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MathematicalPhysicist
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i have two questions from kleppner's book, here it goes (iv'e attached a file which have the sketchs of the two problems):
6.17
A rod of length l and mass m, pivoted at one end, is held by a spring at its midpoint and a spring at its far end, both pulling in opposite directions.
The springs have spring constant k, and at equilibrium their pull is perpendicular to the rod. find the frequency of small oscillations about the equilibrium position.
6.18
Find the period of pendulum consisiting of a disk of mass M and radius R fixed to the end of a rod of length l and mass m. how does the period change if the disk is mounted to the rod by a frictionless bearing so that it is perfectly free to spin.
my attempt at solving the problems
6.17
i got to the next equations which as i see contradict each other:
[tex](1)=N=cos(\theta)[/tex]
[tex](2)=m\frac{d^2x}{dt^2}=2kx-Nsin(\theta)[/tex]
[tex](3)=x=\frac{l\theta}{2}[/tex]
the fourth equation is the equation of the torques around the point of contact of the rod with pivot:
[tex](4)=\frac{mglsin(\theta)}{2}-2kxsin(\frac{\pi}{2}+\theta)=\frac{ml^2\frac{d^2\theta}{dt^2}}{3}[/tex]
but obviously (4) contradicts 1+2, so where did i go wrong here?
6.18
without friction between the bearing and the disk we would have the equation of torques this way:
[tex]-(\frac{mglsin(\theta)}{2}+Mglsin(\theta))=I_0 \alpha[/tex]
where:
[tex]I_0=MR^2/2+ml^2/12[/tex]
and with friction, we would have include f*l.
the torques are calculated around the poivot of the rod with the cieling.
anyway, is something from what i wrote is correct? thwe problems is that for those questions there isn't even an answer clue.
6.17
A rod of length l and mass m, pivoted at one end, is held by a spring at its midpoint and a spring at its far end, both pulling in opposite directions.
The springs have spring constant k, and at equilibrium their pull is perpendicular to the rod. find the frequency of small oscillations about the equilibrium position.
6.18
Find the period of pendulum consisiting of a disk of mass M and radius R fixed to the end of a rod of length l and mass m. how does the period change if the disk is mounted to the rod by a frictionless bearing so that it is perfectly free to spin.
my attempt at solving the problems
6.17
i got to the next equations which as i see contradict each other:
[tex](1)=N=cos(\theta)[/tex]
[tex](2)=m\frac{d^2x}{dt^2}=2kx-Nsin(\theta)[/tex]
[tex](3)=x=\frac{l\theta}{2}[/tex]
the fourth equation is the equation of the torques around the point of contact of the rod with pivot:
[tex](4)=\frac{mglsin(\theta)}{2}-2kxsin(\frac{\pi}{2}+\theta)=\frac{ml^2\frac{d^2\theta}{dt^2}}{3}[/tex]
but obviously (4) contradicts 1+2, so where did i go wrong here?
6.18
without friction between the bearing and the disk we would have the equation of torques this way:
[tex]-(\frac{mglsin(\theta)}{2}+Mglsin(\theta))=I_0 \alpha[/tex]
where:
[tex]I_0=MR^2/2+ml^2/12[/tex]
and with friction, we would have include f*l.
the torques are calculated around the poivot of the rod with the cieling.
anyway, is something from what i wrote is correct? thwe problems is that for those questions there isn't even an answer clue.
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