Consider (unrealistically) a perfectly spherical rotating planet whose g value i

Question

Consider (unrealistically) a perfectly spherical rotating planet whose g value is constant over the surface.Abead lies on a frictionless wire that lies in the north–south direction across the equator. The wire takes the form of an arc of a circle; all points are the same distance from the center of the earth. The bead is released from rest, a short distance from the equator. Because geff does not point directly toward the earth’s center, the bead will head toward the equator and undergo oscillatory motion.
What is the frequency of these oscillations?

Explanation / Answer

Bead lies on a frictionless wire put in N-S direction across equator. If the wire forms an arc of a circle all points are at same distance from the centre of the earth. If a bead is released then bead will head towards the equator then its frequency of oscillations is

It is a displacement of single bead on the string in x axis = 0

Then its displacement along y axis = 0, t

The bead returning to initial value time period = T

-ym sin ( t) = Ym sin ( (t+T))

(t+T) = t + 2

T = 2

= 2/T

Angular frequency

f = / 2

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