In your physics lab, an oscillator is attached to one end of a horizontal string. The other end of the string passes over a frictionless pulley. You suspend a mass M from the free end of the string, producing tension Mg in the string. The oscillator produces transverse waves of frequency f on the string. You don€™t vary this frequency during the experiment, but you try strings with three different linear mass densities m. You also keep a fixed distance between the end of the string where the oscillator is attached and the point where the string is in contact with the pulley€™s rim. To produce standing waves on the string, you vary M; then you measure the node-to-node distance d for each standing wave pattern and obtain the following data:

(a) Explain why you obtain only certain values of d.

(b) Graph µd² (in kg ˆ™ m) versus M (in kg). Explain why the data plotted this way should fall close to a straight line.

(c) Use the slope of the best straight-line fit to the data to determine the frequency f of the waves produced on the string by the oscillator. Take g = 9.80 m/s².

(d) For string A (µ = 0.0260 g/cm), what value of M (in grams) would be required to produce a standing wave with a node-to-node distance of 24.0 cm? Use the value of f that you calculated in part (c).

String B (g/cm) 0.0482 0.0260 0.0260 559 0.0374 0.0374 M (g) d (cm) 365 207 262 249 31.9 24.2 32.0 23.8 48.1