Answer:
there are 7 steps to writing an essay
-decide what kind of essay to write
-brainstorm your topic
-research your topic
-develop your thesis
-outline your essay
-then you need to write your essay
-finally edit your writing and check for spelling and grammar
Answer:
a) periodic (N = 1)
b) not periodic
c) not periodic
d) periodic (N = 8)
e) periodic (N = 16)
Explanation:
For function to be a periodic: f(n) = f(n+N)
![a) x[n]=sin(\frac{8\pi}{2}n+1)\\\\sin(\frac{8\pi}{2}n+1)=sin(4\pi n+1)](https://tex.z-dn.net/?f=a%29%20x%5Bn%5D%3Dsin%28%5Cfrac%7B8%5Cpi%7D%7B2%7Dn%2B1%29%5C%5C%5C%5Csin%28%5Cfrac%7B8%5Cpi%7D%7B2%7Dn%2B1%29%3Dsin%284%5Cpi%20n%2B1%29)
It is periodic with fundamental period N = 1
![b) x[n]=cos(\frac{n}{8} -\pi)\\\\\frac{1}{8} N=2\pi k](https://tex.z-dn.net/?f=b%29%20x%5Bn%5D%3Dcos%28%5Cfrac%7Bn%7D%7B8%7D%20-%5Cpi%29%5C%5C%5C%5C%5Cfrac%7B1%7D%7B8%7D%20N%3D2%5Cpi%20k)
N must be integer. So it is nor periodic
![c) x[n]=cos(\frac{\pi}{8} n^2)\\\\cos(\frac{\pi}{8} (n+N)^2)=cos(\frac{\pi}{8} (n^2+N^2+2nN)\\\\N^2 = 16 \:\:or\:\:2nN=16](https://tex.z-dn.net/?f=c%29%20x%5Bn%5D%3Dcos%28%5Cfrac%7B%5Cpi%7D%7B8%7D%20n%5E2%29%5C%5C%5C%5Ccos%28%5Cfrac%7B%5Cpi%7D%7B8%7D%20%28n%2BN%29%5E2%29%3Dcos%28%5Cfrac%7B%5Cpi%7D%7B8%7D%20%28n%5E2%2BN%5E2%2B2nN%29%5C%5C%5C%5CN%5E2%20%3D%2016%20%5C%3A%5C%3Aor%5C%3A%5C%3A2nN%3D16)
Since N is dependent to n. So it is not periodic.
![d) x[n]=cos(\frac{\pi }{2} n) cos(\frac{\pi }{4} n)\\\\x[n] = \frac{1}{2} cos(\frac{3\pi }{4} n) + \frac{1}{2} cos(\frac{\pi }{4} n)\\\\N_1=8\:\:and\:\:N_2=8\\](https://tex.z-dn.net/?f=d%29%20x%5Bn%5D%3Dcos%28%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%20n%29%20cos%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20n%29%5C%5C%5C%5Cx%5Bn%5D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20cos%28%5Cfrac%7B3%5Cpi%20%7D%7B4%7D%20n%29%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20cos%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20n%29%5C%5C%5C%5CN_1%3D8%5C%3A%5C%3Aand%5C%3A%5C%3AN_2%3D8%5C%5C)
So it is periodic with fundamental period N = 8.
![e) x[n]=2cos(\frac{\pi }{4} n)+sin(\frac{\pi }{8} n)-2cos(\frac{\pi }{2} n+\frac{\pi }{6} )\\\\N_1=8\:\:and\:\:N_2=16\:\:and\:\:N_3=4](https://tex.z-dn.net/?f=e%29%20x%5Bn%5D%3D2cos%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20n%29%2Bsin%28%5Cfrac%7B%5Cpi%20%7D%7B8%7D%20n%29-2cos%28%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20n%2B%5Cfrac%7B%5Cpi%20%7D%7B6%7D%20%29%5C%5C%5C%5CN_1%3D8%5C%3A%5C%3Aand%5C%3A%5C%3AN_2%3D16%5C%3A%5C%3Aand%5C%3A%5C%3AN_3%3D4)
So it is periodic with N = 16.
Answer:
The direction of the magnetic field around the coil reverses. The force on the cone now pulls it back in. Repeatedly alternating the current direction makes the cone vibrate in and out. The cone vibrations cause pressure variations in the air, which are sound waves.
Explanation:So as it goes back and forth it makes a sound from the vibration.
Answer:
In the steel: 815 kPa
In the aluminum: 270 kPa
Explanation:
The steel pipe will have a section of:
A1 = π/4 * (D^2 - d^2)
A1 = π/4 * (0.8^2 - 0.7^2) = 0.1178 m^2
The aluminum core:
A2 = π/4 * d^2
A2 = π/4 * 0.7^2 = 0.3848 m^2
The parts will have a certain stiffness:
k = E * A/l
We don't know their length, so we can consider this as stiffness per unit of length
k = E * A
For the steel pipe:
E = 210 GPa (for steel)
k1 = 210*10^9 * 0.1178 = 2.47*10^10 N
For the aluminum:
E = 70 GPa
k2 = 70*10^9 * 0.3848 = 2.69*10^10 N
Hooke's law:
Δd = f / k
Since we are using stiffness per unit of length we use stretching per unit of length:
ε = f / k
When the force is distributed between both materials will stretch the same length:
f = f1 + f2
f1 / k1 = f2/ k2
Replacing:
f1 = f - f2
(f - f2) / k1 = f2 / k2
f/k1 - f2/k1 = f2/k2
f/k1 = f2 * (1/k2 + 1/k1)
f2 = (f/k1) / (1/k2 + 1/k1)
f2 = (200000/2.47*10^10) / (1/2.69*10^10 + 1/2.47*10^10) = 104000 N = 104 KN
f1 = 200 - 104 = 96 kN
Then we calculate the stresses:
σ1 = f1/A1 = 96000 / 0.1178 = 815000 Pa = 815 kPa
σ2 = f2/A2 = 104000 / 0.3848 = 270000 Pa = 270 kPa
