91Ó°ÊÓ

The vibrating string can be modeled as a standing wave, which is a combination of two sinusoidal waves with equal amplitude and frequency, moving in opposite directions. On this string, we have some points where the amplitude is maximum, which are called antinodes, and points where the amplitude is minimum, which are called nodes. Nodes occur at both ends of the string and at equal intervals between the endpoints. As the string is vibrating in its third overtone, there are four nodes, including the fixed ends of the string. We can represent the standing wave function of the string as follows: \[A(x) = A \sin{(kx)}\] where \(A(x)\) is the amplitude at position \(x\) along the string, and \(k\) is the wave number. #Step 3: Find the wave number \(k\)#

Using the equation for the amplitude at position \(x\), we can find the amplitude at a distance \(\frac{l}{3}\) from one end: \[A\left(\frac{l}{3}\right) = A \sin{\left(\frac{8\pi}{l} \cdot \frac{l}{3}\right)}\] We can now simplify the expression: \[A\left(\frac{l}{3}\right) = A \sin\left(\frac{8\pi}{3}\right)\] Since \(\sin{\left(\frac{8\pi}{3}\right)} = -\sin{\left(\frac{2\pi}{3}\right)} = -\frac{\sqrt{3}}{2}\), we have: \[A\left(\frac{l}{3}\right) = -A \cdot \frac{\sqrt{3}}{2}\] However, we are dealing with an amplitude, which is a scalar quantity, so we will take the absolute value to represent the physical amplitude of the particle: \[A\left(\frac{l}{3}\right) = \left|-A \cdot \frac{\sqrt{3}}{2}\right| = \frac{A\sqrt{3}}{2}\] Now comparing with the given options: (C) \(\frac{\sqrt{3 A}}{2}\) = \(\frac{A\sqrt{3}}{2}\) Thus, the amplitude of the particle at a distance of \(\frac{l}{3}\) from one end is \(\boxed{\text{(C)} \frac{\sqrt{3 A}}{2}}\).