Infinite springs with force constants k,2k, 4k and 8k … respectively are connected in series. What will be the effective force constant of springs, when `k=10N//m`

Explanation

To solve the problem of finding the effective force constant of an infinite series of springs with force constants \( k, 2k, 4k, 8k, \ldots \), we will follow these steps:

### Step-by-step Solution:

1. **Identify the Force Constants**: The force constants of the springs are \( k, 2k, 4k, 8k, \ldots \). We can express these in terms of \( k \):
- \( k_1 = k \)
- \( k_2 = 2k \)
- \( k_3 = 4k \)
- \( k_4 = 8k \)
- And so on...

2. **Use the Formula for Springs in Series**: When springs are connected in series, the effective force constant \( k_{\text{equiv}} \) is given by the formula:
\[ \frac{1}{k_{\text{equiv}}} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} + \ldots \]

3. **Substitute the Force Constants**: Substitute the values of the force constants into the formula:
\[ \frac{1}{k_{\text{equiv}}} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{4k} + \frac{1}{8k} + \ldots \]

4. **Factor Out \( \frac{1}{k} \)**: We can factor out \( \frac{1}{k} \) from the right-hand side:
\[ \frac{1}{k_{\text{equiv}}} = \frac{1}{k} \left( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \right) \]

5. **Recognize the Infinite Geometric Series**: The series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \) is an infinite geometric series where:
- The first term \( a = 1 \)
- The common ratio \( r = \frac{1}{2} \)

6. **Calculate the Sum of the Infinite Series**: The sum \( S \) of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

7. **Substitute Back to Find \( k_{\text{equiv}} \)**: Now substitute the sum back into the equation for \( \frac{1}{k_{\text{equiv}}} \):
\[ \frac{1}{k_{\text{equiv}}} = \frac{1}{k} \cdot 2 \]
Therefore:
\[ k_{\text{equiv}} = \frac{k}{2} \]

8. **Substitute the Value of \( k \)**: Given \( k = 10 \, \text{N/m} \):
\[ k_{\text{equiv}} = \frac{10}{2} = 5 \, \text{N/m} \]

### Final Answer:
The effective force constant of the infinite springs connected in series is \( \boxed{5 \, \text{N/m}} \).