Maximum height from which a cuboid can fall and not be damaged

[cromata]

1. Cuboid of height h is made out of material with density ρ=1000kg/m³ and it can endure highest pressure of σ=90kPa.Cuboid is falling from the height H. When the cuboid hits the ground its centre of mass is at the height h/2.Then starts deformation of the body. During the deformation centre of mass goes from h/2 to h/5. With this information you can find distance traveled by C.M. and average force acting on the body.
Find max H from which cuboid can fall so that pressure is less than σ.

P.S. Sorry for bad English, I hope you can understand the problem.

Homework Equations

F=m dv/dt, m=ρAh (A is size of cuboid surface)

The Attempt at a Solution

F=N-Fg
F=m dv/dt
p*A=ρAh dv*v/dh
p/ρ *∫dh/h=∫dv*v (left integral goes from h/2 to h/5 and right from v to 0, where v=√(2gH))
p/ρ *ln(h/5/h/2)=-v²/2
p/ρ ln(5/2)=gH
H=p/(ρ*g)*ln(5/2)
H=8.41m
I think that this solution should be correct.

 

[haruspex]

v=√(2gH))

What is the total distance the COM descends?

you can find ... average force acting on the body.

No you #%^*+ can't. Sadly, this is a widespread blunder in set questions. To find the average force you need the time for which it applies (and the momentum change), F_avg=∫F.dt/∫dt=Δp/Δt. The question expects you to find it by ΔE/Δx=∫F.dx/∫dx. That will in general give a different result. It will come out the same if the force is constant, but then you would not need to describe it as an average.
Note also that the way the question is worded you would actually want to know the peak force, not the average. The question should tell you to assume the force is constant, thereby avoiding both problems.

Putting that aside, your method is basically right, but wrong in the details, as I hinted above.

Edit: As TSny indicates, I was also wrong in saying your method is basically right.
Treating it as an elastic block could get very messy. The equations would include the possibility of vertical oscillations within it. Maybe try it as a set of horizontal sheets held apart by vertical rods that telescope into themselves under constant friction.
Edit 2: no, that won't work.

 

[TSny]

cromata, you treated $\rho$ as a constant while it deforms.

 

[cromata]

What is the total distance the COM descends?

Total distance is H+3/10*h but it's not really import. Because ground starts to act on the body (and starts to deform it) when it descends distance H.

No you #%^*+ can't. Sadly, this is a widespread blunder in set questions. To find the average force you need the time for which it applies (and the momentum change), Favg=∫F.dt/∫dt=Δp/Δ

I agree with this part.

The question expects you to find it by ΔE/Δx=∫F.dx/∫dx.

So I should have done it like this:?
ΔE=Ek-0=∫Fdx (where Ek is kinetic energy of the body when it hits the floor (just before deformation))
Ek=pA∫dx, Ek=mgH
A*h*ρ**gH=σ*A*(h/2-h/5)
H=3/10 σ/ρg

It will come out the same if the force is constant, but then you would not need to describe it as an average.

Same as what? The first result I posted?

 

[cromata]

cromata, you treated $\rho$ as a constant while it deforms.

Can we assume that cuboid is deformed uniformly? And by that I mean that mass remains uniformally distributed over the body and at the end of deformation V(volume)=2/5V₀(volume before deformation)?

 

[haruspex]

Total distance is H+3/10*h

As I read it, it falls H-h/2 before hitting the ground, then a further 3h/10. When will the peak pressure occur?

 

[haruspex]

Can we assume that cuboid is deformed uniformly?

It may be that the question intends that, but I see no basis for assuming it.

 

[haruspex]

Same as what? The first result I posted?

No, I meant the two methods of calculating average force, the correct Δp/Δt and the incorrect ΔE/Δs, come out the same if the force is constant. But clearly here it is not.

 

[cromata]

It may be that the question intends that, but I see no basis for assuming it.

I think that the problem is stated very poorly(it lacks a lot of information). I don't see a way of finding how ρ changes during the deformation without that kind of info.

 

[haruspex]

I think that the problem is stated very poorly(it lacks a lot of information). I don't see a way of finding how ρ changes during the deformation without that kind of info.

As I mentioned, you could treat it as a uniform elastic block. You would have to consider the deformation of the layer originally at height x to be a function of x and t. If you wish to cheat and find some prior art online, look for springs with mass.

 

[cromata]

As I read it, it falls H-h/2 before hitting the ground, then a further 3h/10.

C.M. starts at H+h/2 and it's on h/2 before hitting the ground, so it travels distance H.

When will the peak pressure occur?

We aren't assuming that force and therefore pressure is constant?

 

[haruspex]

C.M. starts at H+h/2

Sorry, my mistake.

We aren't assuming that force and therefore pressure is constant?

Two parts to that: is it constant throughout the block at a given instant; is it constant for a given layer within the block at all times after touching ground?
I see no good basis for assuming either.

 

[cromata]

Two parts to that: is it constant throughout the block at a given instant; is it constant for a given layer within the block at all times after touching ground?

I think that in reality bottom part of the cuboid should feel the force before upper part. And I'm not sure about second part of the question, i suppose it changes.

 

[haruspex]

bottom part of the cuboid should feel the force before upper part

The lower parts would be bearing providing the upward force to decelerate all above.

about second part of the question, i suppose it changes.

I would expect it to increase as the compression does, just as in a spring.

 

[cromata]

I would expect it to increase as the compression does, just as in a spring.

I agree, but I really think that there isn't enough info to solve this problem (because I don't know which things I can assume to simplify the problem, like uniform deformation and other things we discussed)

 

[TSny]

What is the level of the course?

 

[cromata]

What is the level of the course?

I'm actually helping my friend with solving this problem. He told me now that they are doing harmonic oscillators (it would have been much better if he had told me before what they were doing), so I suppose I should model this system as harmonic oscillator.

 

[haruspex]

I'm actually helping my friend with solving this problem. He told me now that they are doing harmonic oscillators (it would have been much better if he had told me before what they were doing), so I suppose I should model this system as harmonic oscillator.

I.e. a uniform elastic block, as I kept suggesting.
See if https://en.m.wikipedia.org/wiki/Effective_mass_(spring–mass_system) helps.