Solution: (a) We will use the information [1] . The moment of inertia of a thin rod about its center of mass is Irod=12Mrod⋅L2 . The moment of inertia of a thin full disc about an axis through its center and perpendicular to its face is IR=Md2R2 where Md=ρd⋅πR2=4kgm−2⋅3.14⋅(0.48m)2=2.9kg
For further determination, we also determine the moment of inertia of a thin disk with a radius of the hole Ih=Mh2rh2 , Mh=ρd⋅πrh2=4kgm−2⋅3.14⋅(0.2m)2=0.5kg
We determined all the moments of inertia of the constituent parts of our pendulum. Now we must take into account that their axis of rotation is shifted to the pivot point O. We should use the parallel axis theorem for planar case (also known as Huygens–Steiner theorem, or just as Steiner's theorem):
IR=IC+Md2 , where d is the distance from the center of mass C to the reference point R .
The center of rotation a thin rod is shifted on dr=L/2−d0=0.4−0.2=0.2m thus its moment of inertia about pivot point O is
We use negative sign for Mh due to this is holl and its mass was already add in Md and negative sign for dr as the center of thin rod mass is distance upward from pivot O.
(c) The angular frequency w=T2π=2.12⋅3.14=3radian⋅s−1=171°s−1 The period of pendulum you can get from (d)
(d) The period of physical pendulum is T=2πmglI . In our case l=hcm , m=Md−Mh+Mrod =3.6kg if we get g=9.8ms−2 then
T=2⋅3.14⋅3.6⋅9.8⋅0.381.53=2.1s
Answer: (a) The moment of inertia of the compound pendulum at a pivot O is 1.53kgm2
(b) The distance of the center of mass of the compound pendulum from pivot O is 0.38m
(c) The angular frequency of small oscillations is 3 rad/s
(d) The period of small oscillations of pendulum is 2.1s
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