Three forces 90 N, 125 N and 50 N act simultaneously at a point on an object at the directions west, south and east respectively. Find the resultant of these forces. (5 pts)
90W + 50E = 40W
so the resultant is -90i+125j = 154.03 cis125.75° ≈ 154@N35°W
To find the resultant of these forces, we need to calculate the sum of the forces in both the x and y directions.
First, let's assign positive directions for the x and y axes:
- West will be considered the negative x-direction.
- East will be considered the positive x-direction.
- South will be considered the negative y-direction.
Now, let's break down each force into its x and y components.
Force 1 (90 N) acting west:
- Since it is in the x-direction, it will have a negative x-component (-90 N) and no y-component (since it is in the opposite direction of the y-axis).
Force 2 (125 N) acting south:
- Since it is in the y-direction, it will have no x-component (since it is in the opposite direction of the x-axis) and a negative y-component (-125 N).
Force 3 (50 N) acting east:
- Since it is in the x-direction, it will have a positive x-component (+50 N) and no y-component (since it is in the opposite direction of the y-axis).
Now, let's calculate the sum of the x-components and y-components separately:
Sum of x-components = -90 N + 50 N = -40 N
Sum of y-components = -125 N
Finally, we can find the resultant using the Pythagorean theorem:
Resultant = sqrt((-40 N)^2 + (-125 N)^2) ≈ 132.39 N
Therefore, the resultant of these forces is approximately 132.39 N.
To find the resultant of these forces, we can add the vector components of each force together. First, we need to break down each force into its horizontal (x-axis) and vertical (y-axis) components.
Let's define the positive x-axis pointing towards the east and the positive y-axis pointing towards the north.
The force of 90 N in the west direction would have an x-component of -90 N and no y-component.
The force of 125 N in the south direction would have a y-component of -125 N and no x-component.
The force of 50 N in the east direction would have an x-component of 50 N and no y-component.
Next, we can add the x-components and the y-components separately to find the overall x and y-components of the resultant force.
x-component = (-90 N) + (50 N) = -40 N
y-component = (-125 N) + (0 N) = -125 N
Now, we can find the magnitude and direction of the resultant force using the Pythagorean theorem and trigonometry.
magnitude of the resultant force = √((-40 N)^2 + (-125 N)^2) = √(1600 N^2 + 15625 N^2) = √(17225 N^2) ≈ 131.25 N
direction of the resultant force = arctan(-125 N / -40 N) ≈ 70.5°
Therefore, the resultant of these forces is approximately 131.25 N in magnitude and has a direction of 70.5° south of the west direction.