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Oscillations

# Sinusoidal waves 6 cm in amplitude are to be transmitted along a string that has a linear mass density of $\;4 \times 10^{-2} Kg/m\;$ . If the source can deliver a maximum power of 400 W and the string is under a tension of 100 N , what is the highest vibrational frequency at which the source can operate

$(a)\;38\;s^{-1}\qquad(b)\;6.5\;s^{-1}\qquad(c)\;38.5\;s^{-1}\qquad(d)\;37.5\;s^{-1}$

Answer : (d) $\;37.5\;s^{-1}$
Explanation :
Velocity (v) = $\;\sqrt{\large\frac{T}{\mu}}=\sqrt{\large\frac{100}{4 \times 10^{-2}}}=5 \times 10=50\;m/s$
Power (P) = $\;\large\frac{\mu w^2 A^2 v}{2}$
$400=\large\frac{4 \times 10^{-2} \times w^2 \times 6 \times 10^{-2} \times 6 \times 10^{-2} \times 50 \times 50}{2}$
$w^2=\large\frac{2\times 10^{6}}{36}$
$w=\large\frac{\sqrt{2} \times 10^{3}}{6}$
vibrational frequency = $\;\large\frac{w}{2 \pi}=\large\frac{\sqrt{2} \times 10^3}{2 \pi \times 6}$
$=37.5\;s^{-1}$