This is hypothetical question: imagine a tall skyscraper says 1000m tall is lifted by a nign indestructible cables attached only at the top of the building. When the cable is released what happens? Would I immediately see the building starts falling or it will stay at the same height for a brief period of time before falling? I know the building will collapse under its own weight but let's grant this one time scientific miracle and assume the whole building can be lifted in one piece by a piece of cable.
$\begingroup$
$\endgroup$
6
-
1$\begingroup$ Why a skyscraper? Why 1 km tall? I'd expect a tiny bit of "dropped slinky" effect, but skyscrapers aren't very springy. physics.stackexchange.com/q/56833/123208 $\endgroup$– PM 2RingCommented Jan 18 at 8:26
-
1$\begingroup$ Do you mind if one replaces the skyscraper with an elastic beam? Or are you planning/dreaming to drop a skyscraper? $\endgroup$– basicsCommented Jan 18 at 9:34
-
$\begingroup$ How are you holding your system? Also the initial condition acts as a forcing in a linear structure $\endgroup$– basicsCommented Jan 18 at 9:36
-
$\begingroup$ it doesn't have to be a building but it is the most solid and rigid things I can think of and the 1km is for timing $\endgroup$– user6760Commented Jan 18 at 9:40
-
1$\begingroup$ Voting to reopen. Closing reason is absurd. This question is clearly about physics and not about engineering. $\endgroup$– gandalf61Commented Jan 18 at 14:46
|
Show 1 more comment
2 Answers
$\begingroup$
$\endgroup$
6
Would I immediately see the building starts falling or it will stay at the same height for a brief period of time before falling?
Once the cable is released the only force acting on the building is its weight. If we assume that the building is perfectly rigid then all parts of it will immediately start to fall with an acceleration of $g$. Its velocity starts at $0$, but then increases by approximately $10$ metres per second every second.
If the building is not perfectly rigid then its centre of mass will immediately start to fall at an acceleration of $g$, but different parts of the building may behave differently from one another. Initially, the upper part of the building may start to accelerate at a rate greater than $g$, and the lower part of the building may accelerate at a slower rate, causing the building to contract. This may result in stresses which make the building disintegrate. In this situation it is impossible to say exactly how the building will behave without knowing more about how it is constructed.
-
$\begingroup$ I don’t think the COM will start accelerating at g until around h/c or h/2c $\endgroup$ Commented Jan 18 at 19:35
-
-
$\begingroup$ @Jagerber48 If the skyscraper is perfectly rigid then the speed of sound in the skyscraper is infinite. If the skyscraper is not perfectly rigid then we are in the second scenario and all bets are off. $\endgroup$ Commented Jan 18 at 19:46
-
$\begingroup$ No in non rigid scenario not all bets are off. The bet that information travels no faster than the speed of sound is still on. $\endgroup$ Commented Jan 18 at 19:50
-
$\begingroup$ The perfectly rigid scenario is non-physical. $\endgroup$ Commented Jan 18 at 19:51
$\begingroup$
$\endgroup$
The question asks about a skyscraper but the comments clarify that answers can replace the skyscraper with any (colloquially) rigid object. Skyscrapers are made of many materials held together in many ways. I’ll consider a simple long steel I-beam.
No material is perfectly rigid. If it were then You could send signals faster than the speed of light by pushing on one end and instantaneously receiving the motion at the other end, no matter how long the material.
Real materials consist of atoms/molecules held together by electronic bonds. They can be modeled as masses connected by springs. In other words, all materials are like big slinky’s. Stiff materials like metals have very high spring constants and compliant materials have low spring constants. The same way a motion wave travels through a slinky as a wave, motional perturbations of regular materials travel through the material at the finite speed of sound for the material.
In the I beam dropped by a cable scenario what you would see is that, when the I beam is first dangled by the cable it will stretch slightly under its own gravity. When you release the I-beam then right away you will see the top of the I beam start simultaneously falling/re-compressing. The re-compression/falling wave will travel downwards through the beam at the speed of sound until it hits the bottom of the I beam. After that time the entire beam will be falling under the force of gravity. However the beam may be oscillating a little bit as the sound waves bounce off both ends until the energy in the waves is dissipated as heat or acoustic sound.
If you look at the bottom of the I beam you will see no motion until time h/v. this is because until this time the bottom of the I beam feels a downward force due to gravity and an upward spring force from the material above it. Only when that material above it begins falling (because the material above IT begins falling) does he spring force subside and gravity takes over.
This is famously demonstrated in physics 101 courses using slinkies. See e.g. this video https://m.youtube.com/watch?v=eCMmmEEyOO0. The principle is the exact same.
Edit to DIRECTLY address the question: you will see the top of the beam falling IMMEDIATELY after you release it, no delay. But you won’t see the bottom of the beam falling until the decompression wave gets there after time h/v. Incidentally the answer is the same for a building instead of an I beam. The mechanics are just more complicated since so many different materials, bodies, and joints are involved.
