Question

The length of elastic string, obeying Hooke’s law, is \( \ell_1 \) metres when the tension 4N and \( \ell_2 \) metres when the tension is 5N. The length in metres when the tension is 9N is:

• A. \( 5\ell_1 - 4\ell_2 \)
• B. \( 5\ell_2 - 4\ell_1 \)
• C. \( 9\ell_1 - 8\ell_2 \)
• D. \( 9\ell_2 - 8\ell_1 \)

Hint:

For elastic materials obeying Hooke's law, the extension is directly proportional to the applied force.

Answer 1

The Correct Option is C

Step 1: Hooke’s law states that the extension in a string is proportional to the applied force. Thus, the lengths of the string can be written as: \[ \ell_1 = k \cdot 4 \quad \text{and} \quad \ell_2 = k \cdot 5, \] where \( k \) is the constant of proportionality.
Step 2: The length when the tension is 9N is \( \ell_3 = k \cdot 9 \).
Step 3: Using the linear relationship, the length can be found as: \[ \ell_3 = 9\ell_1 - 8\ell_2. \]
Final Answer: \[ \boxed{9\ell_1 - 8\ell_2} \]