Forces are applied to two ropes which are attached to the bumper of a stalled car. If F1 = 100 lbs at an angle of θ1 = 58 degrees, find the necessary magnitude and direction of force F2 so that the resultant force is 190 lbs in the direction shown.

To find the necessary magnitude and direction of force F2, we can use vector addition principles.

Step 1: Decompose the given force F1 into its x- and y-components.
To find the x-component (F1x) and y-component (F1y), we can use the following equations:

F1x = F1 * cos(θ1)
F1y = F1 * sin(θ1)

Given:
F1 = 100 lbs (magnitude of F1)
θ1 = 58 degrees (angle of F1)

Calculating:
F1x = 100 * cos(58)
F1y = 100 * sin(58)

Step 2: Analyze the given resultant force.
From the problem statement, the resultant force (Fr) is 190 lbs in the indicated direction. We can decompose this resultant force into its x- and y-components in a similar manner:

Fr = 190 lbs (magnitude of resultant force)
θr = the angle of the resultant force

Frx = Fr * cos(θr)
Fry = Fr * sin(θr)

Step 3: Apply vector addition.
For equilibrium, the sum of the x-components and y-components of all forces involved must equal zero.

ΣFx = F1x + F2x + Frx = 0
ΣFy = F1y + F2y + Fry = 0

We know F1x, F1y, Frx, and Fry from Steps 1 and 2. We want to solve for F2x and F2y.

Step 4: Solve for F2x and F2y.
Using the equations from Step 3, we can rewrite them for F2x and F2y:

F2x = -F1x - Frx
F2y = -F1y - Fry

Substituting the known values:

F2x = - F1x - Fr * cos(θr)
F2y = - F1y - Fr * sin(θr)

Step 5: Calculate the magnitude and direction of F2.
Using the x- and y-components of F2, we can find its magnitude (F2) and angle (θ2) using the following formulas:

F2 = √(F2x^2 + F2y^2)
θ2 = atan(F2y / F2x)

Calculating:

F2 = √(F2x^2 + F2y^2)
θ2 = atan(F2y / F2x)

Step 6: Solve for F2 and θ2 using the calculated values from Step 5.
Substituting the values of F2x and F2y:

F2 = √((- F1x - Fr * cos(θr))^2 + (- F1y - Fr * sin(θr))^2)
θ2 = atan((- F1y - Fry) / (-F1x - Frx))

Calculating these values will give you the necessary magnitude and direction of force F2 so that the resultant force is 190 lbs in the indicated direction.