The equation of vibration of a stretched string fixed at both ends and vibrating in $5^{\text {th }}$…
The equation of vibration of a stretched string fixed at both ends and vibrating in
$5^{\text {th }}$ harmonic is $Y=3 \sin (0 \cdot 4 x) \cos (200 \pi t)$ where 'x' and 'Y' are in $\mathrm{cm}$ and $\mathrm{t}$ in
second. Length of the string is
$(10 \cdot 5) \pi \mathrm{cm}$
$(8 \cdot 5) \pi \mathrm{cm}$
$(12 \cdot 5) \pi \mathrm{cm}$
$(4 \cdot 5) \pi \mathrm{cm}$
Solution
For $5^{\text {th }}$ harmonic, the string of length $\mathrm{L}$ vibrates in 5 loops
$\frac{5 \lambda}{2}=\mathrm{L}$
Now $\frac{2 \pi}{\lambda}=0.4 \quad \therefore \lambda=\frac{2 \pi}{0.4}=\frac{20 \pi}{4}=5 \pi$
$\therefore \quad L=\frac{5 \times 5 \pi}{2}=\frac{25 \pi}{2}=12.5 \pi$
.