Two springs of spring constants`K_(1)` and `K_(2)` are joined in series. The effective spring constant of the combination is given by

Options

A. `((k_1+k_2))/2`
B. `k_1+k_2`
C. `(k_1k_2)/((k_1+k_2))`
D. `sqrt(k_1k_2)`

Explanation

To find the effective spring constant of two springs joined in series, we can follow these steps:

### Step 1: Understand the System
When two springs with spring constants \( K_1 \) and \( K_2 \) are connected in series, the total elongation \( d \) of the system is the sum of the elongations \( x \) and \( y \) of each spring:
\[ d = x + y \]

### Step 2: Apply Hooke's Law
According to Hooke's Law, the force exerted by a spring is proportional to its elongation:
\[ F = kx \]
For the first spring:
\[ F_1 = K_1 x \]
For the second spring:
\[ F_2 = K_2 y \]

### Step 3: Relate Forces
Since both springs are in series and the same force \( F \) acts on both, we have:
\[ F_1 = F_2 = F \]
Thus, we can write:
\[ F = K_1 x = K_2 y \]

### Step 4: Express Elongations in Terms of Force
From the equations above, we can express the elongations \( x \) and \( y \) in terms of the force \( F \):
\[ x = \frac{F}{K_1} \]
\[ y = \frac{F}{K_2} \]

### Step 5: Substitute Elongations into Total Elongation
Substituting \( x \) and \( y \) into the total elongation equation gives:
\[ d = \frac{F}{K_1} + \frac{F}{K_2} \]

### Step 6: Factor Out the Force
Factoring out \( F \) from the right side, we get:
\[ d = F \left( \frac{1}{K_1} + \frac{1}{K_2} \right) \]

### Step 7: Define Effective Spring Constant
The effective spring constant \( K_{eq} \) is defined by the relationship:
\[ F = K_{eq} d \]
Substituting for \( d \):
\[ F = K_{eq} \left( F \left( \frac{1}{K_1} + \frac{1}{K_2} \right) \right) \]

### Step 8: Solve for Effective Spring Constant
Dividing both sides by \( F \) (assuming \( F \neq 0 \)):
\[ 1 = K_{eq} \left( \frac{1}{K_1} + \frac{1}{K_2} \right) \]
Thus, we can express \( K_{eq} \):
\[ K_{eq} = \frac{1}{\left( \frac{1}{K_1} + \frac{1}{K_2} \right)} \]

### Step 9: Simplify the Expression
This can be simplified to:
\[ K_{eq} = \frac{K_1 K_2}{K_1 + K_2} \]

### Final Answer
The effective spring constant of the combination of springs in series is:
\[ K_{eq} = \frac{K_1 K_2}{K_1 + K_2} \]