Assertion A spring of force constatn k is cut in to two piece having lengths in the ratio 1:2 The force constant of series combination of the two parts is `(3k)/(2)`
The spring connected in series are represented by `k=k_(1)+k_(2)`

Options

A. Both assertion and reson are true and reason is the correct explanation of assertion
B. Both assetion and reason are true but reason is not the correct explanation of assertion
C. Assertion is true but reason is false
D. Both assetion and reason are flase

Answer

To solve the problem, we need to analyze the assertion and the reason provided in the question step by step.

### Step 1: Understanding the Assertion
The assertion states that a spring with a force constant \( k \) is cut into two pieces with lengths in the ratio \( 1:2 \). We need to determine the spring constants of the two pieces after cutting.

### Step 2: Cutting the Spring
Let the total length of the spring be \( L \). If we cut the spring into two pieces in the ratio \( 1:2 \), we can denote the lengths of the two pieces as:
- \( l_1 = \frac{L}{3} \) (for the first piece)
- \( l_2 = \frac{2L}{3} \) (for the second piece)

### Step 3: Finding the Spring Constants
The spring constant \( k \) of a spring is inversely proportional to its length. Therefore, the spring constants \( k_1 \) and \( k_2 \) for the two pieces can be calculated as follows:
- For the first piece:
\[ k_1 = \frac{k \cdot L}{l_1} = \frac{k \cdot L}{\frac{L}{3}} = 3k \]

- For the second piece:
\[ k_2 = \frac{k \cdot L}{l_2} = \frac{k \cdot L}{\frac{2L}{3}} = \frac{3k}{2} \]

### Step 4: Combining the Springs in Series
When two springs are connected in series, the equivalent spring constant \( k_s \) is given by:
\[ \frac{1}{k_s} = \frac{1}{k_1} + \frac{1}{k_2} \]
Substituting the values of \( k_1 \) and \( k_2 \):
\[ \frac{1}{k_s} = \frac{1}{3k} + \frac{2}{3k} = \frac{3}{3k} = \frac{1}{k} \]
Thus, we find:
\[ k_s = k \]

### Step 5: Conclusion
The assertion claims that the force constant of the series combination of the two parts is \( \frac{3k}{2} \), which we found to be incorrect since \( k_s = k \). Therefore, the assertion is false.

### Step 6: Analyzing the Reason
The reason states that the spring constant connected in series is represented by \( k = k_1 + k_2 \). This is also incorrect because, for springs in series, the correct formula is \( \frac{1}{k_s} = \frac{1}{k_1} + \frac{1}{k_2} \).

### Final Answer
Both the assertion and the reason are false.

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