Answer:
Stat PVC = Stat(82+98.5)
Stat PVT = Stat(59+71.5)
Explanation
PVI = 71 + 35
Let G1 = Grade 1; G2 = Grade 2
G1 = +2.1% ; G2 = -3.4%
Highest point of curve at station = 74 + 10
General equation of a curve:
![y = ax^{2} +bx+c\\dy/dx=2ax+b\\](https://tex.z-dn.net/?f=y%20%3D%20ax%5E%7B2%7D%20%2Bbx%2Bc%5C%5Cdy%2Fdx%3D2ax%2Bb%5C%5C)
At highest point of the curve ![dy/dx=o](https://tex.z-dn.net/?f=dy%2Fdx%3Do)
![2ax+b=0\\x=-b/2a\\x=G1L/(G2-G1)\\x=L/2 +(stat 74+10)-(stat 71+35)\\x=L/2 + 275](https://tex.z-dn.net/?f=2ax%2Bb%3D0%5C%5Cx%3D-b%2F2a%5C%5Cx%3DG1L%2F%28G2-G1%29%5C%5Cx%3DL%2F2%20%2B%28stat%2074%2B10%29-%28stat%2071%2B35%29%5C%5Cx%3DL%2F2%20%2B%20275)
![-G1L/(G2-G1) = (L/2 + 275)/100\\L = -2327 ft\\Station PVC = Stat(71+35)+(-2327/2)\\\\Stat PVC = 7135-1163.5\\Stat PVC = Stat(82+98.5)\\](https://tex.z-dn.net/?f=-G1L%2F%28G2-G1%29%20%3D%20%28L%2F2%20%2B%20275%29%2F100%5C%5CL%20%3D%20-2327%20ft%5C%5CStation%20PVC%20%3D%20Stat%2871%2B35%29%2B%28-2327%2F2%29%5C%5C%5C%5CStat%20PVC%20%3D%207135-1163.5%5C%5CStat%20PVC%20%3D%20Stat%2882%2B98.5%29%5C%5C)
Station PVT
![Station PVT = Stat PVI + (L/2)\\Station PVT = Stat(71+35)+(-2327/2)\\Station PVT = 7135-1163.5\\Stat PVT = Stat(59+71.5)](https://tex.z-dn.net/?f=Station%20PVT%20%3D%20Stat%20PVI%20%2B%20%28L%2F2%29%5C%5CStation%20PVT%20%3D%20Stat%2871%2B35%29%2B%28-2327%2F2%29%5C%5CStation%20PVT%20%3D%207135-1163.5%5C%5CStat%20PVT%20%3D%20Stat%2859%2B71.5%29)
Answer:
Frequency = ![3.5294\times 10^{14}s^{-1}](https://tex.z-dn.net/?f=3.5294%5Ctimes%2010%5E%7B14%7Ds%5E%7B-1%7D)
Wavenumber = ![1.1765\times 10^6m^{-1}](https://tex.z-dn.net/?f=1.1765%5Ctimes%2010%5E6m%5E%7B-1%7D)
Energy = ![2.3365\times 10^{-19}J](https://tex.z-dn.net/?f=2.3365%5Ctimes%2010%5E%7B-19%7DJ)
Energy = 1.4579 eV
Energy = ![2.3365\times 10^{-12}erg](https://tex.z-dn.net/?f=2.3365%5Ctimes%2010%5E%7B-12%7Derg)
Explanation:
As we are given the wavelength = 850 nm
conversion used : ![(1nm=10^{-9}m)](https://tex.z-dn.net/?f=%281nm%3D10%5E%7B-9%7Dm%29)
So, wavelength is ![850\times 10^{-9}m](https://tex.z-dn.net/?f=850%5Ctimes%2010%5E%7B-9%7Dm)
The relation between frequency and wavelength is shown below as:
![Frequency=\frac{c}{Wavelength}](https://tex.z-dn.net/?f=Frequency%3D%5Cfrac%7Bc%7D%7BWavelength%7D)
Where, c is the speed of light having value = ![3\times 10^8m/s](https://tex.z-dn.net/?f=3%5Ctimes%2010%5E8m%2Fs)
So, Frequency is:
![Frequency=\frac{3\times 10^8m/s}{850\times 10^{-9}m}](https://tex.z-dn.net/?f=Frequency%3D%5Cfrac%7B3%5Ctimes%2010%5E8m%2Fs%7D%7B850%5Ctimes%2010%5E%7B-9%7Dm%7D)
![Frequency=3.5294\times 10^{14}s^{-1}](https://tex.z-dn.net/?f=Frequency%3D3.5294%5Ctimes%2010%5E%7B14%7Ds%5E%7B-1%7D)
Wavenumber is the reciprocal of wavelength.
