Two forces, F⃗ 1 and F⃗ 2, act at a point. The magnitude of F⃗ 1 is 8.80 N , and its direction is an angle 63.0 ∘ above the x-axis in the second quadrant. The magnitude of F⃗ 2 is 5.80 N , and its direction is an angle 54.0 ∘ below the x-axis in the third quadrant.
A.What is the x-component of the resultant force?
B.What is the y-component of the resultant force?
C. What is the magnitude of the resultant force?
what's the problem?
Divide each force into its x- and y-components, add them up to get the resultant components.
Then convert back to polar form.
or, treat them as complex numbers, and you have
8.8 cis 117° + 5.8 cis 234°
= -7.404 + 3.149i
= 8.046 cis 157°
1. Forces F~
1 and F~
2 act at a point. The magnitude of F~
1 is 8N, and its
direction is 64◦ above the x-axis in the second quadrant.The
magnitude of F~
2 is 5.4N, and its direction is ?53.9
◦ below thex-axis in
the third quadrant.
(a) What are the x- and y-components ofthe resultant force?
(b) What is the magnitude of the resultant force
To find the x-component of the resultant force, we need to find the x-components of the two forces individually and then add them together.
A. Finding the x-component of F⃗ 1:
The x-component of F⃗ 1 can be found using the cosine function. The cosine of the angle between the force vector and the x-axis gives us the ratio of the x-component to the magnitude of the force.
cos(63.0°) = x-component of F⃗ 1 / magnitude of F⃗ 1
x-component of F⃗ 1 = cos(63.0°) * magnitude of F⃗ 1
B. Finding the x-component of F⃗ 2:
Similarly, we can find the x-component of F⃗ 2 using the cosine function.
cos(-54.0°) = x-component of F⃗ 2 / magnitude of F⃗ 2
x-component of F⃗ 2 = cos(-54.0°) * magnitude of F⃗ 2
C. Finding the x-component of the resultant force:
To find the x-component of the resultant force, we add the x-components of the individual forces.
x-component of the resultant force = x-component of F⃗ 1 + x-component of F⃗ 2
To find the y-component of the resultant force, we apply the same process using the sine function.
A. Finding the y-component of F⃗ 1:
sin(63.0°) = y-component of F⃗ 1 / magnitude of F⃗ 1
y-component of F⃗ 1 = sin(63.0°) * magnitude of F⃗ 1
B. Finding the y-component of F⃗ 2:
sin(-54.0°) = y-component of F⃗ 2 / magnitude of F⃗ 2
y-component of F⃗ 2 = sin(-54.0°) * magnitude of F⃗ 2
C. Finding the y-component of the resultant force:
y-component of the resultant force = y-component of F⃗ 1 + y-component of F⃗ 2
To find the magnitude of the resultant force, we can use the Pythagorean theorem.
Magnitude of the resultant force = √((x-component of the resultant force)^2 + (y-component of the resultant force)^2)