The vibrations of a string of length $60\, cm$ fixed at both the ends are represented by the equation $y = 2\,\sin \,\left( {\frac{{4\pi x}}{{15}}} \right)\,\cos \,\left( {96\pi t} \right)$ where $x$ and $y$ are in $cm$. The maximum number of loops that can be formed in it is
Diffcult
Download our app for free and get started
Suppose the string vibrates in $\mathrm{p}$ loops. Wavelength of the $\mathrm{p}^{\text {th }}$ mode of vibration is given by :
$\therefore \lambda_{\mathrm{p}}=\frac{2 l}{\mathrm{p}}$ ........$(i)$
Given that, $y=2 \sin \left(\frac{4 \pi x}{15}\right) \cos (96 \pi t)$
$=2\left[\sin \left(\frac{4 \pi \mathrm{x}}{15}+96 \pi \mathrm{t}\right)+\sin \left(\frac{4 \pi \mathrm{x}}{15}-96 \pi \mathrm{t}\right)\right]$
Comparing it with standard equation, we get
$\mathrm{v}=\frac{\omega}{2 \pi}=\frac{96 \pi}{2 \pi}=48 \mathrm{\,Hz}$
and $ \quad \mathrm{K}=\frac{4 \pi}{15}=\frac{2 \pi}{\lambda_{\mathrm{p}}} \quad$ or
$\lambda_{\mathrm{p}}=\frac{15}{2}$
from eq. $(i)$ we get,
$\frac{15}{2}=\frac{2 \times 60}{\mathrm{p}} $ or $ \mathrm{p}=16$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1If $\vec{u}$ is instantaneous velocity of particle and $\vec{v}$ is velocity of wave, thenView Solution
- 2The displacement of the interfering light waves are ${y_1} = 4\sin \omega \,t$ and ${y_2} = 3\sin \left( {\omega \,t + \frac{\pi }{2}} \right)$. What is the amplitude of the resultant waveView Solution
- 3A plane sound waves passes from medium $1$ into medium $2$. The speed of sound in medium $1$ is $200\,m/s$ and in medium $2$ is $100 \,m/s$. The ratio of amplitude of the transmitted wave to that of incident wave is :-View Solution
- 4An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by $100 Hz,$ then the fundamental frequency of open pipe is .... $Hz$View Solution
- 5Two vibrating tuning forks produce progressive waves given by ${Y_1} = 4\sin 500\pi t$ and ${Y_2} = 2\sin 506\pi t.$ Number of beats produced per minute isView Solution
- 6The speed of sound in hydrogen at $NTP$ is $1270\,m/s$ . Then, the speed in a mixture of hydrogen and oxygen in the ratio $4 : 1$ by volume will be ..... $m/s$View Solution
- 7A cylindrical tube, open at both ends, has a fundamental frequency ${f_0}$ in air. The tube is dipped vertically into water such that half of its length is inside water. The fundamental frequency of the air column now isView Solution
- 8A wave disturbance in a medium is described by $y(x,\,t) = 0.02\cos \,\left( {50\,\pi t + \frac{\pi }{2}} \right)\cos (10\pi x)$, where $ x$ and $y$ are in metres and $t$ in secondsView Solution
- 9$25$ tunning forks are arranged in series in the order of decreasing frequency. Any two successive forks produce $3\, beats/sec.$ If the frequency of the first turning fork is the octave of the last fork, then the frequency of the $21^{st}$ fork is .... $Hz$View Solution
- 10Two vibrating tuning forks produce progressive waves given by ${y_1} = 4\,\sin \,\left( {500\pi t} \right)$ and ${y_2} = 2\,\sin \,\left( {506\pi t} \right)$. These tuning forks are held near the ear of a person. The person will hear $\alpha \, beats/s$ with intensity ratio between maxima and minima equal to $\beta $. Find the value of $\beta - \alpha $View Solution