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[Phys-l] "compound" pendulum



I doubt "compound" is the correct description - if it has a proper name I'd like to know what it is.

I'm picturing a pendulum that in its vertical position looks like this:

m2
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xx
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m1

The connecting rods are ideal (massless, infinitely stiff, no friction, etc). The pivot axis (xx) is not in the center. Distance xx to m2 is L2, distance xx to m1 is L1. I have derived the differential equation describing its motion with a "torque" method and an "energy" method and get the same result. In a nutshell, the "torque" method sums the torques about the pivot axis and sets them equal to I*alpha. The energy method is simply conservation of energy. In both cases the answer I get is

theta'' + (C1/C2)*(g/L1)*sin(theta) = 0

where

C1 = 1 - m2*L2/m1*L1
C2 = 1 + m2*L2^2/m1*L1^2

This reduces correctly to the limiting cases of a simple pendulum {m2=0,L2=0} or just a "balanced rod" that does not oscillate {m1=m2,L1=L2}. Hopefully it is correct :-)

For pedagogical purposes, I'm trying to now solve the problem using a sum of forces approach, but am having a problem conceptualizing and also justifying it. For example, one approach is to consider the free-body diagram for m1. The gravitational force in the theta-direction on m1 is the usual: -m1*g*sin(theta).

After this I'm getting a bit sketchy.

I invoke the infinite stiffness of the rod to claim that the gravitational force on m2 (not a restoring force as I've drawn it) "transmits itself" to the other end of the rod so as to act against the force of gravity on m1. In other words, because of m2, the net force on m1 in the theta direction is really -m1*g*sin(theta) + m2*(L2/L1)*g*sin(theta). This is really just invoking the torque argument again and I think I'm pretty safe doing so in the limit of an infinitely stiff rod. Then I set this net force equal to m1*a1 which is m1*L1*theta''. The result is missing the quantity C2:

theta'' + (C1)*(g/L1)*sin(theta) = 0

At this point I start arguing myself in circles. First I claim "wait the forces have to be equal to something like (m1+m2)*Lcm*theta''" then I start thinking that the sum of forces has to be a single one acting through the center of mass but then why isn't a free-body diagram for m1 not valid. And so on.

I can't see that I need 2 differential equations, 1 for m1 and 1 for m2, because theta for both masses is the same, hence theta'' is the same - besides, I could have done a free-body diagram for m2 instead and tried to solve. I'm ignoring the forces along the length of the pendulum for the same reasons as the simple pendulum (don't need to find the tension in L1 and L2) - can't imagine I'm missing something in that regard.

Am I missing the representation of some internal force? Was the earlier torque or energy approach incorrect instead?


Stefan Jeglinski