So,
![Wavenumber=\frac{1}{Wavelength}=\frac{1}{850\times 10^{-9}m}](https://tex.z-dn.net/?f=Wavenumber%3D%5Cfrac%7B1%7D%7BWavelength%7D%3D%5Cfrac%7B1%7D%7B850%5Ctimes%2010%5E%7B-9%7Dm%7D)
![Wavenumber=1.1765\times 10^6m^{-1}](https://tex.z-dn.net/?f=Wavenumber%3D1.1765%5Ctimes%2010%5E6m%5E%7B-1%7D)
Also,
![Energy=h\times frequency](https://tex.z-dn.net/?f=Energy%3Dh%5Ctimes%20frequency)
where, h is Plank's constant having value as ![6.62\times 10^{-34}J.s](https://tex.z-dn.net/?f=6.62%5Ctimes%2010%5E%7B-34%7DJ.s)
So,
![Energy=(6.62\times 10^{-34}J.s)\times (3.5294\times 10^{14}s^{-1})](https://tex.z-dn.net/?f=Energy%3D%286.62%5Ctimes%2010%5E%7B-34%7DJ.s%29%5Ctimes%20%283.5294%5Ctimes%2010%5E%7B14%7Ds%5E%7B-1%7D%29)
![Energy=2.3365\times 10^{-19}J](https://tex.z-dn.net/?f=Energy%3D2.3365%5Ctimes%2010%5E%7B-19%7DJ)
Also,
![1J=6.24\times 10^{18}eV](https://tex.z-dn.net/?f=1J%3D6.24%5Ctimes%2010%5E%7B18%7DeV)
So,
![Energy=(2.3365\times 10^{-19})\times (6.24\times 10^{18}eV)](https://tex.z-dn.net/?f=Energy%3D%282.3365%5Ctimes%2010%5E%7B-19%7D%29%5Ctimes%20%286.24%5Ctimes%2010%5E%7B18%7DeV%29)
![Energy=1.4579eV](https://tex.z-dn.net/?f=Energy%3D1.4579eV)
Also,
![1J=10^7erg](https://tex.z-dn.net/?f=1J%3D10%5E7erg)
So,
![Energy=(2.3365\times 10^{-19})\times 10^7erg](https://tex.z-dn.net/?f=Energy%3D%282.3365%5Ctimes%2010%5E%7B-19%7D%29%5Ctimes%2010%5E7erg)
![Energy=2.3365\times 10^{-12}erg](https://tex.z-dn.net/?f=Energy%3D2.3365%5Ctimes%2010%5E%7B-12%7Derg)
This question is incomplete, the complete question is;
Determine the design moment strength (ϕMn) for a W21x73 steel beam with a simple span of 18 ft when lateral bracing for the compression flange is provided at the ends only (i.e., Lb = 18 ft). Report the result in kip-ft.
Use Fy=50 ksi and assume Cb=1.0 (if needed).
Answer: the design moment strength for the W21x73 steel beam is 566.25 f-ft
Explanation:
Given that;
section W 21 x 73 steel beam;
now from the steel table table for this section;
Zx = Sx = 151 in³
also given that; fy = 50 ksi and Cb = 1.0
QMn = 0.9 × Fy × Zx
so we substitute
QMn = 0.9 × 50 × 151
QMn = 6795 k-inch
we know that;
12inch equals 1 foot
so
QMn = 6795 k-inch / 12
QMn = 566.25 f-ft
Therefore the design moment strength for the W21x73 steel beam is 566.25 f-ft