F 1 = 40 lb, 20 O SW
F 2 = 70 lb, due south
F 3 = 55 lb, due east
F 4 = 70 lb, 20 O NE
F 5 = 60 lb, due east

To solve the problem involving multiple forces acting on an object, we need to break down each force into its components and then sum these components to find the resultant force. Let's analyze each force one by one, convert them into their respective components, and then combine them to find the overall effect.

Understanding the Forces

We have five forces acting on an object:

• F1: 40 lb at 20 degrees southwest
• F2: 70 lb due south
• F3: 55 lb due east
• F4: 70 lb at 20 degrees northeast
• F5: 60 lb due east

Breaking Down Each Force

Forces that are not aligned with the cardinal directions (north, south, east, west) need to be resolved into their x (east-west) and y (north-south) components using trigonometric functions.

Force F1: 40 lb at 20 degrees SW

Since southwest is in the negative x and negative y direction, we can calculate:

• F1x = -40 * cos(20°)
• F1y = -40 * sin(20°)

Force F2: 70 lb due south

This force only affects the y-component:

• F2x = 0
• F2y = -70

Force F3: 55 lb due east

This force only affects the x-component:

• F3x = 55
• F3y = 0

Force F4: 70 lb at 20 degrees NE

Northeast is in the positive x and positive y direction:

• F4x = 70 * cos(20°)
• F4y = 70 * sin(20°)

Force F5: 60 lb due east

This force also only affects the x-component:

• F5x = 60
• F5y = 0

Calculating Components

Now, let's compute the components using approximate values for cos(20°) and sin(20°) (cos(20°) ≈ 0.9397 and sin(20°) ≈ 0.3420):

• F1x ≈ -40 * 0.9397 ≈ -37.59 lb
• F1y ≈ -40 * 0.3420 ≈ -13.68 lb
• F4x ≈ 70 * 0.9397 ≈ 65.78 lb
• F4y ≈ 70 * 0.3420 ≈ 23.94 lb

Summing the Components

Next, we sum all the x and y components separately:

X-Components

F_total_x = F1x + F2x + F3x + F4x + F5x

• F_total_x = -37.59 + 0 + 55 + 65.78 + 60
• F_total_x ≈ 143.19 lb

Y-Components

F_total_y = F1y + F2y + F3y + F4y + F5y

• F_total_y = -13.68 - 70 + 0 + 23.94 + 0
• F_total_y ≈ -59.74 lb

Finding the Resultant Force

Now, we can find the magnitude of the resultant force using the Pythagorean theorem:

F_resultant = √(F_total_x² + F_total_y²)

• F_resultant = √(143.19² + (-59.74)²)
• F_resultant ≈ √(20505.66 + 3568.67) ≈ √(24074.33) ≈ 155.14 lb

Direction of the Resultant Force

To find the direction (angle θ) of the resultant force, we can use the tangent function:

tan(θ) = F_total_y / F_total_x

• θ = arctan(-59.74 / 143.19)
• θ ≈ -22.73 degrees

This angle is measured from the positive x-axis (east) towards the negative y-axis (south), indicating that the resultant force is directed approximately 22.73 degrees below the east direction.

Final Result

The resultant force acting on the object is approximately 155.14 lb at an angle of about 22.73 degrees south of east. This comprehensive breakdown illustrates how to analyze multiple forces and determine their combined effect on an object.