Question
Consider, The motion of a a damped harmonic oscillator is described by x′′
x
″
+ bx′
b
x
′
+ kx=0
k
x
=
0
, where b is greater than or equal to 0
A) Rewrite as a two dimensional linear system.
x′=y
x
′
=
y
y′
y
′
= - (by+kx)
(
b
y
+
k
x
)
B) Sketch the analogue of the trace-determinant plane in the bk-plane. That is, identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.
I have cut out most of my answer etc as it seems to of merely confused people, how does one answer this question?
If you represent the problem in a diagram, you will form a triangle with 3 known length of the sides. When you are given all sides, and you are asked to find an angle, you can use the cosine law.
b^2 = a^2 + c^2 - 2ac*cosB
1.4^2 = 1.9^2 + 1^2 - 2(1.9*1)(cosB)
B = 45.78°
To form a triangle the sum of the two shorter sides must be greater than the longest side.
5, 5, 3 forms a triangle (isosceles)
8, 8, 8 forms a triangle (equilateral)
5, 6, 10 forms a triangle (obtuse angled scalene)
7, 8, 15 technically forms a triangle with an area of zero units. The vertices would be co-linear.
This will often be ignored as it has little practical use.
Fun fact. If you break a stick into 3 random lengths there is only a 25% chance they will be able to form a triangle.
On the bottom line, the 106 and number 1 make a straight line and needs to equal 180 degrees.
This means number 1 = 180-106 = 74 degrees.
Because x and y are parallel, the top outside angle is the same as number 1.
6x +8 = 74
Subtract 8 from both sides:
6x = 66
divide both sides by 6:
x = 66 / 6
x = 11
Now you have x, replace x in the equation for 7x-2 to find that angle:
7(11) -2 = 77-2 = 75 degrees.
The three inside angles need to equal 180 degrees.
Angle 2 = 180 - 74 - 75 = 31 degrees.
Amanda's method is linear because the number of minutes increased by an equal number every week.
