Question: 35: 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
(1) 1: \sqrt{2}
(2) 1: 2
(3) 1: 1
(4) \sqrt{2}: 1
Answer: Option (1)
Explanation:
The speed of a transverse wave on a stretched string is given by the relation
v = \sqrt{\frac{T}{\mu}},
where T is the tension in the string and \mu is the linear mass density of the string.
Let the initial tension be T. Then the initial speed of the wave is
v_1 = \sqrt{\frac{T}{\mu}}.
If the tension is doubled, the final tension becomes 2T.
The final speed of the wave is
v_2 = \sqrt{\frac{2T}{\mu}}.
Taking the ratio of initial speed to final speed, we get
\frac{v_1}{v_2} = \frac{\sqrt{\frac{T}{\mu}}}{\sqrt{\frac{2T}{\mu}}}.
Simplifying, \frac{v_1}{v_2} = \frac{1}{\sqrt{2}}.
Therefore, the ratio of the initial and final speeds is 1 : \sqrt{2}.
Hence, the correct answer is Option (1).