Suppose Tinkerbell the pixie and Captain Hook are locked in a sealed room. Havin
ID: 2942573 • Letter: S
Question
Suppose Tinkerbell the pixie and Captain Hook are locked in a sealed room. Having been separated from Peter Pan, Tinkerbell finds herself desperately flying around to avoid being caught by the evil Captain Hook. The room is empty but for a 1 kg chandelier suspended from the ceiling by a thin spring, which is stretched 0.4 m to its equilibrium position. Also suppose the chandelier is pulled down 1 m and released, without an initial velocity, to oscillate at t = 0. Assume no damping in the system and g = 10. The 2.5 kg (i.e., she introduces a constant external force of 25 N to the system from the moment she lands onwards) Tinkerbell is getting rather tired and would like to rest on the chandelier, which is beyond the reach of Captain Hook. Find the displacement u(f) of the system at any time given Tinkerbell lands at t = C, where C > 0.Explanation / Answer
To find the spring constant, k, we use the spring equilibrium info (stretched 0.4 m), and F = 0 to find that...
ks - mg = 0
meaning that k = mg/s = 10/0.4 = 25 N/m
Setting up the F for when the motion starts yields...
10 - Fs = ma (general form)
10 - 25y = mŸ (for our vertical situation...)
Ÿ + (25/m)y = (10/m) get this equation when dividing thru by the mass, m.
Ÿ + 25y = 10 (the final equation, noting that m = 1)
We could thus solve this, and use the initial conditions...
y(0) = 1 (position at time 0)
y'(0) = 0 (velocity at time 0)
which would give us a function y(t) that describes the motion, up until tinkerbell lands. Note: this is the general differential equation for the motion WITHOUT tinkerbell landing on the bouncing rig.
Hmmm. I think i would then use the t = C info to then find out the NEW "initial conditions"...the ones which describe the position and velocity of the rig at the time of her landing.
Hmmm. With her landing, the m changes...
m = 1 + 2.5 = 3.5 after that...
Ÿ + (25/3.5)y = (10/3.5)
Solve the new diff eqn, using the new Initial Values (at t = C)!
Let me know what you think!
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