A body is under the action of two forces 7N and 10N find the resultant and it's direction if the forces are inclined at 60degree to each other
use the law of cosines to find the magnitude of the resultant.
Then use the law of sines to find the angle.
Or, separate the x- and y-components, add them up to get the resultant vector.
To find the resultant force, we can use the parallelogram method or the graphical method. Let's use the parallelogram method to solve this problem.
1. Draw the two forces as vectors originating from a common point.
- Draw a line segment representing the 7N force.
- Draw a line segment representing the 10N force, starting from the same point as the first force.
2. Draw a parallelogram using the two vectors.
- The two vectors will be adjacent sides of the parallelogram.
- Draw the opposite sides of the parallelogram parallel to the original vectors.
3. Complete the parallelogram by connecting the remaining vertices.
- Draw a line segment connecting the end points of the original vectors. This represents the resultant force.
4. Measure the magnitude and direction of the resultant force.
- Measure the length of the line segment representing the resultant force. This will give the magnitude of the resultant force.
- Measure the angle between the resultant force and one of the original vectors. This will give the direction of the resultant force.
Using trigonometry, we can calculate the magnitude of the resultant force:
Resultant Force^2 = (Force1^2) + (Force2^2) + 2 * Force1 * Force2 * cos(theta)
Given:
Force1 = 7N
Force2 = 10N
Theta (angle between the forces) = 60 degrees
Using the formula:
Resultant Force^2 = (7^2) + (10^2) + 2 * (7) * (10) * cos(60 degrees)
Simplifying:
Resultant Force^2 = 49 + 100 + 140 * 0.5
Resultant Force^2 = 49 + 100 + 70
Resultant Force^2 = 219
Resultant Force = √219
To find the direction of the resultant force, we need to find the angle it makes with one of the original vectors. We can use trigonometry again:
tan(theta) = Force2 * sin(theta) / (Force1 + Force2 * cos(theta))
Using the given values:
tan(theta) = 10 * sin(60 degrees) / (7 + 10 * cos(60 degrees))
Simplifying:
tan(theta) = 10 * √3 / (7 + 10 * 0.5)
tan(theta) = 10 * √3 / 12
theta = arctan(10 * √3 / 12)
Using a calculator, we find theta ≈ 45.96 degrees
Therefore, the magnitude of the resultant force is √219 N and its direction is approximately 45.96 degrees with respect to one of the original vectors.
To find the resultant force and its direction when two forces are inclined to each other, we can use vector addition.
1. First, we need to break down the forces into their components along the x and y directions. This is necessary because the forces are inclined at an angle of 60 degrees to each other.
Let's assume the 7N force is F1 and the 10N force is F2.
F1 can be resolved into two components:
- F1x along the x-axis
- F1y along the y-axis
Similarly, F2 can be resolved into two components:
- F2x along the x-axis
- F2y along the y-axis
To find these components, we can use trigonometry. Since F1 and F2 are inclined at an angle of 60 degrees, we can use the following formulas:
F1x = F1 * cos(60)
F1y = F1 * sin(60)
F2x = F2 * cos(60)
F2y = F2 * sin(60)
Substituting the given values:
F1x = 7 * cos(60) = 3.5N
F1y = 7 * sin(60) = 6.06N
F2x = 10 * cos(60) = 5N
F2y = 10 * sin(60) = 8.66N
2. Once we have the x and y components of each force, we can find the resultant force along both the x and y directions.
Resultant force along the x-axis:
Rx = F1x + F2x
= 3.5N + 5N
= 8.5N
Resultant force along the y-axis:
Ry = F1y + F2y
= 6.06N + 8.66N
= 14.72N
3. To find the magnitude and direction of the resultant force, we can use the Pythagorean theorem and trigonometry.
Magnitude of the resultant force:
R = sqrt(Rx^2 + Ry^2)
= sqrt((8.5N)^2 + (14.72N)^2)
≈ 17.02N
Direction of the resultant force:
θ = tan^(-1)(Ry / Rx)
= tan^(-1)(14.72N / 8.5N)
≈ 59.47 degrees
Therefore, the magnitude of the resultant force is approximately 17.02N, and the direction of the resultant force is approximately 59.47 degrees.