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What are the rectangular components of force F in the given diagram?
Answer:
To determine the rectangular components of the force F in the given diagram, we need to resolve the force vector into its x, y, and z components. The force F is given as 75 N, and it’s directed at angles, with respect to the axes, provided in the diagram.
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Identify Given Angles:
- Angle between the force and the z-axis: \theta_z = 30^\circ
- Angle in the x-y plane relative to the x-axis: \theta_{xy} = 45^\circ
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Component in the z-direction F_z:
F_z = F \cos(30^\circ)
Since \cos(30^\circ) = \frac{\sqrt{3}}{2} ,
F_z = 75 \times \frac{\sqrt{3}}{2} = 75 \times 0.866 = 64.95 \, \text{N}
- Magnitude of the force in the x-y plane F_{xy}:
Since the force makes a 30^\circ angle with the z-axis, the projection in the x-y plane is:
F_{xy} = F \sin(30^\circ)
Since \sin(30^\circ) = \frac{1}{2} ,
F_{xy} = 75 \times \frac{1}{2} = 37.5 \, \text{N}
- Component in the x-direction F_x:
Given the angle in the x-y plane with respect to the x-axis is \theta_{xy} = 45^\circ :
F_x = F_{xy} \cos(45^\circ)
Since \cos(45^\circ) = \frac{\sqrt{2}}{2},
F_x = 37.5 \times \frac{\sqrt{2}}{2} = 37.5 \times 0.707 = 26.52 \, \text{N}
- Component in the y-direction F_y:
F_y = F_{xy} \sin(45^\circ)
Since \sin(45^\circ) = \frac{\sqrt{2}}{2} ,
F_y = 37.5 \times \frac{\sqrt{2}}{2} = 37.5 \times 0.707 = 26.52 \, \text{N}
Therefore, the rectangular components of the force F are:
F_x = 26.52 \, \text{N}
F_y = 26.52 \, \text{N}
F_z = 64.95 \, \text{N}