Question

Transverse waves are generated in two steel wires A and B with a source of 512 Hz. If their diameters are in the ratio 2 : 1 and tensions are in the ratio 2 : 1, the velocities of the waves are in the ratio (wires A and B) ______.

Options

• `sqrt2 : 1`

• 1 : 2

• `1 : sqrt2`

• 2 : 1

Answer 1

Transverse waves are generated in two steel wires A and B with a source of 512 Hz. If their diameters are in the ratio 2 : 1 and tensions are in the ratio 2 : 1, the velocities of the waves are in the ratio (wires A and B) `underlinebb(1 : sqrt2)`.

Explanation:

Given, f = 512 Hz

`d_A/d_B = 2` and `T_A/T_B = 2`

Because both wires are made of the same material, their densities are the same. However, because their areas differ, their mass per unit length will differ.

⇒ `rho_A = rho_B`

`mu_A/A_A = mu_B/A_B`

⇒ `mu_A/mu_B = A_A/A_B = (d_A/d_B)^2 = 4`

Since the velocity of the wave in a wire is given by,

v = `sqrt(T/mu)`

⇒ `"v"_A/"v"_B = sqrt(T_A/T_B xx mu_B/mu_A)`

= `sqrt(2 xx 1/4) = 1/sqrt2` or 1 : `sqrt2`