

Your friend has slipped and fallen. To help her up, you pull with a force F, as the drawing shows. The vertical component of this force is 130 newtons, and the horizontal component is 150 newtons. Find (a) the magnitude of F and (b) the angle 0.

To solve the problem, we need to determine two things:
1. The magnitude of the force \( F \).
2. The angle \( \theta \) that the force makes with the horizontal.

### Given:
- Vertical component of the force (\( F_y \)) = 130 N
- Horizontal component of the force (\( F_x \)) = 150 N

### Part (a): Magnitude of the force \( F \)

The magnitude of the force \( F \) can be calculated using the Pythagorean theorem, since the vertical and horizontal components form a right triangle:

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

Substitute the given values:

\[
F = \sqrt{(150 \, \text{N})^2 + (130 \, \text{N})^2}
\]

\[
F = \sqrt{22500 \, \text{N}^2 + 16900 \, \text{N}^2}
\]

\[
F = \sqrt{39400 \, \text{N}^2}
\]

\[
F \approx 198.49 \, \text{N}
\]

### Part (b): Angle \( \theta \)

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

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

\[
\theta = \tan^{-1}\left(\frac{130 \, \text{N}}{150 \, \text{N}}\right)
\]

\[
\theta = \tan^{-1}(0.867)
\]

\[
\theta \approx 40.93^\circ
\]

### Final Answers:
- **(a)** The magnitude of the force \( F \) is approximately **198.49 N**.
- **(b)** The angle \( \theta \) is approximately **40.93°**.

Would you like more details on these calculations or have any additional questions?

### Additional Related Questions:
1. How would the angle change if the vertical component increased while keeping the horizontal component constant?
2. What would happen to the magnitude of \( F \) if both components doubled?
3. How can you resolve a force into its components if the angle and magnitude are given?
4. What is the effect of this force angle on the effectiveness of lifting the friend?
5. How would you calculate the work done by this force if the friend is moved horizontally by a certain distance?

**Tip:** Always double-check your trigonometric calculations to ensure you are using the correct function (sine, cosine, or tangent) for the given components and angle.

Learn how to calculate the magnitude and angle of a force with 130N vertical and 150N horizontal components using trigonometry and the Pythagorean theorem. Suitable for high school students studying vectors and trigonometry.

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