Question

If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is:

• A. 1:1
• B. √2:1
• C. 1:√2
• D. 1:2

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The speed of a transverse wave on a stretched string depends on the mechanical properties of the string, specifically its tension and its linear mass density.
Key Formula or Approach:
The speed (\(v\)) of a transverse wave on a string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \(T\) is the tension in the string and \(\mu\) is the linear mass density (mass per unit length).
Step 2: Detailed Explanation:
Let the initial tension be \(T_1 = T\) and the final tension be \(T_2 = 2T\).
The linear mass density \(\mu\) remains constant as it is the same string.
1. Initial speed (\(v_1\)):
\[ v_1 = \sqrt{\frac{T}{\mu}} \]
2. Final speed (\(v_2\)):
\[ v_2 = \sqrt{\frac{2T}{\mu}} = \sqrt{2} \cdot \sqrt{\frac{T}{\mu}} = \sqrt{2} v_1 \]
3. Calculating the ratio of initial speed to final speed:
\[ \frac{v_1}{v_2} = \frac{v_1}{\sqrt{2} v_1} = \frac{1}{\sqrt{2}} \]
Therefore, the ratio \(v_1 : v_2\) is \(1 : \sqrt{2}\).
Step 3: Final Answer:
The ratio of the initial and final speeds of the transverse wave is \(1 : \sqrt{2}\).