Right sorry for the long question, here's what I've done so far.A cable is used to suspend a 800kg safe. The cable is elastic and stretches 20mm when subjected to a tension of 4kN.
a) Find the stiffness of the cable, and it's extension when the safe is hanging in equlibrium.
The safe is being lowered at 6m/s when the motor controlling the cable suddenly jams.
b) Determine the frequency with which the safe vibrates on the end of the cable.
c) Determine also the maximum tension in the cable during the oscillations.
a) In the initial state, where is the extension and is the stiffness constant.
Putting our initial conditions into the above equation, we get , so the stiffness
To find the extension when hanging in equilbrium, we use the fact that .
When the safe is hanging on the cable, this becomes , so the extension
b) Using Newton's Second Law, , resolvign vertically downwards, this gives us
where is the extra extension due occurring with the oscillation motion.
, now we know from above that , so:
,
Now if we let , our DE becomes , solving this gives us.
and
We know that , putting this into our second equation we get , .
Little confused about the initial conditions for though, is the displacement zero, or is it the that I calculated in the first part of the question?
Also how does do I calculate the frequency from this equation?
c) Not sure how to start this one either, guessing it involves inequalities but apart from that not sure.
Thanks in advance for the help