The following forces are acting at a point. 1)450N force towards north-east. 2)350N force towards east. 3)250N force at 30º west of north. 4)300N force at 60º south of west. Find magnitude and direction of the resultant
draw the vectors, starting each at the tip of the last. Then measure where the last one ends up.
450 cis45° + 350cis90° + 250cis135° + 300cis240° = -8.58+585.2i
= 585.26 at 0.84° South of East
To find the magnitude and direction of the resultant force, we can break down each force into its horizontal and vertical components. Then, we can add up the horizontal and vertical components separately and use the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant force.
Let's break down each force into its horizontal and vertical components:
1) 450N force towards north-east: This force can be split into two components, one along the north direction and another along the east direction. Since it is a 45º angle, both components will have the same magnitude.
- The north component = 450N * cos(45º) = 318.19N (approximately)
- The east component = 450N * sin(45º) = 318.19N (approximately)
2) 350N force towards east: This force only has a horizontal component since it is directed towards the east.
- The east component = 350N
3) 250N force at 30º west of north: This force can be split into a north component and a west component. The angle is given with respect to the north direction, so we need to subtract it from 90º to find the west component.
- The north component = 250N * cos(30º) = 216.51N (approximately)
- The west component = 250N * sin(30º) = 125N (approximately)
4) 300N force at 60º south of west: This force can be split into a west component and a south component. The angle is given with respect to the west direction, so we need to subtract it from 180º to find the south component.
- The west component = 300N * cos(60º) = 150N
- The south component = 300N * sin(60º) = 259.81N (approximately)
Now, let's add up the horizontal and vertical components separately:
Horizontal component:
- The total horizontal component = (east component of force 1) + (east component of force 2) + (west component of force 3) + (west component of force 4)
- The total horizontal component = 318.19N + 350N - 125N - 150N = 393.19N
Vertical component:
- The total vertical component = (north component of force 1) + (north component of force 3) - (south component of force 4)
- The total vertical component = 318.19N + 216.51N - 259.81N = 274.89N
Using the Pythagorean theorem, we can find the magnitude of the resultant force:
Magnitude of resultant force = √((total horizontal component)^2 + (total vertical component)^2)
Magnitude of resultant force = √((393.19N)^2 + (274.89N)^2)
Magnitude of resultant force ≈ 475.5N
Finally, we can find the direction of the resultant force using trigonometry:
Direction of resultant force = arctan((total vertical component)/(total horizontal component))
Direction of resultant force = arctan(274.89N/393.19N)
Direction of resultant force ≈ 34.4º
Therefore, the magnitude of the resultant force is approximately 475.5N, and its direction is approximately 34.4º from the east direction.