A 1.5-meter-long steel rod is clamped at both ends and set into longitudinal vibrations, creating standing waves along its length. The speed of longitudinal waves in steel is v = 5000 m / s . The rod

Question

A 1.5-meter-long steel rod is clamped at both ends and set into longitudinal vibrations, creating standing waves along its length. The speed of longitudinal waves in steel is v = 5000 m / s . The rod vibrates in its third harmonic mode.

TestHub · Accepted Answer

Both clamped ends are displacement nodes: $L = \frac{nλ}{2}, f_{n} = \frac{n v}{2 L}$ .

(A) $f_{1} = \frac{v}{2 L} = \frac{5000}{3}$ Hz - correct.

(B) Third harmonic: $f_{3} = \frac{3 v}{2 L} = 5000$ Hz, $ω = 10000 π$ , $λ = \frac{2 L}{3} = 1$ m, $k = 2 π$ : $S = A sin (2 π x) cos (10000 π t)$ - correct.

(C) Nodes at $x = 0, 0.5, 1, 1.5$ m - correct.

(D) Strain, and hence stress, is maximum at the displacement nodes: $x = 0, 0.5, 1, 1.5$ m - correct.

Correct options: A, B, C, D

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