Given Information:
Inductance = L = 5 mH = 0.005 H
Time = t = 2 seconds
Required Information:
Current at t = 2 seconds = i(t) = ?
Energy at t = 2 seconds = W = ?
Answer:
Current at t = 2 seconds = i(t) = 735.75 A
Energy at t = 2 seconds = W = 1353.32 J
Explanation:
The voltage across an inductor is given as
![V(t) = 5(1-e^{-0.5t})](https://tex.z-dn.net/?f=V%28t%29%20%3D%205%281-e%5E%7B-0.5t%7D%29)
The current flowing through the inductor is given by
![i(t) = \frac{1}{L} \int_0^t \mathrm{V(t)}\,\mathrm{d}t \,+ i(0)](https://tex.z-dn.net/?f=i%28t%29%20%3D%20%5Cfrac%7B1%7D%7BL%7D%20%5Cint_0%5Et%20%5Cmathrm%7BV%28t%29%7D%5C%2C%5Cmathrm%7Bd%7Dt%20%5C%2C%2B%20i%280%29)
Where L is the inductance and i(0) is the initial current in the inductor which we will assume to be zero since it is not given.
![i(t) = \frac{1}{0.005} \int_0^t \mathrm{5(1-e^{-0.5t}}) \,\mathrm{d}t \,+ 0\\\\i(t) = 200 \int_0^t \mathrm{5(1-e^{-0.5t}}) \,\mathrm{d}t \\\\i(t) = 200 \: [ {5\: (t + \frac{e^{-0.5t}}{0.5})]_0^t \\i(t) = 200\times5\: \: [ { (t + 2e^{-0.5t} + 2 )] \\](https://tex.z-dn.net/?f=i%28t%29%20%3D%20%5Cfrac%7B1%7D%7B0.005%7D%20%5Cint_0%5Et%20%5Cmathrm%7B5%281-e%5E%7B-0.5t%7D%7D%29%20%5C%2C%5Cmathrm%7Bd%7Dt%20%5C%2C%2B%200%5C%5C%5C%5Ci%28t%29%20%3D%20200%20%5Cint_0%5Et%20%5Cmathrm%7B5%281-e%5E%7B-0.5t%7D%7D%29%20%5C%2C%5Cmathrm%7Bd%7Dt%20%5C%5C%5C%5Ci%28t%29%20%3D%20200%20%5C%3A%20%5B%20%7B5%5C%3A%20%28t%20%2B%20%5Cfrac%7Be%5E%7B-0.5t%7D%7D%7B0.5%7D%29%5D_0%5Et%20%5C%5Ci%28t%29%20%3D%20200%5Ctimes5%5C%3A%20%5C%3A%20%5B%20%7B%20%28t%20%2B%202e%5E%7B-0.5t%7D%20%2B%202%20%29%5D%20%5C%5C)
![i(t) = 1000t +2000e^{-0.5t} -2000\\](https://tex.z-dn.net/?f=i%28t%29%20%3D%201000t%20%2B2000e%5E%7B-0.5t%7D%20-2000%5C%5C)
So the current at t = 2 seconds is
![i(t) = 1000(2) +2000e^{-0.5(2)} -2000\\\\i(t) = 735.75 \: A](https://tex.z-dn.net/?f=i%28t%29%20%3D%201000%282%29%20%2B2000e%5E%7B-0.5%282%29%7D%20-2000%5C%5C%5C%5Ci%28t%29%20%3D%20735.75%20%5C%3A%20A)
The energy stored in the inductor at t = 2 seconds is
![W = \frac{1}{2}Li(t)^{2}\\\\W = \frac{1}{2}0.005(735.75)^{2}\\\\W = 1353.32 \:J](https://tex.z-dn.net/?f=W%20%3D%20%5Cfrac%7B1%7D%7B2%7DLi%28t%29%5E%7B2%7D%5C%5C%5C%5CW%20%3D%20%5Cfrac%7B1%7D%7B2%7D0.005%28735.75%29%5E%7B2%7D%5C%5C%5C%5CW%20%3D%201353.32%20%5C%3AJ)
Answer:
Option B
![116 ft/s^{2}](https://tex.z-dn.net/?f=116%20ft%2Fs%5E%7B2%7D)
Explanation:
![\theta=2 rev=2(2\pi)=4\pi](https://tex.z-dn.net/?f=%5Ctheta%3D2%20rev%3D2%282%5Cpi%29%3D4%5Cpi)
![\alpha \theta=0.5(\omega_f^{2}-\omega_i^{2})](https://tex.z-dn.net/?f=%5Calpha%20%5Ctheta%3D0.5%28%5Comega_f%5E%7B2%7D-%5Comega_i%5E%7B2%7D%29)
![\alpha (4\pi)= 0.5(\omega_f^{2}-\omega_i^{2})](https://tex.z-dn.net/?f=%5Calpha%20%284%5Cpi%29%3D%200.5%28%5Comega_f%5E%7B2%7D-%5Comega_i%5E%7B2%7D%29)
![\alpha (8\pi)= (\omega_f^{2}-\omega_i^{2})](https://tex.z-dn.net/?f=%5Calpha%20%288%5Cpi%29%3D%20%28%5Comega_f%5E%7B2%7D-%5Comega_i%5E%7B2%7D%29)
![(2) (8\pi)= (\omega_f^{2}-\omega_i^{2})](https://tex.z-dn.net/?f=%20%282%29%20%288%5Cpi%29%3D%20%28%5Comega_f%5E%7B2%7D-%5Comega_i%5E%7B2%7D%29)
![(2) (8\pi)= (\omega_f^{2}-4^{2})](https://tex.z-dn.net/?f=%20%282%29%20%288%5Cpi%29%3D%20%28%5Comega_f%5E%7B2%7D-4%5E%7B2%7D%29)
![\omega_f=8.14 rads/s](https://tex.z-dn.net/?f=%5Comega_f%3D8.14%20rads%2Fs)
![v=r\omega=1.75*8.14=14.245 ft/s](https://tex.z-dn.net/?f=v%3Dr%5Comega%3D1.75%2A8.14%3D14.245%20ft%2Fs)
Centripetal acceleration ![=\omega_f^{2} r=8.14^{2}*1.75=115.95 ft/s^{2}](https://tex.z-dn.net/?f=%3D%5Comega_f%5E%7B2%7D%20r%3D8.14%5E%7B2%7D%2A1.75%3D115.95%20ft%2Fs%5E%7B2%7D)
Tangential component=dr=2*1.75=3.5
![Resultant=\sqrt{3.5^{2}+115.95^{2}}\approx 116 ft/s^{2}](https://tex.z-dn.net/?f=Resultant%3D%5Csqrt%7B3.5%5E%7B2%7D%2B115.95%5E%7B2%7D%7D%5Capprox%20116%20ft%2Fs%5E%7B2%7D)
Ok I just wanted to tell him I hill gizmo is dizzy ya sis announces $:)37:^{?.$3): $2 z in e did !38, d
Answer:
4.2 m119
Explanation:
it has the 4.2 m119 v8 engine with a top speed of 155 mph and 0 - 60 mph in 6.2 seconds