Wave Motion 2 Question 45
45. A metallic rod of length $1 m$ is rigidly clamped at its mid-point. Longitudinal stationary waves are set-up in the rod in such a way that there are two nodes on either side of the mid-point. The amplitude of an antinode is $2 \times 10^{-6} m$. Write the equation of motion at a point $2 cm$ from the mid-point and those of the constituent waves in the rod. (Young’s modulus of the material of the rod $=2 \times 10^{11} Nm^{-2} ;$ density $=8000 kg m^{-3}$ )
(1994, 6M)
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Answer:
Correct Answer: 45. $y=2 \times 10^{-6} \sin (0.1 \pi) \sin (25000 \pi t)$,
$y _1=10^{-6} \sin (25000 \pi t-5 \pi x), y _2=10^{-6} \sin (25000 \pi t+5 \pi x)$
Solution:
- Speed of longitudinal travelling wave in the rod will be
$$ v=\sqrt{\frac{Y}{\rho}}=\sqrt{\frac{2 \times 10^{11}}{8000}}=5000 m / s $$
Amplitude at antinode $=2 A$ (Here, $A$ is the amplitude of constituent waves )
$$ \begin{aligned} & =2 \times 10^{-6} m \\ & \therefore \quad A=10^{-6} m \Rightarrow l=\frac{5 \lambda}{2} \\ & \Rightarrow \quad \lambda=\frac{2 l}{5}=\frac{(2)(1.0)}{5} m=0.4 m \end{aligned} $$
Hence, the equation of motion at a distance $x$ from the mid-point will be given by,
$$ \begin{aligned} & \quad \begin{aligned} & y=2 A \sin k x \sin \omega t \\ & \text { Here, } \quad k=\frac{2 \pi}{0.4}=5 \pi \\ & \omega=2 \pi f=2 \pi \frac{v}{\lambda} \\ &=2 \pi\left(\frac{5000}{0.4}\right) rad / s=25000 \pi \\ & \therefore \quad y=\left(2 \times 10^{-6}\right) \sin (5 \pi x) \sin (25000 \pi t) \end{aligned} \end{aligned} $$
Therefore, $y$ at a distance $x=2 cm=2 \times 10^{-2} m$
$$ \begin{aligned} & \text { is } \quad y=2 \times 10^{-6} \sin \left(5 \pi \times 2 \times 10^{-2}\right) \sin (25000 \pi t) \\ & \text { or } \quad y=2 \times 10^{-6} \sin (0.1 \pi) \sin (25000 \pi t) \end{aligned} $$
The equations of constituent waves are
$$ \begin{aligned} & y _1=A \sin (\omega t-k x) \text { and } y _2=A \sin (\omega t+k x) \\ & \text { or } \quad y _1=10^{-6} \sin (25000 \pi t-5 \pi x) \\ & \text { and } \quad y _2=10^{-6} \sin (25000 \pi t+5 \pi x) \end{aligned} $$
