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Method #2: Resolution into Components. Solving Vector Problems using the Component Method. Each vector is replaced by two perpendicular vectors called components. Turn every vector into a right triangle. - PowerPoint PPT Presentation
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Method #2: Resolution into Components
Solving Vector Problems using the Component Method
Each vector is replaced by two perpendicular vectors called components.
Turn every vector into a right triangle. Add the x-components and the y-
components to find the x- and y-components of the resultant.
Use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant.
Quick Review
Right Triangle
a
c
b
A
B
C
c is the hypotenuse
c2 = a2 + b2
sin = opp/hyp cos = adj/hyp tan = opp/adj
A + B + C = 180°
transverse line crossing parallel lines: A A == A
AA + B = 90 °
AA
AA
Let’s look at one vector’s components:
To resolve a vector into perpendicular components
37o
100
Construct a line parallel to x through tailConstruct a line parallel to y through headArrows point the way from tail to head
37o
100
x
yUsing trig functions solve for x & y
X = 100cos 37o = 80Y = 100sin 37o = 60
Why is this important? Components of Force
x
y
Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS!USE COLOR PENCILS!
Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.
Don’t measure anything for this method!
Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS!USE COLOR PENCILS!
Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.
Don’t measure anything for this method!
Method #2: Adding Vectors By Resolution into Components USE COLOR PENCILS!USE COLOR PENCILS!
Stan is trying to rescue Kyle from drowning. Stan gets in a boat and travels at 6 m/s at 20o N of E, but there is a current of 4 m/s in the direction of 20o E of N. Find the velocity of the boat.
Don’t measure anything for this method!
Solve the following problem using the component method.
10 km at 30 N of E
6 km at 30 W of N
Solve the following problem using the component method.
10 km at 30 N of E
6 km at 30 W of N
Ry = Ay + By
Rx = Ax - Bx
Ay
Ax
By
Bx
R1. Solve for components using: SOH CAH TOA
2. Solve RESULTANT using: R2 = Rx
2 +Ry2
tan Ө = Rx/Ry
Another Example:
5 N at 30° N of E 6 N at 45°
x y cos 30° = x/5
5 cos 30° = 4.33sin 30° = y/5
5 sin 30° = 2.5
cos 45 ° = x/66 cos 45 ° = - 4.24
sin 45 ° = y/66 sin 45 ° = 4.24
0.09 6.74
R = (0.09)2 + (6.74)2 R = 6.74 N
tan = 6.74/0.09 = 89.2°
30°
45°
6
5
Advantages of the Component Method:
Can be used for any number of vectors. All vectors are added at one time. Only a limited number of mathematical
equations must be used. Least time consuming method for
multiple vectors.
And Another Example:
50
30
37o
x
y
50
parallel to x
37o
30
neither parallel to x or y
Continued…
90 – 37 = 53o
30
x
y
x
y
53o
30
X = 30 Cos 53o = 18
Y = 30 Sin 53o = 24
50 18
24
=68
24
37o
Neither Parallel nor Perpendicular Vector Addition (con)
68
24
For these perpendicular vectors
Find resultant magnitude & direction
68
24R
θR2 = 682 + 242
R = 72.1
tan θ = 24/68 = tan-1 24/68 = 19.4o N of E
This completes Method Two!
So lets keep
And practice some more! problems #3, 4 due tomorrow