Amassless spring hangs from the ceiling with a small object attached to its lower end. the object is initially held at rest in a position yi
such that the spring is at its rest length. the object is then released from yi
and oscillates up and down, with its lowest position being 10 cm below yi

(a) what is the frequency of the oscillation?
(b) what is the speed of the object when it is 8.0 cm below the initial position?
(c) an object of mass 300 g is attached to the first object, after which the system oscillates with half the original frequency. what is the mass of the first object?
(d) how far below yi is the new equilibrium (rest) position with both objects attached to the spring?

Neptune is the eighth planet from the sun. it was the first planet to get its existence predicted by  mathematical calculations  before it was actually seen through a telescope on sept. 23, 1846. irregularities in the orbit of  uranus  led french astronomer alexis bouvard to suggest that the gravitational pull from another celestial body might be responsible. german astronomer johann galle then relied on subsequent calculations to   spot neptune via telescope. previously, astronomer galileo galilei sketched the planet, but he mistook it for a star due to its slow motion. in accordance with all the other planets seen in the sky, this new world was given a name from greek and roman mythology — neptune, the roman god of the sea.

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