Will a sliding box fall over when it stops?

In summary: I think one can model this as moving and then suddenly stopping when sticking occurs (if not there will still be sliding.) What happens to the kinetic energy? I believe it becomes rotational kinetic energy about the leading corner of the block that acts as a pivot.
  • #36
Referring to #33, I followed a different route. Starting from
##\ddot\phi(\frac 13-(\mu \sin(\phi)- \cos(\phi))\cos(\phi))=(\mu \sin(\phi)- \cos(\phi))(1-{\dot\phi}^2\sin(\phi)))## and with ##\dot \phi =0##,
I got ##\ddot\phi=\frac{\mu \sin(\phi)- \cos(\phi)}{\frac 13-(\mu \sin(\phi)- \cos(\phi))\cos(\phi)}## which I plotted for two "typical" values of ##\mu## (see below). From the plots it is clear that the condition must be ##\mu \sin(\phi) > \cos(\phi)##. Specifically, additional plots (not shown) indicate that the denominator stays positive for ##\mu## less than about 4/3 which is above what one would expect for blocks sliding on surfaces. So I think that's it.

PhiDDot.png
 

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  • #37
kuruman said:
I plotted for two "typical" values of μ
Yes, it looks sort of reasonable for those, except why does it peak for μ=1 and φ<π/2?
And it gets really weird for larger μ. Try μ=1.6.
 
  • #38
I was incommunicado for several days because of hardware problems. I thought about this some more and here is what I have.
haruspex said:
Ahem.. correcting my algebra..

For the condition for starting/continuing to tip we can take ##\dot\phi=0## and ##\ddot\phi>0##.
Will also assume φ≤π/2, since the development in post 18 might be invalid else.
From the equation in post #18:
##(\frac 13-(\mu\sin(\phi)-\cos(\phi))\cos(\phi))(\mu\sin(\phi)-\cos(\phi))>0##
Writing ##\mu=\tan(\alpha)## we can break this into two cases.
1. ##\mu\sin(\phi)>\cos(\phi)## (i.e. φ+α>π/2)
##\frac 13>(\tan(\alpha)\sin(\phi)-\cos(\phi))\cos(\phi)##
##\frac 23>\tan(\alpha)\sin(2\phi)-1-\cos(2\phi)##
##\frac 53>\tan(\alpha)\sin(2\phi)-\cos(2\phi)##
##5\cos(\alpha)>3(\sin(\alpha)\sin(2\phi)-\cos(\alpha)\cos(2\phi))##
##5\cos(\alpha)+3\cos(\alpha+2\phi)>0##
Looks weird, but..
2. φ+α<π/2
As above but with all inequalities reversed. I'd say the conclusion for this case is it can't happen.

Still looks mighty strange...
Case 1 is always the case if there is to be tipping. Here is the reason. Suppose the block is just sliding with friction but not tipping, like a book sliding across a table on its cover. The torque equation about the CM is ##\mu N h-Ns=0##. Here ##s## is the distance from the point vertically below the CM to the point where the ##N## acts on the block. As ##\mu## increases, ##s## must increase to compensate. However ##s## cannot increase past ##w## the half width of the base. At the threshold, ##\mu N h-Nw=0##. Tipping occurs if ##w## is less that the threshold value, ##w<\mu h##. Let ##\phi_0## be the angle between ##A## and the horizontal. Then ##\cos(\phi_0)=w/A## and ##\sin(\phi_0)=h/A##. The tipping condition becomes ##\cos(\phi_0)<\mu \sin(\phi_0)## or ##\mu \sin(\phi_0)-\cos(\phi_0)>0.## Thus if the condition is satisfied and the block starts tipping, ##\phi## increases and ##\mu \sin(\phi)-\cos(\phi)>0## remains positive since the sine increases and the cosine decreases but remains positive for ##\phi_0<\phi<\pi/2##. Now since the numerator is always positive, one has to ensure that the denominator be also positive.
##\frac 13>(\tan(\alpha)\sin(\phi)-\cos(\phi))\cos(\phi)##
##\frac 13>\frac{(\sin(\alpha)\sin(\phi)-\cos(\phi)\cos(\alpha))}{cos(\alpha)}\cos(\phi)##
##\cos(\phi+\alpha)\cos(\phi)>-\frac {\cos(\alpha)} {3}.##
 
Last edited:
<h2>1. What factors determine whether a sliding box will fall over when it stops?</h2><p>The main factors that determine whether a sliding box will fall over when it stops are its height, mass, and center of gravity. A taller and heavier box with a higher center of gravity is more likely to fall over compared to a shorter and lighter box with a lower center of gravity.</p><h2>2. Does the surface on which the box is sliding have an impact on whether it will fall over when it stops?</h2><p>Yes, the surface on which the box is sliding can have an impact on whether it will fall over when it stops. A rough or uneven surface can cause the box to lose its balance and fall over, while a smooth and flat surface can help the box maintain its stability.</p><h2>3. Can the speed at which the box is sliding affect whether it will fall over when it stops?</h2><p>Yes, the speed at which the box is sliding can affect whether it will fall over when it stops. A higher speed can cause the box to have more momentum, making it more likely to fall over when it stops suddenly.</p><h2>4. Are there any external forces that can cause a sliding box to fall over when it stops?</h2><p>Yes, there are external forces that can cause a sliding box to fall over when it stops. For example, if there is a strong wind blowing against the box, it can cause it to lose its balance and fall over when it stops.</p><h2>5. Is there a way to prevent a sliding box from falling over when it stops?</h2><p>Yes, there are ways to prevent a sliding box from falling over when it stops. One way is to ensure that the box is placed on a stable and level surface. Another way is to lower the box's center of gravity by adding weight to the bottom or by placing heavier items inside the box.</p>

1. What factors determine whether a sliding box will fall over when it stops?

The main factors that determine whether a sliding box will fall over when it stops are its height, mass, and center of gravity. A taller and heavier box with a higher center of gravity is more likely to fall over compared to a shorter and lighter box with a lower center of gravity.

2. Does the surface on which the box is sliding have an impact on whether it will fall over when it stops?

Yes, the surface on which the box is sliding can have an impact on whether it will fall over when it stops. A rough or uneven surface can cause the box to lose its balance and fall over, while a smooth and flat surface can help the box maintain its stability.

3. Can the speed at which the box is sliding affect whether it will fall over when it stops?

Yes, the speed at which the box is sliding can affect whether it will fall over when it stops. A higher speed can cause the box to have more momentum, making it more likely to fall over when it stops suddenly.

4. Are there any external forces that can cause a sliding box to fall over when it stops?

Yes, there are external forces that can cause a sliding box to fall over when it stops. For example, if there is a strong wind blowing against the box, it can cause it to lose its balance and fall over when it stops.

5. Is there a way to prevent a sliding box from falling over when it stops?

Yes, there are ways to prevent a sliding box from falling over when it stops. One way is to ensure that the box is placed on a stable and level surface. Another way is to lower the box's center of gravity by adding weight to the bottom or by placing heavier items inside the box.

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