A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
The speed of sound in steel is ³ m/s (or 5060 m/s).
Explanation
This question tests understanding of standing waves in a rod clamped at its middle. When a rod is clamped at its middle, the clamped point becomes a displacement node (since it's fixed), and both free ends become displacement antinodes. For longitudinal vibrations, each half of the rod corresponds to one-quarter of a wavelength (λ/4). The fundamental frequency relationship v = fλ is then applied to find the speed of sound in steel.
Solution Steps
Step 1: Given - Length of steel rod, L = 100 cm = 1 m. Since the rod is clamped at its middle, each half = L/2 = 50 cm = 0.5 m. Fundamental frequency, f = 2.53 kHz = 2530 Hz.
Step 2: For a rod clamped at middle, the middle point is a displacement node and both ends are displacement antinodes. For fundamental mode of longitudinal vibrations, distance from node to antinode = λ/4.
Step 3: Therefore, L/2 = λ/4, which gives λ = 2L = m = 2 m.
Step 4: Speed of sound in steel: v = f × λ = 2530 Hz × 2 m = 5060 m/s or ³ m/s.