Two equal forces of magnitude 'p' each are angled first at 60° later at 120°. The ratio of magnitude of their resultants is

Options

A. `1:sqrt(3)`
B. `sqrt(3):1`
C. `1:1`
D. `1:2`

Answer

To solve the problem of finding the ratio of the magnitudes of the resultant forces when two equal forces of magnitude 'p' are angled first at 60° and later at 120°, we can follow these steps:

### Step 1: Calculate the Resultant for 60°
1. **Identify the forces and the angle**: Let the two forces be \( F_1 = p \) and \( F_2 = p \) with an angle \( \theta_1 = 60° \) between them.
2. **Use the formula for the resultant**: The magnitude of the resultant \( R_1 \) of two forces can be calculated using the formula:
\[ R_1 = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta_1)} \]
3. **Substitute the values**:
\[ R_1 = \sqrt{p^2 + p^2 + 2 \cdot p \cdot p \cdot \cos(60°)} \]
Since \( \cos(60°) = \frac{1}{2} \):
\[ R_1 = \sqrt{p^2 + p^2 + 2 \cdot p^2 \cdot \frac{1}{2}} = \sqrt{2p^2 + p^2} = \sqrt{3p^2} = \sqrt{3}p \]

### Step 2: Calculate the Resultant for 120°
1. **Identify the angle**: Now, let’s consider the same forces with an angle \( \theta_2 = 120° \).
2. **Use the formula for the resultant**: The magnitude of the resultant \( R_2 \) is given by:
\[ R_2 = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta_2)} \]
3. **Substitute the values**:
\[ R_2 = \sqrt{p^2 + p^2 + 2 \cdot p \cdot p \cdot \cos(120°)} \]
Since \( \cos(120°) = -\frac{1}{2} \):
\[ R_2 = \sqrt{p^2 + p^2 + 2 \cdot p^2 \cdot (-\frac{1}{2})} = \sqrt{2p^2 - p^2} = \sqrt{p^2} = p \]

### Step 3: Find the Ratio of the Resultants
1. **Calculate the ratio**:
\[ \text{Ratio} = \frac{R_1}{R_2} = \frac{\sqrt{3}p}{p} = \sqrt{3} \]
2. **Express the ratio**:
\[ R_1 : R_2 = \sqrt{3} : 1 \]

### Final Answer
The ratio of the magnitudes of their resultants is \( \sqrt{3} : 1 \).

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