Physics - Wave on String Question with Solution | TestHub

The amplitude of a standing wave at position $x$ is $A_x=A_{max}|\sin (kx)|$, where $k=n\pi /L$ for a string fixed at both ends. The maximum kinetic energy of a small segment $dx$ is $d{K}_{max}=\frac{1}{2}(\mu dx)(\omega A_x{)}^2$. At $x=L/12$, this is $K_0$:$$K_0=\frac{1}{2}\mu dx{\omega }^2{A}_{max}^2{\sin }^2\left(\frac{n\pi (L/12)}{L}\right)=\frac{1}{2}\mu dx{\omega }^2{A}_{max}^2{\sin }^2\left(\frac{n\pi }{12}\right)$$ At an antinode,$A_x=A_{max}$, so the maximum kinetic energy is $4K_0$:$$4K_0=\frac{1}{2}\mu dx{\omega }^2{A}_{max}^2$$ Dividing the expression for$K_0$ by the expression for $4K_0$:$$\frac{K_0}{4K_0}={\sin }^2\left(\frac{n\pi }{12}\right)⟹{\sin }^2\left(\frac{n\pi }{12}\right)=\frac{1}{4}$$

Taking the square root,$\sin \left(\frac{n\pi }{12}\right)=\pm \frac{1}{2}$. Possible values for$\frac{n\pi }{12}$ are $\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6},\dots$. For $n$ to be an integer: If$\frac{n\pi }{12}=\frac{\pi }{6}$, then $n=2$. If$\frac{n\pi }{12}=\frac{5\pi }{6}$, then $n=10$. If$\frac{n\pi }{12}=\frac{7\pi }{6}$, then $n=14$. Thus,$n=2$ is a possible integer value.