6.1 Basic Concepts and Formulae
239
about the vertical wire and released the oscillator would execute oscillations in the
horizontal plane. For small twists the restoring torque will be proportional to the
angular displacement
τ = −Cθ
(6.26)
where C is known as torsional constant. The time period of oscillations is given by
T = 2π
l
C
(6.27)
Coupled Harmonic Oscillators
Two equal masses connected by a spring and two other identical springs fixed to
rigid supports on either side, Fig. 6.2, permit the masses to jointly undergo SHM
along a straight line, so that the system corresponds to two coupled oscillators. The
equation of motion for mass m 1 is
m ¨
x 1 + k(2x 1 − x 2 ) = 0
(6.28)
Fig. 6.2
and that for m 2 is
m ¨
x 2 + k(2x 2 − x 1 ) = 0
(6.29)
Equations (6.28) and (6.29) are coupled equations.
Assuming x 1 = A 1 sin ωt and x 2 = A 2 sin ωt
(6.28) and (6.29) become
¨
x 1 = −ω
2 A 1 sin ωt = −ω
2 x 1
(6.30)
¨
x 2 = −ω
2 A 2 sin ωt = −ω
2 x 2
(6.31)
Inserting (6.30) and (6.31) in (6.28) and (6.29), we get on rearrangement
(2k − mω
2
)x 1 − kx 2 = 0
(6.32)
− kx 1 + (2k − mω
2
)x 2 = 0
(6.33)