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.