Question

If a string suspended from the ceiling is given a downward force $F_1$, its length becomes $L_1$. Its length is $L_2$ if the downward force is $F_2$. What is its actual length?

• A. L1+L2/2
• B. √ L1L2
• C. F2L1+F1L2 / F2+ F1
• D. F2L1- F1L2 / F2- F1

Answer 1

The Correct Option is D

Given: A string suspended from the ceiling is subjected to two downward forces \( F_1 \) and \( F_2 \), resulting in lengths \( L_1 \) and \( L_2 \) respectively. We are asked to find the actual length of the string.

Concept: The actual length \( L \) of the string is related to the forces applied to it. The string follows Hooke's law, which states that the extension or compression in the string is proportional to the applied force. When two different forces \( F_1 \) and \( F_2 \) cause different extensions \( L_1 \) and \( L_2 \), we can use the principle of superposition to calculate the actual length of the string. 

Approach: - The forces \( F_1 \) and \( F_2 \) are proportional to the extensions \( L_1 \) and \( L_2 \), respectively. This means the relationship between force and extension can be expressed as: \[ \frac{F_2 - F_1}{L_2 - L_1} = \text{constant}. \] This constant is the stiffness of the string. - The actual length \( L \) is the length corresponding to zero force, which can be calculated by the formula: \[ L = \frac{F_2 L_1 - F_1 L_2}{F_2 - F_1}. \] 

Final Answer: The actual length of the string is given by: \[ L = \frac{F_2 L_1 - F_1 L_2}{F_2 - F_1}. \]