Serway & Jewett — Superposition and Standing Waves: Problem 44

Review. The top end of a yo-yo string is held stationary. The yo-yo itself is much more massive than the string. It starts from rest and moves down with constant acceleration \(0.800 \text{ m/s}^2\) as it unwinds from the string. The rubbing of the string against the edge of the yo-yo excites transverse standing-wave vibrations in the string. Both ends of the string are nodes even as the length of the string increases. Consider the instant \(1.20 \text{ s}\) after the motion begins from rest. (a) Show that the rate of change with time of the wavelength of the fundamental mode of oscillation is \(1.92 \text{ m/s}\). (b) What if? Is the rate of change of the wavelength of the second harmonic also \(1.92 \text{ m/s}\) at this moment? Explain your answer. (c) What if? The experiment is repeated after more mass has been added to the yo-yo body. The mass distribution is kept the same so that the yo-yo still moves with downward acceleration \(0.800 \text{ m/s}^2\). At the \(1.20\text{-s}\) point in this case, is the rate of change of the fundamental wavelength of the string vibration still equal to \(1.92 \text{ m/s}\)? Explain. (d) Is the rate of change of the second harmonic wavelength the same as in part (b)? Explain.