Infinite springs with force constant`k , 2k, 4k and 8 k .... respectively are connected in series. The effective force constant of the spring will b

Options

A. 2k
B. K
C. k/2
D. 2048

Answer

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

### Step 1: Identify the pattern of the spring constants
The spring constants are given as:
- First spring: \( k \)
- Second spring: \( 2k \)
- Third spring: \( 4k \)
- Fourth spring: \( 8k \)
- Continuing this pattern, the \( n \)-th spring constant can be expressed as \( 2^{n-1} k \).

### Step 2: Write the formula for the effective spring constant in series
For springs connected in series, the effective spring constant \( k_{\text{eff}} \) can be calculated using the formula:
\[ \frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} + \ldots \]
In our case, we have an infinite series of springs.

### Step 3: Substitute the values of spring constants into the formula
Substituting the values of the spring constants into the formula, we get:
\[ \frac{1}{k_{\text{eff}}} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{4k} + \frac{1}{8k} + \ldots \]
This can be simplified as:
\[ \frac{1}{k_{\text{eff}}} = \frac{1}{k} \left( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \right) \]

### Step 4: Recognize the series as a geometric series
The series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \) is a geometric series where the first term \( a = 1 \) and the common ratio \( r = \frac{1}{2} \). The sum of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values, we have:
\[ S = \frac{1}{1 - \frac{1}{2}} = 2 \]

### Step 5: Substitute the sum back into the equation for \( k_{\text{eff}} \)
Now substituting the sum back into the equation for \( k_{\text{eff}} \):
\[ \frac{1}{k_{\text{eff}}} = \frac{1}{k} \cdot 2 \]
Thus,
\[ k_{\text{eff}} = \frac{k}{2} \]

### Final Answer
The effective force constant of the infinite series of springs is:
\[ k_{\text{eff}} = \frac{k}{2} \]