Determine the magnitude and coordinate direction angles of the resultant force, and sketch this vector on the coordinate system
The Correct Answer and Explanation is :
To determine the magnitude and coordinate direction angles of the resultant force acting on the bracket, follow these steps:
1. Resolve Each Force into Cartesian Components:
Assuming the forces ( F_1 ) and ( F_2 ) are given with their magnitudes and angles relative to the coordinate axes, decompose each force into its x, y, and z components.
For a force ( F ) with magnitude ( |F| ) and direction angles ( \alpha ), ( \beta ), and ( \gamma ) with the x, y, and z axes respectively:
- ( F_x = |F| \cos(\alpha) )
- ( F_y = |F| \cos(\beta) )
- ( F_z = |F| \cos(\gamma) )
2. Calculate the Components of Each Force:
Using the given magnitudes and angles, compute the components for ( F_1 ) and ( F_2 ).
3. Determine the Resultant Force Components:
Sum the corresponding components of ( F_1 ) and ( F_2 ) to find the resultant force ( F_R ):
- ( F_{Rx} = F_{1x} + F_{2x} )
- ( F_{Ry} = F_{1y} + F_{2y} )
- ( F_{Rz} = F_{1z} + F_{2z} )
4. Compute the Magnitude of the Resultant Force:
Use the Pythagorean theorem to find the magnitude of ( F_R ):
[ |F_R| = \sqrt{F_{Rx}^2 + F_{Ry}^2 + F_{Rz}^2} ]
5. Determine the Coordinate Direction Angles:
The direction angles ( \alpha_R ), ( \beta_R ), and ( \gamma_R ) of the resultant force with respect to the x, y, and z axes can be found using:
- ( \cos(\alpha_R) = \frac{F_{Rx}}{|F_R|} )
- ( \cos(\beta_R) = \frac{F_{Ry}}{|F_R|} )
- ( \cos(\gamma_R) = \frac{F_{Rz}}{|F_R|} )
Then, calculate the angles:
- ( \alpha_R = \cos^{-1}\left(\frac{F_{Rx}}{|F_R|}\right) )
- ( \beta_R = \cos^{-1}\left(\frac{F_{Ry}}{|F_R|}\right) )
- ( \gamma_R = \cos^{-1}\left(\frac{F_{Rz}}{|F_R|}\right) )
6. Sketch the Resultant Vector:
On the coordinate system, draw the vector ( F_R ) using its components:
- Start at the origin.
- Move ( F_{Rx} ) units along the x-axis.
- From that point, move ( F_{Ry} ) units along the y-axis.
- Finally, move ( F_{Rz} ) units along the z-axis.
The endpoint represents the tip of the resultant vector. Draw a line from the origin to this point to represent ( F_R ).
Explanation:
By resolving each force into its Cartesian components, we can systematically combine them to find the overall effect on the bracket. Summing the components allows us to determine the exact direction and magnitude of the resultant force. Calculating the coordinate direction angles provides insight into the orientation of this force in three-dimensional space. Sketching the vector helps visualize how the combined forces act on the bracket, ensuring a comprehensive understanding of the system’s mechanics.