### Q.14:- A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 × 10^{–2} kg and its linear mass density is 4.0 × 10^{–2} kg m^{–1}. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?

**Answer:-**

**
**(a) Mass of the wire,

*m*= 3.5 × 10

^{–2}kg

Linear mass density,

*μ = m/l*= 4.0 × 10

^{-2}kg m

^{-1}

Frequency of vibration,

*= 45 Hz∴ length of the wire, l =*

**v***m*/

*μ*= 3.5 × 10

^{–2 }/ 4.0 × 10

^{-2 }= 0.875 m

The wavelength of the stationary wave (

*λ*) is related to the length of the wire by the relation:

λ = 2

*l*/

*m*

where,

*n*= Number of nodes in the wire

For fundamental node,

*n*= 1:

λ = 2

*l*

λ = 2 × 0.875 = 1.75 m

The speed of the transverse wave in the string is given as:

*v*= ν

*λ*= 45 × 1.75 = 78.75 m/s

(b) The tension produced in the string is given by the relation:

*T *= *v*^{2}*µ*

= (78.75)^{2} × 4.0 × 10^{–2} = 248.06 N