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Given:
- Force \( F_1 = 3 \, \text{N} \)
- Force \( F_2 = 2 \, \text{N} \)
- Force \( F_3 = 3 \, \text{N} \)
- Force \( F_4 = 1.8 \, \text{N} \)
Calculations:
To calculate the torque about the z-axis for each force:
For \( F_1 \):
\[ \tau_1 = |\vec{r}_1 \times \vec{F}_1| = |\vec{r}_1| \cdot |\vec{F}_1| \cdot \sin(\theta_1) \]For \( F_2 \):
\[ \tau_2 = |\vec{r}_2 \times \vec{F}_2| = |\vec{r}_2| \cdot |\vec{F}_2| \cdot \sin(\theta_2) \]For \( F_3 \):
\[ \tau_3 = |\vec{r}_3 \times \vec{F}_3| = |\vec{r}_3| \cdot |\vec{F}_3| \cdot \sin(\theta_3) \]For \( F_4 \):
\[ \tau_4 = |\vec{r}_4 \times \vec{F}_4| = |\vec{r}_4| \cdot |\vec{F}_4| \cdot \sin(\theta_4) \]Since the position vectors are in the xy-plane, the angle \( \theta \) between each position vector and the z-axis is 90 degrees. Therefore, \( \sin(\theta) = 1 \) for all forces.
Given the magnitudes of the forces and assuming they act perpendicular to the position vectors, we can compute the torques directly using the given information.
Now, let's calculate each torque and sum them up to find the total torque about the z-axis.
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