https://file.aiquickdraw.com/mathbot/file/1726913289384zusfdv4g.jpg

To solve this problem, we need to find the force \( F_1 \) that prevents the system from rotating around point \( B \), which acts as the axis of rotation.

### Given:
- \( F_3 = 20 \, \text{N} \), acting vertically upward at a distance of 2 meters from point \( B \).
- \( F_2 = 10 \, \text{N} \), acting at an angle of \( 30^\circ \) at a distance of 3 meters from point \( B \).
- \( F_1 \), whose magnitude and direction are to be found, acts at an angle of \( 150^\circ \) at a distance of 2 meters from point \( B \).

### Approach:
To maintain rotational equilibrium, the sum of moments around point \( B \) must be zero:
\[
\sum \tau_B = 0
\]
Where \( \tau \) is the torque, which is calculated as:
\[
\tau = r \times F \times \sin(\theta)
\]
Here, \( r \) is the distance from point \( B \), \( F \) is the force, and \( \theta \) is the angle between the force and the lever arm.

### Step 1: Calculate the torque due to each force

1. **Torque from \( F_3 \)**:
   \[
   \tau_3 = 2 \, \text{m} \times 20 \, \text{N} \times \sin(90^\circ) = 40 \, \text{Nm}
   \]
   This torque is counterclockwise.

2. **Torque from \( F_2 \)**:
   \[
   \tau_2 = 3 \, \text{m} \times 10 \, \text{N} \times \sin(30^\circ) = 3 \times 10 \times 0.5 = 15 \, \text{Nm}
   \]
   This torque is also counterclockwise.

3. **Torque from \( F_1 \)**:
   The torque due to \( F_1 \) will be clockwise:
   \[
   \tau_1 = 2 \, \text{m} \times F_1 \times \sin(150^\circ) = 2 \times F_1 \times 0.5 = F_1 \, \text{Nm}
   \]

### Step 2: Set up the equation for rotational equilibrium

Since the system is in equilibrium, the sum of the torques must be zero. Counterclockwise torques are positive, and clockwise torques are negative:
\[
40 + 15 - F_1 = 0
\]
\[
F_1 = 55 \, \text{N}
\]

Thus, the force \( F_1 \) that prevents the system from rotating is \( 55 \, \text{N} \).

Let me know if you'd like further clarification or a detailed explanation of any part.

---

**Follow-up questions:**

1. What is the principle behind calculating torque in rotational systems?
2. How does the direction of the force affect the torque in this case?
3. What role does the distance from the axis of rotation play in determining torque?
4. Can you explain how the angles \( 30^\circ \) and \( 150^\circ \) affect the torque calculation?
5. How would the solution change if \( F_2 \) or \( F_3 \) were increased?

**Tip:** Always remember that the torque's effectiveness depends on both the force applied and the perpendicular distance from the axis of rotation.

Find the magnitude of force F1 that prevents the system from rotating around point B as shown in the diagram.

Learn how to calculate the force F1 that prevents a system from rotating around point B in this detailed physics problem involving torques, equilibrium, and trigonometry. Suitable for high school physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation