Coupled Oscillators in a String

[elsamp123]

Homework Statement

A string of length 3L and negligible mass is attached to two fixed supports at its ends.
The tension in the string is T.

(a) A particle of mass m is attached at a distance L from the left end of the string.
What is the period for small transverse oscillations of m?
(b) A second particle of mass m is connected to the string a distance L=2 from the
right end of the string. Each of the three string segments have a tension T. Describe how
the two masses will oscillate for each of the normal modes of oscillation, and determine the
frequency ω for each of the two normal modes.

Thank you so very much! =)

 

[lippyka]

Homework EquationsT = (1/2)mv^2/LThe Attempt at a Solution (a) The period of oscillation is given by: T = 2π√(mL/T). (b) For the first normal mode, the two masses will oscillate in phase, with the middle segment of the string having zero tension. The frequency for this mode is given by ω = √(2T/mL). For the second normal mode, the two masses will oscillate out of phase, with the middle segment of the string having maximum tension. The frequency for this mode is given by ω = √(3T/mL).