A particle is dropped from a height equal to the radius of the earth above the tunnel dug through the earth as shown in the figure.

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Let r is distance from center, If r > R ​ F = r 2 GM m ​ ( Not SHM ) r < R ​ F = R 3 GM m r ​ ( SHM ) ​ ⇒ But motion is periodic to find velocity at center, Apply COME, ​ 2 1 ​ m ( 0 2 ) − 2 R GM m ​ = 2 1 ​ m v C 2 ​ − 2 3 ​ R GM m ​ ​ v C ​ = R 2 GM ​ ​ ​

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