One of the two rectangular components of a force is `20 N` and it makes an angle of `30^(@)` with the force. The magnitude of the other component is

Options

A. `(10)/(sqrt3)`
B. `(20)/(sqrt3)`
C. `(15)/(sqrt3)`
D. `(40)/(sqrt3)`

Answer

To find the magnitude of the other rectangular component of the force, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- One component of the force, \( F_1 = 20 \, \text{N} \)
- Angle with the force, \( \theta = 30^\circ \)

2. **Define the Components of the Force:**
- Let \( F \) be the total force.
- The horizontal component of the force can be expressed as:
\[ F_x = F \cos \theta \]
- The vertical component of the force can be expressed as:
\[ F_y = F \sin \theta \]
- We know that one of the components (let's say the vertical component) is \( F_y = 20 \, \text{N} \).

3. **Set Up the Equation:**
- Since \( F_y = F \sin \theta \), we can write:
\[ 20 = F \sin(30^\circ) \]

4. **Calculate \( \sin(30^\circ) \):**
- We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]

5. **Substitute and Solve for \( F \):**
- Substitute \( \sin(30^\circ) \) into the equation:
\[ 20 = F \cdot \frac{1}{2} \]
- Rearranging gives:
\[ F = 20 \cdot 2 = 40 \, \text{N} \]

6. **Find the Other Component:**
- Now we need to find the other component, which is the horizontal component \( F_x \):
\[ F_x = F \cos(30^\circ) \]
- We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
- Substitute \( F \) into the equation:
\[ F_x = 40 \cdot \frac{\sqrt{3}}{2} = 20\sqrt{3} \, \text{N} \]

### Final Answer:
The magnitude of the other component is \( 20\sqrt{3} \, \text{N} \).

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