b, c, and d are real input coefficients. Develop a matlab program to find all roots of equation (1) using the methods discussed in the Numerical Analysis class. Your program can not use the matlab built-in functions fzero and roots. You should turn in a .m file cubic24903674.m which contains a matlab function of the form function [rts,info] = cubic24903674(C) where, C = (a, b, c, d) is the input vector of coefficients, rts is the vector of roots and info is your output message. Your program will be stress-tested against cubic equations that may have: 1. (40 points) random roots; or 2. (20 points) very large or very small roots; or 3. (20 points) multiple roots or nearly multiple roots; or 4. (20 points) less than 3 roots or more than 3 roots. You will receive credit for a test polynomial only if your program gets the number of roots correctly, and only then will each correct root (accurate to within a relative error of at most 10^(−6) , as compared to the roots function in matlab) receive additional credit. Your program will receive 0 points if the strings fzero or roots (both in lower case letters) show up anywhere in your .m file.

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Please write clearly, show all work in an organized fashion, and circle answers. 1) Using the data shown in Figures 6.14 (at 25oC) and 6.21, combine both curves onto one plot, being careful to correctly plot the modulus, yield strength, tensile (ultimate) strength, and ductility. Discuss how the modulus, yield strength, and ductility compare for pure iron (figure 6.14) vs. the alloy steel. 2) The equation for the effect of grain size on yield strength is given by: y = I +kD-0.5 where y is the yield stress, I is the intrinsic resistance of the lattice to dislocation motion, k is the “blocking parameter” which measures the effectiveness of grain boundaries in blocking dislocation motion, and D is the grain diameter. Use this equation to determine the change in yield strength of a typical steel when the grain size is increased from 10micron to 50 micron (1 micron = 10-6 m), due to grain growth. . I = 150 MN/m2 and k = 0.70 MN/m1.5 . 3) Using the data shown in Callister Figure 7.19, draw an approximate stress-strain curve for the 1040 steel at 0% cold work and at 30% cold work, clearly indicating the yield strength, ductility, and tensile strength of the steel before and after cold-working (Young’s modulus of steel E = 250 MPa). 4) A fatigue test is carried out on a steel having an ultimate strength of 289 MPa. The number of cycles required to break the specimen at different stresses are given below: Stress Amplitude Fatigue Life (MPa) (cycles) 223 4.5 x 104 209 2.4 x 105 192 8.0 x 105 178 1.5 x 106 175 2.7 x 106 168 7.8 x 106 168 >1.0 x 107 (did not break) 165 >2.6 x 107 162 >2.2 x 107 a) Plot the data on linear-log scale, preferably with a computerized figure-plotting program. b) Determine the average fatigue strength at 106 cycles (hint: use curve-fitting software to fit the line). c) What is the ratio of the fatigue strength at 106 cycles to the ultimate strength? e) If you plan to use this material for 108 cycles, what is the maximum fatigue strength you would recommend (assuming 20% fluctuations in stress amplitude). Callister Homework Problems: 7.22, 8.4, 8.12 (see next page)

## Please write clearly, show all work in an organized fashion, and circle answers. 1) Using the data shown in Figures 6.14 (at 25oC) and 6.21, combine both curves onto one plot, being careful to correctly plot the modulus, yield strength, tensile (ultimate) strength, and ductility. Discuss how the modulus, yield strength, and ductility compare for pure iron (figure 6.14) vs. the alloy steel. 2) The equation for the effect of grain size on yield strength is given by: y = I +kD-0.5 where y is the yield stress, I is the intrinsic resistance of the lattice to dislocation motion, k is the “blocking parameter” which measures the effectiveness of grain boundaries in blocking dislocation motion, and D is the grain diameter. Use this equation to determine the change in yield strength of a typical steel when the grain size is increased from 10micron to 50 micron (1 micron = 10-6 m), due to grain growth. . I = 150 MN/m2 and k = 0.70 MN/m1.5 . 3) Using the data shown in Callister Figure 7.19, draw an approximate stress-strain curve for the 1040 steel at 0% cold work and at 30% cold work, clearly indicating the yield strength, ductility, and tensile strength of the steel before and after cold-working (Young’s modulus of steel E = 250 MPa). 4) A fatigue test is carried out on a steel having an ultimate strength of 289 MPa. The number of cycles required to break the specimen at different stresses are given below: Stress Amplitude Fatigue Life (MPa) (cycles) 223 4.5 x 104 209 2.4 x 105 192 8.0 x 105 178 1.5 x 106 175 2.7 x 106 168 7.8 x 106 168 >1.0 x 107 (did not break) 165 >2.6 x 107 162 >2.2 x 107 a) Plot the data on linear-log scale, preferably with a computerized figure-plotting program. b) Determine the average fatigue strength at 106 cycles (hint: use curve-fitting software to fit the line). c) What is the ratio of the fatigue strength at 106 cycles to the ultimate strength? e) If you plan to use this material for 108 cycles, what is the maximum fatigue strength you would recommend (assuming 20% fluctuations in stress amplitude). Callister Homework Problems: 7.22, 8.4, 8.12 (see next page)

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