1-) A spring with spring constant 18.5N/m hangs from the ceiling. A 570g ball is

Question

1-) A spring with spring constant 18.5N/m hangs from the ceiling. A 570g ball is attached to the spring and allowed to come to rest. It is then pulled down 7.00cm and released. What is the time constant if the ball's amplitude has decreased to 3.00cm after 57.0 oscillations?

2-) A 56.0kg cheerleader uses an oil-filled hydraulic lift to hold four 110kg football players at a height of 1.30m. If her piston is 19.0cm in diameter, what is the diameter of the football players' piston?

3-) A 0.500m -diameter vat of liquid is 2.20m deep. The pressure at the bottom of the vat is 1.30atm. What is the mass of the liquid in the vat?

Explanation / Answer

The natural oscillation frequency of the system is

f = 1/ (2pi) * sqrt (18.5/0.57) = 1.185 Hz

57 oscillations take

t = 57/1.185 = 48 sec

A model to be used for the oscillations is given by

y(t) = A exp (-t/tau) * cos (wt)

from the initial conditions, the equation above is re-written as

y(t) = 7*exp(-t/tau)*cos(wt)

The initial amplitude is 7 cm. After 57secs, the amplitude becomes 3, thus the time constant tau can be calculated as

tau = -57 / ln(3/7) = 48.3 secs

2)

Pascals Principle says that the pressure at one end equals the pressure at the other end. Thus

F1/A1 = F2/A2

you have enough information for:
F1 = m1g
F2 = 4m2g
A1 = pi d

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