Sound and Wave - Standing Wave

Small-amplitude standing waves of wavelength λ occur on a string with tension T, mass per unit μ, and length L. One end of string is fixed and the other end is attached to a ring of mass M that slides on a frictionless rod, as shown in the figure above. When gravity is neglected, which of the following conditions correctly determines the wavelength? (You might want to consider the limiting cases M → 0 and M → ∞.)

A. μ/M = (2π/λ) cot (2πL/λ)
B. μ/M = (2π/λ) tan (2πL/λ)
C. μ/M = (2π/λ) sin (2πL/λ)
D. λ = 2L/n, n = 1, 2, 3, ...
E. λ = 2L/(n + ½), n = 1, 2, 3, ...

(GR9677 #85)

Solution:

Consider the limiting cases M → 0 and M → ∞.

D and E do not depend on M → FALSE

If M → 0 , μ/M → ∞

C. FALSE because sin (2πL/λ)  cannot go to infinity

tan α = sin α / cos α → can go to infinity if cos α = 0

cot α = cos α / sin α → can go to infinity if sin α = 0

If M → ∞ , μ/M → 0, ring will not move = nodes on left and right side
The only possible wavelength: λ = 2L

2πL/λ = 2πL/2L = π

sin π = 0 → cot π = ∞ and tan π = 0 

 

Answer: B