A body has a time of 3 s under the action of one force and 4 s under the action of another force. Calculate the time period when both the forces act in the same direction simultaneously. Assume that the body undergoes the same displacement, under the action of the two different forces.

Explanation

To solve the problem, we need to find the time period \( T_3 \) when both forces act in the same direction simultaneously. We are given:

- Time period \( T_1 = 3 \) seconds (for force \( F_1 \))
- Time period \( T_2 = 4 \) seconds (for force \( F_2 \))

### Step-by-step Solution:

1. **Understanding the relationship between force and time period:**
The time period \( T \) is related to the angular frequency \( \omega \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Therefore, we can express the angular frequencies for the two forces:
\[ \omega_1 = \frac{2\pi}{T_1} = \frac{2\pi}{3} \]
\[ \omega_2 = \frac{2\pi}{T_2} = \frac{2\pi}{4} = \frac{\pi}{2} \]

2. **Using the relationship of angular frequencies:**
When both forces act simultaneously, the total angular frequency \( \omega_3 \) can be expressed as:
\[ \omega_3^2 = \omega_1^2 + \omega_2^2 \]

3. **Calculating \( \omega_1^2 \) and \( \omega_2^2 \):**
\[ \omega_1^2 = \left(\frac{2\pi}{3}\right)^2 = \frac{4\pi^2}{9} \]
\[ \omega_2^2 = \left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4} \]

4. **Finding a common denominator to add \( \omega_1^2 \) and \( \omega_2^2 \):**
The common denominator for \( \frac{4\pi^2}{9} \) and \( \frac{\pi^2}{4} \) is 36. Thus:
\[ \omega_1^2 = \frac{4\pi^2}{9} = \frac{16\pi^2}{36} \]
\[ \omega_2^2 = \frac{\pi^2}{4} = \frac{9\pi^2}{36} \]

5. **Adding \( \omega_1^2 \) and \( \omega_2^2 \):**
\[ \omega_3^2 = \omega_1^2 + \omega_2^2 = \frac{16\pi^2}{36} + \frac{9\pi^2}{36} = \frac{25\pi^2}{36} \]

6. **Finding \( \omega_3 \):**
\[ \omega_3 = \sqrt{\frac{25\pi^2}{36}} = \frac{5\pi}{6} \]

7. **Calculating the time period \( T_3 \):**
Now, we can find the time period \( T_3 \):
\[ T_3 = \frac{2\pi}{\omega_3} = \frac{2\pi}{\frac{5\pi}{6}} = \frac{2 \cdot 6}{5} = \frac{12}{5} \text{ seconds} \]

### Final Answer:
The time period when both forces act in the same direction simultaneously is:
\[ T_3 = \frac{12}{5} \text{ seconds} \approx 2.4 \text{ seconds} \]