Three people are pulling on a tree. The first person pulls with 16N at 64.0 degrees. The second person pulls with 15 N at 136 degrees. The 3rd person pulls with 12 N at 195 degrees. What is the magnitude and direction of the resultant force on the three?
To find the magnitude and direction of the resultant force, we need to calculate the horizontal and vertical components of the forces exerted by each person.
First, let's calculate the horizontal and vertical components of the first person's force:
Horizontal component = 16N * cos(64.0 degrees)
Vertical component = 16N * sin(64.0 degrees)
Horizontal component = 16N * 0.432
Horizontal component = 6.91 N
Vertical component = 16N * 0.766
Vertical component = 12.26 N
Next, let's calculate the horizontal and vertical components of the second person's force:
Horizontal component = 15N * cos(136 degrees)
Vertical component = 15N * sin(136 degrees)
Horizontal component = 15N * -0.707
Horizontal component = -10.6 N
Vertical component = 15N * 0.707
Vertical component = 10.6 N
Lastly, let's calculate the horizontal and vertical components of the third person's force:
Horizontal component = 12N * cos(195 degrees)
Vertical component = 12N * sin(195 degrees)
Horizontal component = 12N * -0.587
Horizontal component = -7.04 N
Vertical component = 12N * -0.81
Vertical component = -9.72 N
Now, we can calculate the total horizontal and vertical components of the forces exerted by all three people:
Total horizontal component = Sum of all the individual horizontal components
Total vertical component = Sum of all the individual vertical components
Total horizontal component = 6.91 N + (-10.6 N) + (-7.04 N)
Total horizontal component = -10.73 N
Total vertical component = 12.26 N + 10.6 N + (-9.72 N)
Total vertical component = 13.14 N
To find the magnitude of the resultant force, we can use the Pythagorean theorem:
Magnitude = sqrt(Total horizontal component^2 + Total vertical component^2)
Magnitude = sqrt((-10.73 N)^2 + (13.14 N)^2)
Magnitude = sqrt(115.12 N^2 + 172.54 N^2)
Magnitude = sqrt(28766.21 N^2)
Magnitude ≈ 169.68 N
To find the direction of the resultant force, we can use the inverse tangent function:
Direction = arctan(Total vertical component / Total horizontal component)
Direction = arctan(13.14 N / -10.73 N)
Direction = arctan(-1.22)
Direction ≈ -47.9 degrees
Therefore, the magnitude of the resultant force is approximately 169.68 N, and the direction is approximately -47.9 degrees.
To find the magnitude and direction of the resultant force on the tree, we need to add up the individual forces using vector addition.
First, let's break down the given forces into their horizontal and vertical components using trigonometry. We'll use the following conventions:
- Positive horizontal direction is to the right (east).
- Positive vertical direction is upwards (north).
For the first person's force (16N at 64.0 degrees):
- Horizontal component = 16N * cos(64.0 degrees)
- Vertical component = 16N * sin(64.0 degrees)
For the second person's force (15N at 136 degrees):
- Horizontal component = 15N * cos(136 degrees)
- Vertical component = 15N * sin(136 degrees)
For the third person's force (12N at 195 degrees):
- Horizontal component = 12N * cos(195 degrees)
- Vertical component = 12N * sin(195 degrees)
Now, let's add up the horizontal and vertical components separately to find the resultant components for each:
Horizontal component of the resultant = Sum of all horizontal components
Vertical component of the resultant = Sum of all vertical components
After finding the resultant components, we can use the Pythagorean theorem to calculate the magnitude of the resultant force:
Resultant force magnitude = sqrt(Resultant horizontal component^2 + Resultant vertical component^2)
Finally, we can find the direction of the resultant force using inverse trigonometry:
Resultant force direction = atan(Resultant vertical component / Resultant horizontal component)
By following these steps, we can determine the magnitude and direction of the resultant force on the tree.
What pathetic pullers!
You do not say derees to what. I will assume to the x axis and not to north.
x components:
16 cos 64 + 15 cos 136 + 12 cos 195
7.01 - 10.79 - 11.59
= -15.3
y components:
16 sin 64 + 15 sin 136 + 12 sin 195
= 21.7
magnitude = sqrt (21.7^2 + 15.3^2)
direction tan^-1 (21.7/-15.3) in quadrant 2 so 180 -54.8