The ratio of the radii of gyration of a circular disc about a tangential axis in the Plane of the disc and of a circular ring of the same radius about the tangential axis in the Plane of the ring is
Answers
Answer:
It is given,
The radius of the circular disc = The radius of circular ring
The moment of inertia of the circular disc about the tangential axis in the plane of the disc,
Id = 5/4 [Md* R²] …… [here Md = mass of the disc & R = radius of the disc] ..…. (i)
The moment of inertia of the circular ring about the tangential axis in the plane of the ring,
Ir = 3/2 [Mr* R²] …… [here Mr = mass of the ring & R = radius of the disc] …....... (ii)
We know the formula for the moment of inertia for angular motion is given as,
I = MK² …… [here K = radius of gyration]
⇒ K = √[I/M] ….. (iii)
Let the radius of gyration of disc and ring be denoted as “Kd” & “Kr” respectively.
Therefore, based on eq. (iii) & substituting the values from (i) & (ii), we get
The ratio of radii of gyration of the circular disc & circular ring is,
= Kd/Kr
= √[{Id/Md} * {Mr/Ir}]
= √[{(5/4)*(Md* R²)/Md} * {Mr/(3/2*Mr*R²)}]
cancelling the similar terms
= √[(5/4) / (3/2)]
= √[5/6]
Thus , the ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about the tangential axis in the plane of the ring is √5 : √6.
the ans is √5/√6 .
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