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117. Problem 16.80P (HRW) A wheel is free to rotate about its fixed axle. A spring is attached to one of its spokes a distance r from the axle (see the diagram). Assuming that the wheel is a hoop of mass m and radius R, we have to obtain (a) the angular frequency of small oscillations of this system in terms of m, R, r, and the spring constant k. We have to find how the result changes if (b) r = R and (c) r = 0.
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Solution: Click For PDF Version (a) Rotational inertia of a hoop of mass m and radius R is
So the equation of motion of the hoop will be
So the hoop will execute SHM with angular frequency
(b) When r = R, angular frequency of SHM will be
(c) when r = 0, hoop will rotate freely without any acceleration. |
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Let us assume that because of the spring attached to a spoke at distance r
from the axle hoop undergoes small oscillations. As the force constant of the
spring is k, when the hoop has rotated clockwise by angle
it will be pulled by the spring with force
. Torque
on the hoop due to this force will be
