(20 pts) A spring is hanging from the ceiling and the bottom end of the spring i
ID: 1656608 • Letter: #
Question
(20 pts) A spring is hanging from the ceiling and the bottom end of the spring is at y-0, see ig.(). When Joseph Anderson hangs a 2.10 cm diameter lead ball on the end of the spring, the spring stretches so that the bottom end of the spring is at y-2.45 cm, see Fig.(i). Leticia Castro now pulls the ball down until the bottom of the spring is at y-3.95 cm, see Fig.(ilil); she then relea time t-0 and the ball oscillates in simple harmonic motion. Take the acceleration of gravity to be g-9.80 m/s2; incidentally, the density of lead is 11.34 gm/cm. 3 ,y = 0.0 cm- y=2.45 cm y=3.95 cm- a) (2 pts) What is the amplitude A of the oscillation?Explanation / Answer
a. given diameter of the ball, d = 2.1 cm
volume = pi*d^3/6 = pi*(2.1)^3/6 = 4.849 cm^3
density of lead, rho = 11.34 gm/cm^3
so, mass of ball = density * volume = 11.34*4.849 = 54.98766 g
when the ball is attached to the spring, the equilibrium position of the spring shifts by xo = 2.45 cm
so if force constant of the spring is k
kxo = mg
k = 54.98766*10^-3 * 9.81/2.45*10^-2 = 22.0175 N/m
Given amplitude, A = 3.95 - 2.45 = 1.5 cm ( the extra strech from equilibrium position)
natural frequency of oscilation, w = sqroot(k/m) = 20.01 rad/s
equation of motion taking the given origin as reference
y = 1.5*cos(wt) + 2.45 in cm
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