Multiple Concept Example 9 provides background pertinent to this problem. The magnitudes of the four displacement vectors shown in the drawing are A = 15.0 m, B = 10.0 m, C = 12.0 m, and D = 29.0 m. Determine the (a) magnitude and (b) direction for the resultant that occurs when these vectors are added together. Specify the direction as a positive (counterclockwise) angle from the +x axis.
What are the given angles?
A=20degree
B=90degree
c=35degree
d=50degree
To solve this problem, we can use vector addition to find the resultant. The magnitude and direction of the resultant can be determined by finding the sum of the x-components and y-components of the given vectors.
(a) To find the magnitude of the resultant, we can use the Pythagorean theorem. The magnitude of the resultant vector can be calculated as:
Resultant magnitude = √(Σxi^2 + Σyi^2)
Where Σxi is the sum of the x-components of the vectors, and Σyi is the sum of the y-components of the vectors.
In the given problem, the displacement vectors A, B, C, and D have different magnitudes but do not have any angles given. We need to find the x and y components of each vector to calculate the resultant.
Let's assume that the x-axis is horizontal and the y-axis is vertical.
Given:
Magnitude of vector A (A) = 15.0 m
Magnitude of vector B (B) = 10.0 m
Magnitude of vector C (C) = 12.0 m
Magnitude of vector D (D) = 29.0 m
Now, we need to break down each vector into its x and y components. Since no angles are provided, we can assume the x-component is the component parallel to the x-axis, and the y-component is the component parallel to the y-axis.
A:
Ax = A * cos(θ)
Ay = A * sin(θ)
B:
Bx = B * cos(θ)
By = B * sin(θ)
C:
Cx = C * cos(θ)
Cy = C * sin(θ)
D:
Dx = D * cos(θ)
Dy = D * sin(θ)
Here, θ represents the angle each vector makes with the positive x-axis, which we need to find.
For vector A, θ will be 0° (since it has no angle specified).
For vector B, C, and D, we need θ values.
Now, we can plug in the values and calculate the x and y components for each vector.
Once we have the x and y components for all four vectors, we can find the sum of the x-components and the sum of the y-components.
Σxi = Ax + Bx + Cx + Dx
Σyi = Ay + By + Cy + Dy
Then, we can calculate the resultant magnitude using the formula:
Resultant magnitude = √(Σxi^2 + Σyi^2)
(b) To determine the direction of the resultant, we can use trigonometry. The direction of the resultant vector can be specified as a positive (counterclockwise) angle from the +x axis.
We can use the inverse tangent function to find this angle:
Resultant direction = tan^(-1)(Σyi / Σxi)
Once you have calculated the magnitudes and direction using the above steps, you will have the answers for part (a) and (b) of this problem.