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

(N/A) Given:
Mass of the wire,$m = 3.5 \times 10^{-2} \;kg$
Linear mass density,$\mu = 4.0 \times 10^{-2} \;kg \;m^{-1}$
Frequency of vibration,$f = 45 \;Hz$
Length of the wire,$l = \frac{m}{\mu} = \frac{3.5 \times 10^{-2}}{4.0 \times 10^{-2}} = 0.875 \;m$
For the fundamental mode of vibration,the length of the wire $l$ is equal to half the wavelength,i.e.,$l = \frac{\lambda}{2}$.
Therefore,$\lambda = 2l = 2 \times 0.875 = 1.75 \;m$.
$(a)$ The speed of the transverse wave $(v)$ is given by:
$v = f \lambda = 45 \times 1.75 = 78.75 \;m/s$.
$(b)$ The tension $(T)$ in the string is given by the relation $v = \sqrt{\frac{T}{\mu}}$,so $T = v^2 \mu$.
$T = (78.75)^2 \times 4.0 \times 10^{-2} = 6201.5625 \times 0.04 = 248.06 \;N$.