The mechanics of two pendulums coupled by a stressed spring is discussed, and the behavior for small oscillations is described. When the system is in its highest symmetry configuration, the pendulums are independent and the normal frequencies are degenerate.

1.
N. W.
Ashcroft
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6.
Reference 2, Chap. 2.
7.
In the large stretch approximation the rest length of the spring can be neglected.
8.
A system satisfying this condition must have small kc and large |d0| so that their product equals |F0|. The small displacements of the bodies during the oscillations (compared with the large |d0|) implies that the force of the spring remains approximately F0 .
9.
In this pendulum the constant force S(|S|=mg/cosθ0) plays the role of the weight mg in ordinary pendulums. The difference can be seen as a replacement of the gravitational field g by g/cosθ0. The frequency ωp is obtained by replacing g by g/cosθ0 in the relation for an ordinary pendulum (ω=g/L). The value θ0=90° is not attainable because it would require infinite tension in the spring and imply that ωp+.
10.
For example, when θ0=0° the motion of the center of mass is only horizontal.
11.
The rectilinear trajectories (segments) can be obtained from the initial conditions of ẋ1=ẋ2=0.
12.
The experimental devices were by Pasco and the sensors were managed by SCIENCE WORKSHOP software (www.pasco.com/).
13.
The angular displacements from θ0 were set using the sensors and chosen to be <5°.
14.
The mass of the pendulum rods was neglected in deriving Eqs. (12), (14), and (15). Better agreement with experiment is obtained using the equations of motion θ̈1=(lcmMg/Icosθ0)θ1(kcl2/I)[(θ1+θ2)cos2θ0+(θ1θ2)sin2θ0] and θ̈2=(lcmMg/Icosθ0)θ2(kcl2/I)[(θ2+θ1)cos2θ0+(θ2θ1)sin2θ0]. Here θ1 and θ2 are the angular displacements of the pendulums from θ0; lcm is the distance between the pivot and the center of mass of each pendulum; l is the distance between the pivot and the point where the spring is jointed; and M and I are the mass and moment of inertia (relative to the pivot) of each pendulum. The result for the normal frequencies is ωs=(lcmMg/Icosθ0)+2(kcl2/I)cos2θ0 andωd=(lcmMg/Icosθ0)+2(kcl2/I)sin2θ0 [see Fig. 7(b)]. The values of the parameters were lcm=30.1cm, l=34.5cm, M=103.3g, and I=102300gcm2.
15.
Reference 3, Chap. 48.
16.
The derivation is as follows. An initial position with ycm=xcm is chosen.Because the circular motion must have the angular velocity ωp2+ωc2 (the degeneracy frequency), the initial vertical component of the velocity must be the product of this angular velocity and xcm: ẏcm=ωp2+ωc2xcm (having chosen the anticlockwise direction). From geometry it follows that ẋcm=ẏcm.
17.
This procedure comes from the fact that in this case the pendulums are independent and oscillate with frequency ωp2+ωc2. Therefore, if they start at the same distance (x1=x2) from the origin, pendulum 2, after 3/4 of a oscillation, passes the origin with the velocity ωp2+ωc2x1.
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