A horizontal force of 1000 N acts toward the right from a point and a vertical force of 3000 N acts downward from the same point and perpendicular to the horizontal force. Find the resultant force of the two forces and the angle this resultant makes with the horizontal axis.

To find the resultant force of the two given forces and the angle it makes with the horizontal axis, we can use the following steps:

### 1. Given Data

- Horizontal force \( F_x = 1000 \, \text{N} \) (toward the right)
- Vertical force \( F_y = 3000 \, \text{N} \) (downward)

### 2. Resultant Force Calculation

The resultant force \( F_R \) is the vector sum of the horizontal and vertical forces. It can be calculated using the Pythagorean theorem:

\[
F_R = \sqrt{F_x^2 + F_y^2}
\]

Substituting the given values:

\[
F_R = \sqrt{(1000 \, \text{N})^2 + (3000 \, \text{N})^2}
\]

\[
F_R = \sqrt{1000000 \, \text{N}^2 + 9000000 \, \text{N}^2}
\]

\[
F_R = \sqrt{10000000 \, \text{N}^2}
\]

\[
F_R = 3162.28 \, \text{N}
\]

### 3. Angle with the Horizontal Axis

The angle \( \theta \) that the resultant force makes with the horizontal axis can be found using the tangent function:

\[
\tan(\theta) = \frac{F_y}{F_x}
\]

\[
\theta = \tan^{-1}\left(\frac{3000 \, \text{N}}{1000 \, \text{N}}\right)
\]

\[
\theta = \tan^{-1}(3)
\]

\[
\theta \approx 71.57^\circ
\]

### Final Answer:

- The resultant force \( F_R \) is approximately **3162.28 N**.
- The angle \( \theta \) with the horizontal axis is approximately **71.57°**.

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Do you want further details or have any questions?

Here are 5 related questions for deeper understanding:

1. How would the resultant force change if the horizontal force was doubled?
2. What if both forces acted in the same direction—how would the angle be affected?
3. Can you find the components of the resultant force if the angle is given instead?
4. How would the result differ if the vertical force was acting upward instead of downward?
5. What is the significance of the angle in real-world applications such as construction or engineering?

**Tip:** When working with vectors, always keep track of the direction (angles) as they are crucial in determining the resultant force and its orientation.

Learn how to calculate the resultant force and the angle it makes with the horizontal axis. Solve a physics problem involving vectors and trigonometry with a step-by-step solution. Suitable for students in Grades 9-12.

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