If two forces 4n and 5n incline at angle 45 degree to each other find the resultant and direction
To find the resultant of the two forces, you can use the concept of vector addition. This involves breaking down the forces into their component vectors, adding the corresponding components, and then combining them to find the magnitude and direction of the resultant vector.
Step 1: Convert the given forces into their component vectors.
- Force 1 (4N) inclined at an angle of 45 degrees: Resolve it into its horizontal and vertical components.
- Horizontal component: F1x = 4N * cos(45°) = 4N * 0.7071 ≈ 2.828N
- Vertical component: F1y = 4N * sin(45°) = 4N * 0.7071 ≈ 2.828N
- Force 2 (5N) inclined at an angle of 45 degrees: Resolve it into its horizontal and vertical components.
- Horizontal component: F2x = 5N * cos(45°) = 5N * 0.7071 ≈ 3.536N
- Vertical component: F2y = 5N * sin(45°) = 5N * 0.7071 ≈ 3.536N
Step 2: Add the corresponding components of the two forces to find the resultant.
- Horizontal component of the resultant: Rx = F1x + F2x = 2.828N + 3.536N = 6.364N
- Vertical component of the resultant: Ry = F1y + F2y = 2.828N + 3.536N = 6.364N
Step 3: Use the Pythagorean theorem to find the magnitude of the resultant vector (R).
- Magnitude of the resultant: R = √(Rx^2 + Ry^2) = √(6.364N^2 + 6.364N^2) = √(40.5216N^2) ≈ 6.364N
Step 4: Use trigonometry to find the direction of the resultant vector.
- Angle of the resultant (θ): θ = arctan(Ry/Rx) = arctan(6.364N/6.364N) = arctan(1) ≈ 45°
Therefore, the magnitude of the resultant force is approximately 6.364N, and its direction is approximately 45 degrees.