A mass m is suspended from the two coupled springs connected in series. The force constant for spring are `k_(1)` and `k_(2)`. The time period of the suspended mass will be:

Options

A. `T=2pisqrt((m)/(k_(1)-k_(2)))`
B. `T=2pisqrt((mk_(1)k_(2))/(k_(1)+k_(2)))`
C. `T=2pisqrt((m)/(k_(1)+k_(2)))`
D. `T=2pisqrt((m(k_(1)+k_(2)))/(k_(1)k_(2)))`

Explanation

To find the time period of a mass \( m \) suspended from two coupled springs connected in series with force constants \( k_1 \) and \( k_2 \), we can follow these steps:

### Step 1: Understand the formula for the time period
The time period \( T \) of a mass-spring system is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k_{\text{equivalent}}}} \]
where \( k_{\text{equivalent}} \) is the equivalent spring constant of the system.

### Step 2: Determine the equivalent spring constant for springs in series
For two springs connected in series, the equivalent spring constant \( k_{\text{equivalent}} \) can be calculated using the formula:
\[ \frac{1}{k_{\text{equivalent}}} = \frac{1}{k_1} + \frac{1}{k_2} \]
This can be rearranged to find \( k_{\text{equivalent}} \):
\[ k_{\text{equivalent}} = \frac{k_1 k_2}{k_1 + k_2} \]

### Step 3: Substitute \( k_{\text{equivalent}} \) into the time period formula
Now, we substitute the expression for \( k_{\text{equivalent}} \) into the time period formula:
\[ T = 2\pi \sqrt{\frac{m}{\frac{k_1 k_2}{k_1 + k_2}}} \]

### Step 4: Simplify the expression
This can be simplified further:
\[ T = 2\pi \sqrt{\frac{m (k_1 + k_2)}{k_1 k_2}} \]

### Final Result
Thus, the time period \( T \) of the suspended mass is:
\[ T = 2\pi \sqrt{\frac{m (k_1 + k_2)}{k_1 k_2}} \]