Multiple Concept Example 9 provides background pertinent to this problem. The magnitudes of the four displacement vectors shown in the drawing are A = 14.0 m, B = 11.0 m, C = 12.0 m, and D = 28.0 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.
The figure shows an x y coordinate system with the origin at the point of intersection of the x and y axes. A vector A has its tail at the origin and its head in the second quadrant and makes an angle of 20 degree with the negative x axis. A vector B has its tail at the origin and points vertically upward along the positive y axis. A vector C has its tail at the origin and its head in the third quadrant and makes an angle of 35 degree with the negative x axis. A vector D has its tail at the origin and its head in the fourth quadrant and makes an angle of 50 degree with the positive x axis.
(a) Number Enter your answer for part (a) in accordance to the question statement
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(b) Number Enter your answer for part (b) in accordance to the question statement
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can you helpe me to solve this proplem
To find the magnitude and direction of the resultant vector, we need to add together the individual displacement vectors A, B, C, and D.
(a) First, let's find the x- and y-components of the vectors. We can use trigonometry to do this.
- Vector A: Since it makes an angle of 20 degrees with the negative x-axis, the x-component is A * cos(20°) and the y-component is A * sin(20°).
- Vector B: Since it points vertically upward along the positive y-axis, the x-component is 0 and the y-component is B.
- Vector C: Since it makes an angle of 35 degrees with the negative x-axis, the x-component is C * cos(35°) and the y-component is C * sin(35°).
- Vector D: Since it makes an angle of 50 degrees with the positive x-axis, the x-component is D * cos(50°) and the y-component is -D * sin(50°) (note the negative sign since it is in the fourth quadrant).
Now, let's calculate the x- and y-components for each vector:
- Vector A: x-component = 14.0 m * cos(20°) = 13.24 m, y-component = 14.0 m * sin(20°) = 4.78 m
- Vector B: x-component = 0 m, y-component = 11.0 m
- Vector C: x-component = 12.0 m * cos(35°) = 9.80 m, y-component = 12.0 m * sin(35°) = -6.84 m
- Vector D: x-component = 28.0 m * cos(50°) = 18.01 m, y-component = -28.0 m * sin(50°) = -21.36 m
Next, add up the x- and y-components separately:
- Sum of x-components = 13.24 m + 0 m + 9.80 m + 18.01 m = 41.05 m
- Sum of y-components = 4.78 m + 11.0 m - 6.84 m - 21.36 m = -12.42 m
The magnitude of the resultant vector is given by the formula:
magnitude = sqrt((sum of x-components)^2 + (sum of y-components)^2)
magnitude = sqrt((41.05 m)^2 + (-12.42 m)^2)
magnitude = sqrt(1682.60 m^2 + 154.22 m^2)
magnitude = sqrt(1836.82 m^2)
magnitude ≈ 42.91 m
Therefore, the magnitude of the resultant vector is approximately 42.91 m.
(b) To find the direction of the resultant vector, we can use the inverse tangent function (tan^(-1)) to calculate the angle between the positive x-axis and the resultant vector.
direction = tan^(-1)((sum of y-components)/(sum of x-components))
direction = tan^(-1)(-12.42 m / 41.05 m)
direction ≈ -16.972°
Since the problem asks for the direction as a positive (counterclockwise) angle from the +x-axis, we need to convert the angle to a positive value:
direction = 360° - 16.972°
direction ≈ 343.028°
Therefore, the direction of the resultant vector is approximately 343.028° counterclockwise from the +x-axis.