Question

A string of length L is under tension, and the speed of a wave in the string is v. What will be the speed of a wave in the string if the length is increased to 2L but with no change in the mass or tension

Answer 1

Answer:

The speed increases by square root 2 times.

Explanation:

Let the wave velocity initially be 'v' with tension in the string as 'T' of mass 'm'.

Given:

Initial length of the string (L₁) = L

Final length of the string (L₂) = 2L

Wave velocity in a stretched string of length 'L' is given as:

$v=\sqrt{\frac{LT}{m}}$

From the above equation we can conclude that if tension 'T' and mass 'm' remains the same, the wave velocity is directly proportional to the square root of length of the stretched string. Therefore,

$v=k\sqrt L\\\\\frac{v_1}{\sqrt{L_1}}=\frac{v_2}{\sqrt{L_2}}$

Now, plug in the given values and solve for v₂ in terms of v₁. This gives,

$v_2=\frac{v}{\sqrt{L}}\times \sqrt{2L}\\\\v_2=\sqrt2 v$

Therefore, the speed of the wave is increased by square root 2 times.