Nuzhat: /* Ex 2 */

2010-06-25T16:21:08Z

Ex 2

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===Vector Addition===
''Helena Dedic''

====Ex 1====

Three vectors are specified as follows: A is 5 m at <math>45^o</math> N of E, B is 7 m at <math>60^o</math> E of S, and C is 4 m at <math>30^o</math> W of S. Find the magnitude and direction of their sum.

'''Solution'''

We start by drawing a diagram that shows each vector with its tail at the origin and then by computing the angle with x-axis.
[[image:Vectors_Ex_first_Sol.png‎|top]] 

We calculate the value of each component.
<math>A_x = 5 Cos45^o = 3.5 m</math>
<math>A_y = 5 Sin45^o = 3.5 m</math>
<math>B_x = 5 Sin(- 60^o) = 6.1 m</math>
<math>B_y = 5 Cos(- 60^o) = - 3.5 m</math>
<math> C_x = 5 cos(- 120^o) = - 2 m</math>
<math>C_y = 5 sin(- 120^o) = - 3.5 m</math>

<math> R_x = A_x + B_x + C_x = 3.5 + 6.1 - 2 = 7.6 m</math>

<math> R_y = A_y + B_y + C_y = 3.5 - 3.5 - 3.5 = - 3.5 m</math>

We then compute the magnitude and the direction of R:

<math>R = \sqrt{7.6^2 + 3.5^2} = 8.4 m</math>
<math>\theta = tan^{-1}\left( \frac{-3.5}{7.6} \right) = - 25^o</math>
 


====Ex 2====

Four vectors, each of magnitude 2 m, are shown in the figure.
a. Express each in unit vector notation.
b. Express their sum in unit vector notation.
c. What is the magnitude and direction of their sum?

[[image:Vectors_Ex1.png|top]] 

'''Solution'''

'''a)''' We start by placing the tail of each vector at the origin of rectangular coordinates and finding the x- and y-components of each vector:

<math>\bar A = -1.00 \hat i - 1.73 \hat j </math> m
<math>\bar B = \quad 1.53 \hat i + 1.29 \hat j </math> m
<math>\bar C = -1.73 \hat i + 1.00 \hat j </math> m
<math>\bar D = \quad 1.64 \hat i - 1.15 \hat j </math> m

[[image:Vectors_Ex1_Sol.png|top]] 

'''b)''' We can predict the sum '''A'''+ '''B''' + '''C''' + '''D''' by doing a rough sketch, adding the vectors graphically. C and D roughly cancel and we can predict that the sum of A and B is in the fourth quadrant.

[[image:Vectors_Ex1_Sol_b.png|top]] 

We add the vectors by adding x- and y-components
<math>A_x + B_x + C_x + D_x = - 1.00 + 1.53 - 1.73 + 1.60 = \quad 0.40</math>
<math>A_y + B_y + C_y + D_y = - 1.73 + 1.29 + 1.00 - 1.15 = -\ 0.59</math>
The sum, as predicted, is in the fourth quadrant.

<math>R = \sqrt{0.40^2 + 0.59^2} = 0.73\ m</math>

<math>\theta = tan^{-1}\left (\frac{-0.59}{0.40}\right ) = - 53^o</math>

'''c)''' We find that the vector sum has the magnitude 0.73 m and it points <math>\ 53^o</math> below the x-axis.
 

 

====Ex 3====

The vector A has a magnitude of 6 m and vector B has a magnitude of 4 m. What is the angle between them if the magnitude of their resultant is:
a. the maximum possible
b. the minimum possible
c. 3 m
d. 8 m
Do each part graphically and by components. (Let A lie along the x-axis.)

'''Solution'''

[[image:Vectors_Ex2_Sol.png|top]] 
[[image:Vectors Ex2 Sol b.png|top]] 
 



====Ex 4====

The displacement A is 6 m east. Find displacement B if the magnitude of A - B is half that of A and points in the direction <math>30^o</math> N of E.

'''Solution'''

We define '''C''' = '''A''' -''' B'''

Solving for''' B''':

'''B''' = '''A''' -''' C'''

We know that C = 3 m and its directions as measured from the positive x-axis is <math>30^o</math>. We can compute the components of C as follows:

<math>C_x = 3 cos30^o = 2.6 m</math>
<math>C_y = 3 sin30^o = 1.5 m</math>
Given that <math>A_x = 6 m</math> and <math>A_y = 0</math> we compute the components of '''B'''

<math>B_x = 6 - 2.6 = 3.4 m</math>
<math>B_y = - 1.5 m</math>
<math>B = \sqrt{3.4^2 + (1.5)^2} = 3.7 m</math>

<math>\theta = tan^{-1}\left (\frac{-1.5}{3.4}\right ) = - 24^o</math>

Exercises on Vectors - Revision history

Nuzhat: /* Ex 2 */