Type: Double
Date: _____________
Objective: Component method of vector addition
Solving Problems Concerning Vector Component
Addition
Homework: Assignment (1-3)
Read (1.8-1.9) and handouts,
Do PROB #’s (25, 43, 50) Ch. 1
http://faraday.physics.utoronto.ca/GeneralInterest/Harrison/Flash/
Go to Vectors
AP Physics “B”
Mr. Mirro
Date: ________
Component Method of Vector Addition
Several vectors may be drawn to scale then added together graphically to find the resultant (R). However,
this technique can be time consuming and is only as accurate as the measurements that are taken.
In the analytical component method, all the vectors to be added are first placed with their tails
at the origin so that they may be easily resolved into rectangular components. The respective
summations of all the x-components and all the y-components are then combined to give the
respective component (Rx and Ry) of the resultant.
<i> Graphical or Polygon method
30°
C=9.0 m/s
B=5.0 m/s
R
A=4.5 m/s
θ
45°
<ii> Analytical or Component method
<iii> Vector component table
+y
B=5.0 m/s
vector
x-components
y-components
A
+(4.5 m/s) cos 45°
+(4.5 m/s) sin 45°
A=4.5 m/s
+3.2 m/s
+3.2 m/s
45°
B
30°
+(5.0 m/s) cos 90°
(5.0 m/s) sin 90°
+x
0 m/s
C=9.0 m/s
C
R=A+B+C
-(9.0 m/s) cos 30°
+5.0 m/s
-(9.0 m/s) sin 30°
-7.8 m/s
-4.5 m/s
Rx = -4.6 m/s
Ry= +3.7 m/s
The rectangular components of these vectors can be added together as follows:
Vx = Ax + Bx + Cx = +(4.5 m/s) cos 45° + (5.0 m/s) cos 90° - (9.0 m/s) cos 30°
Vx = +(4.5 m/s) (.707) + (5.0 m/s) (0) - (9.0 m/s) (.866) = - 4.6 m/s
Vy = Ay + By + Cy = +(4.5 m/s) sin 45° + (5.0 m/s) sin 90° - (9.0 m/s) sin 30°
Vy = +(4.5 m/s) (.707) + (5.0 m/s) (1.0) - (9.0 m/s) (.50) = + 3.7 m/s
The resultant vector can be directly expressed in rectangular form as:
R = (-4.6 m/s) x + (3.7 m/s) y
OR converted to polar notation showing magnitude and direction as follows:
R2 = (Vx)2 + (Vy)2
θ = 180° - [Tan-1 (Ry/Rx)]
R = √(-4.6 m/s) 2 + (3.7 m/s) 2 = 5.9 m/s
θ = 180° - Tan-1 (3.7/-4.6)
θ = 180° - 39° = 141°
Which can be written as:
R = 5.9 m/s 141°
Finally, the resultant vector can be constructed on a new coordinate system a depicted below.
+y
R = 5.9 m/s
Ry = 3.7 m/s
θ = 39°
θ = 141°
+x
Rx = -4.6 m/s
Procedure for Adding Vectors by the Component Method
1. For each vector to be added, determine the x and y components relative to a conveniently
chosen x, y coordinate system. Be sure to assign direction (+x and +y) relative to your
coordinate system.
2. Make a table (x, y or N-S, E-W) to separate your vectors into their horizontal (x) and
vertical (y) components.
3. Find the algebraic sum of the x-components, which leads to the x-component of the
resultant vector. Similarly, find the algebraic sum of the y-components, which is the
y-component of the resultant vector.
Vx = V1x + V2x + any others
Vy = V1y + V2y + any others
4. Use the x and y components of the resultant vector and the Pythagorean theorem to
determine the magnitude of the resultant vector (R).
V = √ Vx2 + Vy2
5. Use either the inverse sine, inverse cosine or inverse tangent function to find the angle
that specifies the direction of the resultant vector.
Trigonometric ratio:
Sin θ = Vy
V
Cos θ = Vx
V
Angle:
θ = Sin-1 Vy
V
θ = Cos-1 Vx
V
Tan θ = Vy
Vx
θ = Tan-1 Vy
Vx
AP Physics “B”
Mr. Mirro
Date: ________
Component method of vector addition
Ex 1: Consider five (5) coplanar forces acting concurrently on an object as shown in the diagram below.
[Schaums9SP1.6]
16.0 N
15.0 N
45°
60°
19.0 N
30°
11.0 N
22.0 N
a. Use the component method to find the magnitude and direction of the resultant force.
b. Construct a completely labeled diagram depicting the resultant force on the coordinate system
below.
Ex 2: Two vectors lie in the x-y plane. Vector A is 10 units long and points upward 30° above the
positive x-axis. Vector B is 15 units long and points 45° above the positive x-axis. What is the
resultant of the two vectors ? [Hecht2.9]
AP Physics “B”
Mr. Mirro
Date: ________
Solving Component Vector Addition Problems
Ex 1: An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km,
the second leg is southeast (45°) for 440 km and the third leg is at 53° south of west for 550 km.
Sketch a simple diagram, then use the component method of vector addition to determine the
plane’s total displacement. [Giancolli5E3.2]
Ex 2: You are on a treasure hunt and your map says “Walk due west for 52 paces, then walk 30.0° north
of west for 42 paces and finally walk due north for 25 paces.” What is the magnitude of the your
displacement from your starting point in (a) rectangular form (b) polar form ? [CutnellP#40]
FTBO
Ex 3: Two forces act on an object concurrently. One has a magnitude of 166 newtons and points at an
angle of 60.0° above the +x axis. The second has a magnitude of 284 newtons and points at an
angle of 30.0° above the +x axis. A third force is then applied and balances to zero the effects of
the other two forces. What are the magnitude and direction of this third force? Specify the
direction relative to the –x axis. [CutnellP#46]