A massless spring is hanging vertically from the ceiling. A mass m is attached to the bottom end of the spring and released from rest. How close to its final resting position is the mass when βt = 1 given that the mass finally comes to rest a distance d below the point from which it was released and the oscillator is critically damped.
At the final rest position, the weight of the mass is exactly balanced by the Restoring force of the spring.
Now the weight of mass
and the magnitude of restoring force at maximum displacement is
then equating above two, we get
............................(1)
Alo we know that the natural frequency of the oscillation is
or using equation 1, ...................(2)
If y(t) denotes the displacement from rest. then for critical damping we know that displacement is given by
...........................(3)
Now taking positive y in downward direction thus displacement at t=0s is
and since it started from rest, thus initial velocity will be
Now putting t=0 in equation 3, we get
.....................(4)
Similarly, we'll get after putting the initial conditions,
......................(5)
Putting above two equations(4 and 5) in equation 3, we get
Then putting time, t=1 in above we'll get displacement
Now using equation 2 in above,
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