Serway & Jewett — Superposition and Standing Waves: Problem 40

Review. For the arrangement shown in Figure P17.40, the inclined plane and the small pulley are frictionless; the string supports the object of mass \(M\) at the bottom of the plane; and the entire string has mass \(m\). The system is in equilibrium, and the vertical part of the string has a length \(h\). We wish to study standing waves set up in the vertical section of the string. (a) What analysis model describes the object of mass \(M\)? (b) What analysis model describes the waves on the vertical part of the string? (c) Find the tension in the string. (d) Model the shape of the string as one leg and the hypotenuse of a right triangle. Find the whole length of the string. (e) Find the mass per unit length of the string. (f) Find the speed of waves on the string. (g) Find the lowest frequency for a standing wave on the vertical section of the string. (h) Evaluate this result for \(M = 1.50\text{ kg}\), \(m = 0.750\text{ g}\), \(h = 0.500\text{ m}\), and \(\theta = 30.0^\circ\). (i) Find the numerical value for the lowest frequency for a standing wave on the sloped section of the string.