(Figure 1) shows four electrical charges located at the corners of a rectangle. Like charges, you will recall, repel each other while opposite charges attract. Charge B
exerts a repulsive force (directly away from B
) on charge A
of 3.3 N
. Charge C
exerts an attractive force (directly toward C
) on charge A
of 6.6 N
. Finally, charge D
exerts an attractive force of 2.2 N
on charge A
.
Figure1 of 1
Four charged particles are labeled from 'A' to D and placed in the vertices of a rectangle. Charges 'A' and B are positive, charges C and D are negative. Horizontal sides 'A' B and C D have a length of 141 centimeters. Vertical sides 'A' C and B C have a length of 100 centimeters. What is the direction of the net force  F⃗ net
exerted on charge  A
? Use the  xy
-plane with the origin at  A
and with  x
-axis directed to  B
.
Express your answer in degrees measured clockwise from the negative  y
-axis.

To determine the direction of the net force \(\mathbf{F}_{\text{net}}\) on charge \(A\), we'll break down the problem into the following steps:

1. **Identify the Forces:**
   - The force exerted by charge \(B\) on \(A\) is repulsive and acts directly to the right (along the positive \(x\)-axis). Let's denote this force as \(\mathbf{F}_B = 3.3 \, \text{N}\).
   - The force exerted by charge \(C\) on \(A\) is attractive and acts directly downward (along the negative \(y\)-axis). Let's denote this force as \(\mathbf{F}_C = 6.6 \, \text{N}\).
   - The force exerted by charge \(D\) on \(A\) is attractive and acts diagonally toward \(D\). Let's denote this force as \(\mathbf{F}_D = 2.2 \, \text{N}\).

2. **Determine the Components of the Forces:**
   - \(\mathbf{F}_B\) acts entirely along the \(x\)-axis: \(\mathbf{F}_B = 3.3 \, \text{N} \, \hat{i}\).
   - \(\mathbf{F}_C\) acts entirely along the \(y\)-axis: \(\mathbf{F}_C = -6.6 \, \text{N} \, \hat{j}\).
   - \(\mathbf{F}_D\) has both \(x\) and \(y\) components. The angle between the diagonal \(AD\) and the horizontal axis can be determined using the dimensions of the rectangle.

3. **Calculate the Angle for Force \(D\):**
   - The length \(AB = 141 \, \text{cm}\) and \(AC = 100 \, \text{cm}\).
   - The distance \(AD\) (the diagonal) can be calculated using the Pythagorean theorem:
     \[
     AD = \sqrt{AB^2 + AC^2} = \sqrt{141^2 + 100^2} = \sqrt{19881 + 10000} \approx 174.92 \, \text{cm}
     \]
   - The angle \(\theta\) between the diagonal \(AD\) and the horizontal side \(AB\) is:
     \[
     \theta = \tan^{-1}\left(\frac{AC}{AB}\right) = \tan^{-1}\left(\frac{100}{141}\right) \approx 35.1^\circ
     \]

   Therefore, the components of \(\mathbf{F}_D\) are:
   \[
   \mathbf{F}_{Dx} = 2.2 \, \cos(35.1^\circ) \, \text{N} \quad \text{(towards the negative \(x\)-axis)}
   \]
   \[
   \mathbf{F}_{Dy} = -2.2 \, \sin(35.1^\circ) \, \text{N} \quad \text{(towards the negative \(y\)-axis)}
   \]

4. **Sum the Components to Find the Net Force:**
   - Net force in the \(x\)-direction:
     \[
     F_{\text{net},x} = F_B + F_{Dx} = 3.3 + 2.2 \times (-\cos(35.1^\circ)) \approx 3.3 - 1.8 = 1.5 \, \text{N}
     \]
   - Net force in the \(y\)-direction:
     \[
     F_{\text{net},y} = F_C + F_{Dy} = -6.6 + 2.2 \times (-\sin(35.1^\circ)) \approx -6.6 - 1.3 = -7.9 \, \text{N}
     \]

5. **Determine the Angle of the Net Force:**
   - The angle \(\phi\) of the net force from the negative \(y\)-axis can be found using:
     \[
     \tan(\phi) = \frac{F_{\text{net},x}}{-F_{\text{net},y}} = \frac{1.5}{7.9} \approx 0.19
     \]
     \[
     \phi \approx \tan^{-1}(0.19) \approx 10.7^\circ
     \]
     This angle is measured clockwise from the negative \(y\)-axis.

Thus, the direction of the net force on charge \(A\) is approximately **10.7° clockwise from the negative \(y\)-axis**.

Would you like more details on this solution? Here are some related questions:

1. How would the direction change if the magnitude of force \(C\) were reduced?
2. What if charge \(D\) were moved closer to charge \(A\)?
3. How does the position of charge \(B\) affect the net force on \(A\)?
4. What if charges \(B\) and \(C\) were switched?
5. How would the net force be affected if charge \(A\) were also negative?

**Tip:** When dealing with forces in multiple directions, always break them into their components and handle each direction separately.

Learn how to determine the direction of the net force on charge A placed in a rectangular arrangement of electrical charges. This problem involves calculating forces using vector components, applying the Pythagorean theorem to find diagonal distances, and using trigonometric functions to determine force angles. Suitable for high school students.

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