Two forces are in the ratio of 5 : 2. The maximum and minimum of their resultants are in the ratio is

Options

A. `5:2`
B. `2:5`
C. `7:3`
D. `3:7`

Answer

To solve the problem of finding the ratio of the maximum and minimum resultant of two forces given in the ratio of 5:2, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Forces**:
Let the two forces be \( F_1 \) and \( F_2 \). According to the ratio given, we can express them as:
\[ F_1 = 5x \quad \text{and} \quad F_2 = 2x \]
where \( x \) is a common multiplicative factor.

2. **Calculate the Maximum Resultant**:
The maximum resultant occurs when the two forces are in the same direction (i.e., \( \theta = 0^\circ \)). The formula for the resultant \( R \) is:
\[ R_{\text{max}} = F_1 + F_2 = 5x + 2x = 7x \]

3. **Calculate the Minimum Resultant**:
The minimum resultant occurs when the two forces are in opposite directions (i.e., \( \theta = 180^\circ \)). The formula for the resultant in this case is:
\[ R_{\text{min}} = |F_1 - F_2| = |5x - 2x| = 3x \]

4. **Find the Ratio of Maximum to Minimum Resultant**:
Now, we need to find the ratio of the maximum resultant to the minimum resultant:
\[ \text{Ratio} = \frac{R_{\text{max}}}{R_{\text{min}}} = \frac{7x}{3x} = \frac{7}{3} \]

5. **Conclusion**:
Therefore, the ratio of the maximum and minimum resultant of the two forces is:
\[ \text{Ratio} = 7 : 3 \]

### Final Answer:
The maximum and minimum of their resultants are in the ratio \( 7 : 3 \).

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