A steel rod of length 1 m is clamped at its middle. Take Y = 2 × 10¹¹ Pa and ρ = 8 g cm⁻³. Find the fundamental longitudinal frequency and the first overtone frequency.
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A steel rod of length 1 m is clamped at its middle. Take Y = 2 × 10¹¹ Pa and ρ = 8 g cm⁻³. Find the fundamental longitudinal frequency and the first overtone frequency.

G1 Physics · Accepted Answer

Step 1: Longitudinal-wave speed in steel is v = √(Y/ρ)=√[(2×10¹¹)/(8×10³)] = 5000 m s⁻¹. Step 2: A rod clamped at its middle has a node at the centre and free ends as antinodes; the fundamental has L=λ/2, so f₁=v/(2L)=2500 Hz. Step 3: Only odd multiples satisfy the same centre-node/free-end pattern; first overtone = 3f₁ = 7.5 kHz. Final: 2.5 kHz, 7.5 kHz.