Eisersity Physics Problems 14.81 A 200).ha bucket containing 10.0 kg of water is

Question

Eisersity Physics Problems 14.81 A 200).ha bucket containing 10.0 kg of water is hanging from a vertical ideal spring of force constant 25 N/m and oscilating up and down with an amplitude of 3.00 cm. Suddenly the bucket springs a leak in the bottom such that water drops out at a steady rate of 2.00 g/s. When the bucket, is half ll, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. Is the period getting longer or shorter? (c) What is the shortest period this system can have?

Explanation / Answer

We know that the time period is given by the relation

T = 2? ?(m/K)
and we known that dm/dt = rate of the water leaking out = -2x10-3 kg/s, negative because the water is leaking,

Also the rate of change of the period is given by dT/dt

a) When the bucket is half full, then m = 5.0 kg (weight of half water) + 2.0 kg (weight of bucket) = 7.0 kg

also, spring constant = K = 125 N/m

thus time period is

T = 2? x ?(7.0/125) = 1.47 s-1 = Hz

B) dT/dt = ?/?(mK) dm/dt

Therefore, dT/dt = ? x (-2x10-3)/ (?(7.0x125)) = - 0.212 x 10-3 m/s

C) Since dT/dt is negative, the period is getting shorter

D) for the shortest period, the bucket is empty, so m=2.0 kg.

Therefore, T = 2? x ?(2.0/125) = 0.79 Hz

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