Oscillations and Waves

Question 1

Two identical point masses P and Q, suspended from two separate massless springs of spring constants k₁ and k₂, respectively, oscillate vertically. If their maximum speeds are the same, the ratio (AQ/AP) of the amplitude AQ of mass Q to the amplitude AP of mass P is

Accepted Answer

√(k₁/k₂). In simple harmonic motion the maximum speed is v_max = Aω, with ω = √(k/m). Equal maximum speeds and equal masses therefore give AP√k₁ = AQ√k₂, so AQ/AP = √(k₁/k₂).

Answer

k₂/k₁

Answer

k₁/k₂

Answer

√(k₂/k₁)

Question 2

A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to:

Accepted Answer

f. An open pipe of length L has fundamental frequency f = v/2L. Dipping it halfway leaves an air column of length L/2 that is now closed at the water surface, and a closed pipe has f' = v/4L'. Substituting L' = L/2 gives f' = v/2L, exactly the original f.

Answer

f/2

Answer

3f/2

Answer

2f

Question 3

For sound waves, if the number of nodes for the 5th harmonic of an open-ended pipe is n and that for the 9th harmonic of the same pipe with one of its ends closed is m, the ratio n/m is

Accepted Answer

1. An open pipe's Nth harmonic carries N nodes, so n = 5. A closed pipe supports only odd harmonics, and counting them in order — 1st, 3rd, 5th, 7th, 9th — the 9th harmonic is the fifth of these and carries 5 nodes, so m = 5 as well and the ratio is 1.

Answer

5/9

Answer

9/5

Answer

3/5

Q1.Two identical point masses P and Q, suspended from two separate massless springs of spring constants k₁ and k₂, respectively, oscillate vertically. If their maximum speeds are the same, the ratio (AQ/AP) of the amplitude AQ of mass Q to the amplitude AP of mass P is

Q2.A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to:

Q3.For sound waves, if the number of nodes for the 5th harmonic of an open-ended pipe is n and that for the 9th harmonic of the same pipe with one of its ends closed is m, the ratio n/m is