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 derive the relationship step by step.

### Step 1: Understand the time period of a spring-mass system
The time period \( T \) of a mass \( m \) attached to a spring with spring constant \( k \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

### Step 2: Write the equations for the two springs
For the first spring with spring constant \( k_1 \), the time period \( T_1 \) is:
\[ T_1 = 2\pi \sqrt{\frac{m}{k_1}} \]
Squaring both sides gives:
\[ T_1^2 = 4\pi^2 \frac{m}{k_1} \quad \text{(Equation 1)} \]

For the second spring with spring constant \( k_2 \), the time period \( T_2 \) is:
\[ T_2 = 2\pi \sqrt{\frac{m}{k_2}} \]
Squaring both sides gives:
\[ T_2^2 = 4\pi^2 \frac{m}{k_2} \quad \text{(Equation 2)} \]

### Step 3: Find the equivalent spring constant for springs in series
When two springs are connected in series, the equivalent spring constant \( k \) is given by:
\[ \frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2} \]
This can be rearranged to:
\[ k = \frac{k_1 k_2}{k_1 + k_2} \]

### Step 4: Write the time period for the equivalent spring constant
The time period \( T \) for the equivalent spring constant \( k \) is:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Squaring both sides gives:
\[ T^2 = 4\pi^2 \frac{m}{k} \]

### Step 5: Substitute the value of \( k \)
Using the expression for \( k \) from the series connection, we substitute it into the equation for \( T^2 \):
\[ T^2 = 4\pi^2 \frac{m}{\frac{k_1 k_2}{k_1 + k_2}} = 4\pi^2 \frac{m(k_1 + k_2)}{k_1 k_2} \]

### Step 6: Relate \( T^2 \) to \( T_1^2 \) and \( T_2^2 \)
Now, we can use Equations 1 and 2 to express \( \frac{m}{k_1} \) and \( \frac{m}{k_2} \):
From Equation 1:
\[ \frac{m}{k_1} = \frac{T_1^2}{4\pi^2} \]
From Equation 2:
\[ \frac{m}{k_2} = \frac{T_2^2}{4\pi^2} \]
Substituting these into the equation for \( T^2 \):
\[ T^2 = 4\pi^2 \left(\frac{T_1^2}{4\pi^2} + \frac{T_2^2}{4\pi^2}\right) = T_1^2 + T_2^2 \]

### Final Result
Thus, the time period of the block oscillating with the combination of the two springs in series is:
\[ T = \sqrt{T_1^2 + T_2^2} \]