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Consider this example:

images

  • Treat G as 10.
  • The blue blocks provide lift of 10N, or -1KG.
  • The red block has a weight of 20N, or 2KG.
  • The black bar has no weight.
  • All have their center of gravity in their center.
  • All three have infinite strength.
  • We ignore air resistance.
  • We assume all the pieces are joined.

In the first example, all three are centered, and in line with each other.

In the second, the red block is centered between the blue blocks.

My question is which would be the most stable, i.e. less likely to roll into its more natural state with the lift above the weight, where the center of gravity in the red block to shift to one side. Which would allow the most shift before flipping?

The second one? By how much?

My initial thoughts were about center of gravity, but i'm not sure this even applies anymore, is it more about torque from the systems weight and lift once it has passed the last stable point?

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4 Answers 4

up vote 2 down vote accepted

You could think of it in terms of torque. It's going to require a greater shear force in the second example, to overcome the torque provided by the lifts being farther apart. Since torque is the cross-product of F, the force and r, displacement vector. r is greater in the second example for the lifts, so more external torque will be required to cause the system to rotate (and become unstable)

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An easy way to think about it is to take your diagram, and then move the gravity vector left or right by some angle.

Which configuration can tolerate a larger change in the direction of gravity without one of the blocks or the composite tipping over?

enter image description here

Actually, I'm assuming there is no glue here, so the actual CG of the group on the left should be somewhat to the left of center, because when it tips far enough to the left, the rightmost blue block ceases to have any weight on it, so it is no longer part of the composite. That makes the necessary angle even smaller.

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Jkej was correct in stating that both systems are unstable as stated. Any shift, no matter how small would cause either system to flip. In order for a system to be stable there has to be a restorative force which would tend to bring the system back to equilibrium. In this case, there is no restorative force. Imagine that both systems are tilted by 1 degree. In that case there would be a net torque on each system of: $$10N * cos(1 deg) h $$ Where h is the height from the bottom of the blue block to the top of the red block. And the direction of the torque would be away from the initial position.

This may seem counter intuitive as one could imagine carrying a 2 kg box with two hands each lifting 10 N, and that system seems perfectly stable; the box doesn't suddenly flip out of one's hands. However, the reason the box doesn't flip out of one's hands is that people provide feedback by varying the amount of lift force as the box tips. If the box tips to the left, then the left hand provides more force, creating a net torque that counter balances the destabilizing torque.

In the case of a human, the feedback is non-trivial to model. However, I believe the system described below will provide insight into how the spread of supports affects stability.

Suppose you replace your to blue blocks with springs with one end anchored to the bar, and one end anchored to the ground. In it's balanced equilibrium state each spring would supply 10 N of lift just as the blue blocks had. However, now when the red block and bar are rotated by 1 deg the system behaves differently. Assuming the springs have the same stiffness, the magnitude of the displacement would be equal at each end (in order to have a net upward force of 20 N). The destabilizing torque due to gravity would be $$ 10 N * cos(1deg)h$$ where h is the height of the red box (the blue boxes are no longer moving so they no longer contribute to the instability) Now there is also a restorative torque due to the springs being compressed different amounts. $$ k*cos(1deg)(\frac{w}2)^2$$ Where w is the width between the springs, and k is the spring stiffness.

As long as the restorative force grows faster than the destabilizing force the system will try to move back to equilibrium. (To determine if it's actually stable requires looking at any dampening conditions, but the way it's written right now it would just oscillate like a pendulum) Since the restorative force has a positive relationship with width, a wider separation would not require as stiff springs to make the system not flip. However, if we try to examine this system using the OP criteria fro stability, we find that the system either tends to restore itself upright, or it flips, regardless of how far the block is perturbed. This scenario is fairly typical in analyzing simple systems.

Going back to the example of holding a box, if you place your hands very close together it requires much more force (like a greater spring stiffness) to keep a box from toppling than if you place your hands far apart.

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Both cases are unstable. Any small shift in the position of red will cause the black bar to flip. The second case will flip a little slower because the system has a larger moment of inertia.

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