A uniform horizontal rod of length 40 cm and mass 12 kg is supported by two identical wires as shown in figure (15-E9). Where should a mass of 48 kg be placed on the rod so that the same tuning fork may excite the wire on left into its fundamental vibrations and that on right into its first overtone? Take g = 10 m s⁻¹.

Three resonant frequencies of the string are 90, 150 and 210 Hz. (A) Find the highest possible fundamental frequency of vibration of this string. (B) Which harmonics of the fundamental are the given frequencies? (C) Which overtones are these frequencies? (D) If The length of the string is 80 cm, what would be the speed of a transverse wave on this string?

Two wires are kept right between the same pair of supports. The tensions in the wires are in the ratio 2:1, the radii are in the ratio 3:1 and the densities are in the ratio 1 : 2. Find the ratio of their fundamental frequencies.

Figure (15-E10) shows an aluminium wire of length 60 cm joined to the steel wire of length 80 cm and stretched between two fixed supports. The tension produced is 40 N. The cross-sectional area of the steel wire is 10 mm² and that of the aluminium wire is 30 mm². What could be the minimum frequency of a tuning fork which can produce standing waves in the system with joint as a node? The density of aluminium is 26 cm⁻³ and that of steel is 7.8 g cm⁻³.

A string of length L fixed at both ends vibrates in its fundamental mode at a frequency v and a maximum amplitude A. (A) Find the wavelength and the wave number k. (B) Take the origin at one end of the string and the X-axis along the string. Take the Y-axis along The direction of the displacement. Take t = 0 at the instant when the middle point of the string passes through its mean position and is going towards the positive y-direction. Write the equation describing the standing wave.