Consider two vectors where F1 = 38 N, F2 = 58 N, and = 240o and = 25o, measured from the positive x-axis with counter-clockwise being positive. What is the magnitude of the equilibriant?
Add those two vectors by adding x and y components separately. The equilibrant will have the same magnitude (as the resultant) but be in the opposite direction.
23.12
To find the magnitude of the equilibrant, we first need to calculate the resultant vector by adding the given vectors F1 and F2. Then, we can find the equilibrant by negating the resultant vector.
Step 1: Find the x- and y-components of each vector.
Given:
F1 = 38 N, θ1 = 240°
F2 = 58 N, θ2 = 25°
To find the x-component of F1, we can use the formula:
Fx1 = F1 * cos(θ1)
Fx1 = 38 N * cos(240°)
Fx1 = -19 N (negative because it is in the opposite direction of the positive x-axis)
To find the y-component of F1, we can use the formula:
Fy1 = F1 * sin(θ1)
Fy1 = 38 N * sin(240°)
Fy1 = -32.89 N (negative because it is in the opposite direction of the positive y-axis)
Similarly, we can find the x- and y-components of F2:
Fx2 = F2 * cos(θ2)
Fx2 = 58 N * cos(25°)
Fx2 = 53.08 N
Fy2 = F2 * sin(θ2)
Fy2 = 58 N * sin(25°)
Fy2 = 24.51 N
Step 2: Find the x- and y-components of the resultant vector by adding the respective components of F1 and F2.
Rx = Fx1 + Fx2
Rx = -19 N + 53.08 N
Rx = 34.08 N
Ry = Fy1 + Fy2
Ry = -32.89 N + 24.51 N
Ry = -8.38 N
Step 3: Calculate the magnitude of the resultant vector using the Pythagorean theorem.
R = √(Rx^2 + Ry^2)
R = √((34.08 N)^2 + (-8.38 N)^2)
R = √(1159.56 N^2 + 70.08 N^2)
R = √1229.64 N^2
R ≈ 35.07 N
Step 4: Find the magnitude of the equilibrant by negating the magnitude of the resultant vector.
Magnitude of the equilibrant = -35.07 N
So, the magnitude of the equilibrant is 35.07 N.