Determine the magnitude and coordinate direction angles of the resultant force acting at A

Determine the magnitude and coordinate direction angles of the resultant force acting at A

The Correct Answer and Explanation is :

iturn0search3To determine the magnitude and coordinate direction angles of the resultant force acting at point A, follow these steps:

1. Resolve Each Force into Cartesian Components:

Assume forces F1\mathbf{F}_1 and F2\mathbf{F}_2 are acting at point A with known magnitudes and directions.

• **Force F1\mathbf{F}_1:**
  • Magnitude: F1F_1
  • Direction angles: α1\alpha_1, β1\beta_1, γ1\gamma_1
  The components of F1\mathbf{F}_1 are: F1x=F1cos⁡α1,F1y=F1cos⁡β1,F1z=F1cos⁡γ1F_{1x} = F_1 \cos\alpha_1, \quad F_{1y} = F_1 \cos\beta_1, \quad F_{1z} = F_1 \cos\gamma_1
• **Force F2\mathbf{F}_2:**
  • Magnitude: F2F_2
  • Direction angles: α2\alpha_2, β2\beta_2, γ2\gamma_2
  The components of F2\mathbf{F}_2 are: F2x=F2cos⁡α2,F2y=F2cos⁡β2,F2z=F2cos⁡γ2F_{2x} = F_2 \cos\alpha_2, \quad F_{2y} = F_2 \cos\beta_2, \quad F_{2z} = F_2 \cos\gamma_2

2. Calculate the Components of the Resultant Force:

Sum the corresponding components of F1\mathbf{F}_1 and F2\mathbf{F}_2: Rx=F1x+F2x,Ry=F1y+F2y,Rz=F1z+F2zR_x = F_{1x} + F_{2x}, \quad R_y = F_{1y} + F_{2y}, \quad R_z = F_{1z} + F_{2z}

3. Determine the Magnitude of the Resultant Force:

Use the Pythagorean theorem in three dimensions: R=Rx2+Ry2+Rz2R = \sqrt{R_x^2 + R_y^2 + R_z^2}

4. Find the Coordinate Direction Angles of the Resultant Force:

The direction angles α\alpha, β\beta, and γ\gamma are calculated as: cos⁡α=RxR,cos⁡β=RyR,cos⁡γ=RzR\cos\alpha = \frac{R_x}{R}, \quad \cos\beta = \frac{R_y}{R}, \quad \cos\gamma = \frac{R_z}{R}

Then, determine the angles: α=cos⁡−1(RxR),β=cos⁡−1(RyR),γ=cos⁡−1(RzR)\alpha = \cos^{-1}\left(\frac{R_x}{R}\right), \quad \beta = \cos^{-1}\left(\frac{R_y}{R}\right), \quad \gamma = \cos^{-1}\left(\frac{R_z}{R}\right)

Explanation:

Resolving each force into its Cartesian components allows for straightforward vector addition. By summing the respective components, we obtain the components of the resultant force vector R\mathbf{R}. The magnitude of R\mathbf{R} is found using the three-dimensional extension of the Pythagorean theorem, which accounts for all spatial dimensions. The coordinate direction angles α\alpha, β\beta, and γ\gamma describe the orientation of R\mathbf{R} relative to the x, y, and z axes, respectively. These angles are essential for understanding the spatial direction of the resultant force, which is crucial in applications like structural analysis and mechanical design.

By following this method, one can accurately determine both the magnitude and the precise orientation of the resultant force acting at point A, ensuring a comprehensive understanding of the force’s impact on the system.