A particle under the action of one SHM has a period of 3 s and under the effect of another it has a period 4s. Its time period under the combined action of both the SHM's in the same direction is (0.4x)second. Find value of x?

Answer

To solve the problem, we need to find the value of \( x \) given the periods of two simple harmonic motions (SHMs) acting on a particle. Let's break down the solution step by step.

### Step 1: Identify the given periods
We are given two periods:
- Period \( T_1 = 3 \) seconds
- Period \( T_2 = 4 \) seconds

### Step 2: Relate the periods to spring constants
The time period \( T \) of a simple harmonic motion is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( m \) is the mass of the particle and \( k \) is the spring constant.

From the periods, we can express the spring constants \( k_1 \) and \( k_2 \) for the two SHMs:
1. For \( T_1 = 3 \) seconds:
\[ T_1 = 2\pi \sqrt{\frac{m}{k_1}} \implies k_1 = \frac{4\pi^2 m}{T_1^2} = \frac{4\pi^2 m}{3^2} = \frac{4\pi^2 m}{9} \]

2. For \( T_2 = 4 \) seconds:
\[ T_2 = 2\pi \sqrt{\frac{m}{k_2}} \implies k_2 = \frac{4\pi^2 m}{T_2^2} = \frac{4\pi^2 m}{4^2} = \frac{4\pi^2 m}{16} \]

### Step 3: Find the effective spring constant
When two SHMs act in the same direction, the effective spring constant \( k_{\text{effective}} \) is the sum of the individual spring constants:
\[ k_{\text{effective}} = k_1 + k_2 = \frac{4\pi^2 m}{9} + \frac{4\pi^2 m}{16} \]

### Step 4: Calculate the combined spring constant
To add these fractions, we need a common denominator:
- The least common multiple of 9 and 16 is 144.

Now, convert each term:
\[ k_1 = \frac{4\pi^2 m}{9} = \frac{64\pi^2 m}{144} \]
\[ k_2 = \frac{4\pi^2 m}{16} = \frac{36\pi^2 m}{144} \]

Adding these gives:
\[ k_{\text{effective}} = \frac{64\pi^2 m}{144} + \frac{36\pi^2 m}{144} = \frac{100\pi^2 m}{144} \]

### Step 5: Find the resultant time period
Now, we can find the time period \( T_{\text{effective}} \) for the combined SHM:
\[ T_{\text{effective}} = 2\pi \sqrt{\frac{m}{k_{\text{effective}}}} = 2\pi \sqrt{\frac{m}{\frac{100\pi^2 m}{144}}} = 2\pi \sqrt{\frac{144}{100\pi^2}} = 2\pi \cdot \frac{12}{10\pi} = \frac{24}{10} = 2.4 \text{ seconds} \]

### Step 6: Relate to the given equation
According to the problem, this time period is also given as:
\[ T_{\text{effective}} = 0.4x \]
Setting this equal to our calculated time period:
\[ 0.4x = 2.4 \]

### Step 7: Solve for \( x \)
To find \( x \):
\[ x = \frac{2.4}{0.4} = 6 \]

### Final Answer
Thus, the value of \( x \) is \( \boxed{6} \).

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