Mechanics of two pendulums coupled by a stressed spring

N. W.

 

Ashcroft

and

N. D.

 

Mermin

, (, , ), Chap. 22.

F. S.

 

Crawford

, Jr., (, , ), Chap. 1.

R. P.

 

Feynman

,

R. B.

 

Leighton

, and

M.

 

Sands

, (, , ), Vol. , Chap. 49.

L. O.

 

Olsen

, “,”  , – ().

M. J.

 

Moloney

, “,”  , – ().

Reference 2, Chap. 2.

In the large stretch approximation the rest length of the spring can be neglected.

A system satisfying this condition must have small $kc$ and large $|d⃗0|$ so that their product equals $|F⃗0|$⁠. The small displacements of the bodies during the oscillations (compared with the large $|d⃗0|$⁠) implies that the force of the spring remains approximately $F⃗0$ .

In this pendulum the constant force $S⃗$ $(|S⃗|=mg/cosθ0)$ plays the role of the weight $mg⃗$ 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 $ωp→+∞$⁠.

For example, when $θ0=0°$ the motion of the center of mass is only horizontal.

The rectilinear trajectories (segments) can be obtained from the initial conditions of $ẋ1=ẋ2=0$⁠.

The experimental devices were by Pasco and the sensors were managed by SCIENCE WORKSHOP software (www.pasco.com/).

The angular displacements from $θ0$ were set using the sensors and chosen to be $<5°$⁠.

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 $θ̈1=−(lcmMg/Icosθ0)θ1−(kcl2/I)[(θ1+θ2)cos2θ0+(θ1−θ2)sin2θ0]$ and $θ̈2=−(lcmMg/Icosθ0)θ2−(kcl2/I)[(θ2+θ1)cos2θ0+(θ2−θ1)sin2θ0]$⁠. 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 $ωs=(lcmMg/Icosθ0)+2(kcl2/I)cos2θ0$ and$ωd=(lcmMg/Icosθ0)+2(kcl2/I)sin2θ0$ [see Fig. 7(b)]. The values of the parameters were $lcm=30.1cm$⁠, $l=34.5cm$⁠, $M=103.3g$⁠, and $I=102300gcm2$⁠.

Reference 3, Chap. 48.

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 $ẋcm=−ẏcm$⁠.

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 $ωp2+ωc2x1$⁠.