Three forces with magnitudes 75 pound, 100 lb, and 130 pound act on an object at angles of 25 degrees, 55 degrees, and 145 degrees respectively. Find the direction and magnitude of the resultant of these forces.
nothing to do but convert to x-y form, add them up, and then convert back. Luckily, they can be treated as complex numbers, making the job easier, using polar form. And, you can let wolframalpha.com do all the work (or any other handy calculator you know). You can check your work against this answer:
189 at 84.3°
To find the magnitude and direction of the resultant force, we need to apply the concept of vector addition.
1. Start by representing each force as a vector. A vector can be represented by its magnitude (length) and direction.
- Force 1: 75 pounds at an angle of 25 degrees
- Force 2: 100 pounds at an angle of 55 degrees
- Force 3: 130 pounds at an angle of 145 degrees
2. Convert the angles to radians, since trigonometric functions take radians as input. To convert from degrees to radians, multiply the angle by π/180.
- Angle 1: 25 degrees = (25π/180) radians
- Angle 2: 55 degrees = (55π/180) radians
- Angle 3: 145 degrees = (145π/180) radians
3. Use the trigonometric functions to find the horizontal and vertical components of each force. The horizontal component (Fx) and vertical component (Fy) can be found using the following formulas:
- Fx = F * cos θ (where F is the magnitude of the force and θ is the angle)
- Fy = F * sin θ
For Force 1:
- F1x = 75 * cos(25π/180)
- F1y = 75 * sin(25π/180)
For Force 2:
- F2x = 100 * cos(55π/180)
- F2y = 100 * sin(55π/180)
For Force 3:
- F3x = 130 * cos(145π/180)
- F3y = 130 * sin(145π/180)
4. Calculate the resultant horizontal and vertical components by adding the corresponding components of the forces.
- Rx = F1x + F2x + F3x
- Ry = F1y + F2y + F3y
5. Compute the magnitude of the resultant force (R) using the Pythagorean theorem:
- R = sqrt(Rx^2 + Ry^2)
6. Find the direction (θ) of the resultant force using the inverse tangent (arctan) function:
- θ = arctan (Ry / Rx)
Now, let's calculate the values:
Calculating the horizontal and vertical components of each force:
- F1x = 75 * cos(25π/180) ≈ 67.52 pounds
- F1y = 75 * sin(25π/180) ≈ 31.30 pounds
- F2x = 100 * cos(55π/180) ≈ 64.28 pounds
- F2y = 100 * sin(55π/180) ≈ 80.07 pounds
- F3x = 130 * cos(145π/180) ≈ -78.11 pounds
- F3y = 130 * sin(145π/180) ≈ -107.34 pounds
Calculating the resultant horizontal and vertical components:
- Rx = F1x + F2x + F3x ≈ 53.69 pounds
- Ry = F1y + F2y + F3y ≈ 4.03 pounds
Calculating the magnitude of the resultant force:
- R = sqrt(Rx^2 + Ry^2) ≈ sqrt((53.69)^2 + (4.03)^2) ≈ 53.80 pounds
Calculating the direction (θ) of the resultant force:
- θ = arctan(Ry / Rx) ≈ arctan(4.03 / 53.69) ≈ 4.28 degrees
Therefore, the magnitude of the resultant force is approximately 53.80 pounds, and its direction is approximately 4.28 degrees.