The length of an elastic string is a metre when the longitudinal tension is 4 N and b metre when the longitudinal tension is 5 N. The length of the string in metre when the longitudinal tension is 9 N is

Options

A. 4a-5b
B. 5b-4a
C. 9b-9a
D. a+b

Answer

To solve the problem, we need to determine the length of an elastic string under different tensions using the information provided. Let's break it down step by step.

### Step 1: Define the Variables
Let:
- \( L \) = original length of the string (when no tension is applied)
- \( A \) = length of the string when tension is 4 N
- \( B \) = length of the string when tension is 5 N
- \( K \) = spring constant of the string
- \( F \) = applied tension

### Step 2: Write the Equations for Length
According to Hooke's Law, the elongation \( x \) of the string due to the applied tension \( F \) can be expressed as:
\[ x = \frac{F}{K} \]

Thus, the new lengths can be expressed as:
1. For 4 N tension:
\[ A = L + \frac{4}{K} \]
2. For 5 N tension:
\[ B = L + \frac{5}{K} \]

### Step 3: Rearranging the Equations
From the above equations, we can express \( L \) in terms of \( A \) and \( B \):
1. Rearranging the first equation:
\[ L = A - \frac{4}{K} \]
2. Rearranging the second equation:
\[ L = B - \frac{5}{K} \]

### Step 4: Set the Two Expressions for \( L \) Equal
Since both expressions represent \( L \), we can set them equal to each other:
\[ A - \frac{4}{K} = B - \frac{5}{K} \]

### Step 5: Solve for \( K \)
Rearranging the equation gives:
\[ A - B = -\frac{1}{K} \]
Thus,
\[ \frac{1}{K} = B - A \]

### Step 6: Substitute \( K \) Back into the Length Equation for 9 N Tension
Now we need to find the length of the string when the tension is 9 N:
\[ L' = L + \frac{9}{K} \]
Substituting \( L \) from either equation (let's use the first):
\[ L' = \left(A - \frac{4}{K}\right) + \frac{9}{K} \]
\[ L' = A + \frac{5}{K} \]

### Step 7: Substitute \( K \) into the Length Equation
Now substitute \( K \) from the previous step:
\[ L' = A + 5(B - A) \]
\[ L' = A + 5B - 5A \]
\[ L' = 5B - 4A \]

### Final Answer
The length of the string when the longitudinal tension is 9 N is:
\[ L' = 5B - 4A \]