Question

A standing wave $y=A\sin \left(\frac{20}{3}\pi x\right)\cos \left(1000\pi t\right)$ is maintained in a taut string where $y$ and $x$ are expressed in meters. The distance between the successive points oscillating with the amplitude $A/2$ across a node is equal to :

• A. $2.5cm$
• B. $25cm$
• C. $5cm$
• D. $10cm$

Answer 1

The correct option is C $5cm$

A standing wave is basically an equation of simple harmonic motion having amplitude as a function of $x$.
Similarly in the given equation of wave :
$y=A\sin \left(\frac{20\pi }{3}x\right)\cos \left(1000\pi t\right)$
Here the Amplitude $A\left(x\right)=A\sin \left(\frac{20\pi }{3}x\right)$
where x is the distance of any point from the node
Now, The distance between the point oscillating with the amplitude A/2 above the node is equal to :
$\frac{A}{2}=A\sin \left(\frac{20\pi }{3}x\right)\frac{1}{2}=\sin \left(\frac{20\pi }{3}x\right)\sin \frac{\pi }{6}=\sin \left(\frac{20\pi }{3}x\right)\frac{\pi }{6}=\frac{20\pi }{3}xx=\frac{1}{40}m$
In standing wave there are two such points with amplitude ($\frac{A}{2}$) one above the centre line and another below the centre line.
The distance between the successive points oscillating with the amplitude A/2 across a node is equal to =$2\times x$
$2\times \frac{1}{40}=\frac{1}{20}mor5cm$
Hence option C is correct