Simple harmonic motion + rate of changes

[Carnauba]

A bucket of mass 2.0 kg containing 10 kg of water is hanged on a vertical ideal spring with constant 125 N/m, oscillating up and down with an amplitude equal to 3.0 cm.
Suddenly arises a leaky in the bottom of the bucket so that the water flows at constant rate of 2.0 grams/s. When the bucket is half full:

a) determine the period of oscillation and the rate at which the period varies with time.
b) What is the shortest period that this system can have?

 

[Simon Bridge]

Welcome to PF:
To get the best out of this forum, please show us how you have attempted the problem yourself. Do you, for instance, know the equation for the period of a mass on a spring?Your English is pretty good! Just a few tips...

the bucket is "hung" not "hanged" ... "hanged" means you've killed it via a popular execution method and is usually reserved for humans. "hung" is fine for inanimate objects and lumps of meat, thus: "The man was hanged, then he hung there."

"the bucket springs a leak" or "the bucket starts leaking" ... this is a tricky one since it relied on idiom. It is reasonable to say "the bucket becomes leaky" or "a leak arises in the bottom" but pretty unusual.