When a force `F_1` acts on a particle, frequency is 6 Hz and when a force `F_2` acts, frequency is 8 Hz. What is the frequency when both the forces act simultaneously in the same direction?

Answer

To solve the problem, we need to find the frequency of a particle when two forces \( F_1 \) and \( F_2 \) are acting on it simultaneously. We know the frequencies associated with each force acting individually: \( n_1 = 6 \) Hz when \( F_1 \) acts and \( n_2 = 8 \) Hz when \( F_2 \) acts.

### Step-by-Step Solution:

1. **Understanding the Relationship Between Force and Frequency**:
The force acting on a particle in oscillatory motion is related to its frequency. The relationship can be expressed as:
\[ F \propto n^2 \]
where \( F \) is the force and \( n \) is the frequency.

2. **Expressing Forces in Terms of Frequencies**:
For the forces \( F_1 \) and \( F_2 \):
\[ F_1 = k \cdot n_1^2 \quad \text{and} \quad F_2 = k \cdot n_2^2 \]
where \( k \) is a constant that depends on the mass of the particle and other parameters.

3. **Finding the Resultant Force**:
When both forces act simultaneously, the resultant force \( F \) can be expressed as:
\[ F = F_1 + F_2 = k \cdot n_1^2 + k \cdot n_2^2 = k(n_1^2 + n_2^2) \]

4. **Relating Resultant Force to Resultant Frequency**:
The resultant frequency \( n \) when both forces act is given by:
\[ F = k \cdot n^2 \]
Thus, we can set the two expressions for \( F \) equal to each other:
\[ k(n_1^2 + n_2^2) = k \cdot n^2 \]

5. **Cancelling the Constant**:
Since \( k \) is common in both sides, we can cancel it out:
\[ n^2 = n_1^2 + n_2^2 \]

6. **Substituting the Known Frequencies**:
Substitute \( n_1 = 6 \) Hz and \( n_2 = 8 \) Hz into the equation:
\[ n^2 = 6^2 + 8^2 = 36 + 64 = 100 \]

7. **Calculating the Resultant Frequency**:
Taking the square root gives:
\[ n = \sqrt{100} = 10 \text{ Hz} \]

### Final Answer:
The frequency when both forces act simultaneously in the same direction is **10 Hz**.

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