Let `T_(1)` and `T_(2)` be the time periods of two springs A and B when a mass m is suspended from them separately. Now both the springs are connected in parallel and same mass m is suspended with them. Now let T be the time period in this position. Then

Options

A. `T = T_(1)+T_(2)`
B. `T=(T_(1)T_(2))/(T_(1)+T_(2))`
C. `T^(2)= T_(1)^(2)+T_(2)^(2)`
D. `1/T^(2)=1/T_(1)^(2)+1/T_(2)^(2)`

Answer

To solve the problem, we need to establish a relationship between the time periods \( T_1 \), \( T_2 \), and \( T \) when two springs are connected in parallel. Here’s a step-by-step solution:

### Step 1: Understand the Time Period of a Spring
The time period \( T \) of a mass \( m \) suspended from a spring with spring constant \( k \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

### Step 2: Write the Time Periods for Each Spring
For spring A with spring constant \( k_A \):
\[ T_1 = 2\pi \sqrt{\frac{m}{k_A}} \]
For spring B with spring constant \( k_B \):
\[ T_2 = 2\pi \sqrt{\frac{m}{k_B}} \]

### Step 3: Combine the Springs in Parallel
When springs A and B are connected in parallel, the equivalent spring constant \( k_{eq} \) is the sum of the individual spring constants:
\[ k_{eq} = k_A + k_B \]

### Step 4: Write the Time Period for the Combined Springs
The time period \( T \) for the combined system is given by:
\[ T = 2\pi \sqrt{\frac{m}{k_{eq}}} = 2\pi \sqrt{\frac{m}{k_A + k_B}} \]

### Step 5: Square the Time Period Equations
Now, we will square the equations for \( T_1 \), \( T_2 \), and \( T \):
\[ T_1^2 = 4\pi^2 \frac{m}{k_A} \]
\[ T_2^2 = 4\pi^2 \frac{m}{k_B} \]
\[ T^2 = 4\pi^2 \frac{m}{k_A + k_B} \]

### Step 6: Take the Reciprocals
Taking the reciprocal of each squared time period:
\[ \frac{1}{T_1^2} = \frac{k_A}{4\pi^2 m} \]
\[ \frac{1}{T_2^2} = \frac{k_B}{4\pi^2 m} \]
\[ \frac{1}{T^2} = \frac{k_A + k_B}{4\pi^2 m} \]

### Step 7: Add the Reciprocals of \( T_1^2 \) and \( T_2^2 \)
Adding the reciprocals of \( T_1^2 \) and \( T_2^2 \):
\[ \frac{1}{T_1^2} + \frac{1}{T_2^2} = \frac{k_A}{4\pi^2 m} + \frac{k_B}{4\pi^2 m} = \frac{k_A + k_B}{4\pi^2 m} \]

### Step 8: Relate to the Time Period \( T \)
From the equation for \( T \):
\[ \frac{1}{T^2} = \frac{k_A + k_B}{4\pi^2 m} \]
Thus, we can conclude:
\[ \frac{1}{T^2} = \frac{1}{T_1^2} + \frac{1}{T_2^2} \]

### Final Result
This leads us to the final relationship:
\[ \frac{1}{T^2} = \frac{1}{T_1^2} + \frac{1}{T_2^2} \]

### Conclusion
The correct answer is:
\[ \text{Option 4: } \frac{1}{T^2} = \frac{1}{T_1^2} + \frac{1}{T_2^2} \]

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