Practice question

**Longitudinal vibrations in rods** clamped at middle have fundamental with node at clamp and antinodes at ends, f₁ = v/(2L), v speed of sound in material (m/s). This relation allows v extraction from measured f₁ and length L, e.g., v = 2L·f₁. For rod clamped at middle, fundamental: v₁ = (v/2L) . 2000 = (v/2 × 1.5) ⇒ v = 2000 × 3 = 6000 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 6000 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.