You and your team are exploring an antiquated research facility in the mountains of southern Argentina that had been abandoned in the 1960s. You come to a giant locked door that has no visible handles or actuators, but you find a hand-held device nearby that has two buttons, labeled “Open” and “Close.” It looks like some kind of crude remote control, but when you push the buttons they just click and nothing happens. You open the device’s top cover and inspect it. The internal mechanism resembles an old acoustic remote control called the “Space Command 600” that your parents had for their ancient TV. Pushing a button on the remote actuated a small hammer on the inside that struck the end of an aluminum rod, about 2 or 3 cm in length. The rod vibrated and emitted an ultrasonic sound wave that actuated an electrical circuit in the TV that was sensitive to that frequency. The TV remote had three buttons, and therefore three rods that vibrated at different frequencies. The first frequency turned the TV on and off , the second made the tuning dial click to the next station, and the third made the dial turn in the opposite direction. You pull off the cover of the device and fi nd that it has places for two ¼-inch diameter rods, but both are missing. 

However, written on the inside of the cover of the device is the following: “Open = 95.50 kHz” and “Close = 102.50 kHz.” Your team members search and eventually find a long piece of ¼-inch diameter aluminum rod.

(a) To what lengths must you cut the rod in order to get the remote to work properly (i.e., so that the fundamental frequencies of the rods match those utilized by the remote)?

(b) Suppose you had instead found a titanium rod. What lengths would be required in that case? (Young’s moduli are Y_Al = 6.9 × 10¹⁰ N/m² and Y_Ti = 1.2 × 10¹¹ N/m²; the mass densities are ρ_Al = 2700 kg/m³ and ρ_Ti = 4500 kg/m³.)