The time period of oscillations of a block attached to a spring is `t_(1)`. When the spring is replaced by another spring, the time period of the block is `t_(2)`. If both the springs are connected in series and the block is made to oscillate using the combination, then the time period of the block is

Options

A. `T = t_1 + t_2`
B. `T^2 = t_1^2 + t_2^2`
C. `T^(-1) = t_1^(-1) + t_2^(-1)`
D. `T^(-2) = t_1^(-2) + t_2^(-2)`

Answer

To solve the problem, we need to find the time period of oscillation of a block attached to two springs connected in series. We will denote the time periods of the individual springs as \( t_1 \) and \( t_2 \), and we will derive the time period \( T \) for the combination of the two springs.

### Step 1: Understand the Time Period Formula
The time period \( T \) of a mass \( m \) attached to a spring with spring constant \( k \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
For the first spring with spring constant \( k_1 \), the time period is:
\[ t_1 = 2\pi \sqrt{\frac{m}{k_1}} \]
For the second spring with spring constant \( k_2 \), the time period is:
\[ t_2 = 2\pi \sqrt{\frac{m}{k_2}} \]

### Step 2: Find the Effective Spring Constant for Springs in Series
When two springs are connected in series, the effective spring constant \( k \) can be calculated using the formula:
\[ \frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2} \]
This implies:
\[ k = \frac{k_1 k_2}{k_1 + k_2} \]

### Step 3: Substitute the Effective Spring Constant into the Time Period Formula
Now, we can substitute the effective spring constant \( k \) into the time period formula for the combined system:
\[ T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{m}{\frac{k_1 k_2}{k_1 + k_2}}} \]
This simplifies to:
\[ T = 2\pi \sqrt{\frac{m (k_1 + k_2)}{k_1 k_2}} \]

### Step 4: Relate \( T \) to \( t_1 \) and \( t_2 \)
Now, we can express \( T \) in terms of \( t_1 \) and \( t_2 \):
From the equations for \( t_1 \) and \( t_2 \):
\[ t_1^2 = 4\pi^2 \frac{m}{k_1} \quad \text{and} \quad t_2^2 = 4\pi^2 \frac{m}{k_2} \]
Thus, we can express \( \frac{m}{k_1} \) and \( \frac{m}{k_2} \) as:
\[ \frac{m}{k_1} = \frac{t_1^2}{4\pi^2} \quad \text{and} \quad \frac{m}{k_2} = \frac{t_2^2}{4\pi^2} \]

### Step 5: Substitute Back into the Time Period Equation
Substituting these into the expression for \( T \):
\[ T^2 = 4\pi^2 \left( \frac{m}{k} \right) = 4\pi^2 \left( \frac{m (k_1 + k_2)}{k_1 k_2} \right) \]
This leads to:
\[ T^2 = \frac{t_1^2 t_2^2}{t_1^2 + t_2^2} \]

### Final Result
Thus, the time period \( T \) for the block oscillating with both springs in series is given by:
\[ T = \sqrt{t_1^2 + t_2^2} \]