5.5 – Stability

Stability is a measure of a body’s ability to return to its original position after being tilted slightly.

Objects on Solid Surfaces

An object will topple over if it is rotated too far.

Question 1
Mike is pushing horizontally onto the surface of the block.  Given that the block has uniform density and weighs 500 N, calculate the minimum initial force that Mike must exert in the position shown in order to tilt the blocks.

Firstly, identify the forces acting on the box.

Taking moments about the pivot (the bottom right corner of the box):

Clockwise moment = force x distance

= F x 1.50 m

= 1.5F N m

Anticlockwise moment = force x distance

= 500 x 0.55 m

=275 N m

Applying the principle of moments;

1.5F = 275

F = 183 N
Example 2A
If Mike lets go of the box from the following position, what will happen?

Because he lets go there will only be the weight force acting on the box. The line of action of the weight is to the left of the pivot (i.e. the CoG still lies directly above the base area) thus the weight will produce anti-clockwise moments on the box bringing it back to its original position.
Example 2B
If Mike lets go of the box from the following position, what will happen?

Because he lets go there will again only be the weight force acting on the box. However, this time the line of action of the weight is to the right of the pivot (i.e. the CoG no longer lies directly above the base area) thus the weight will produce clockwise moments on the box making it fall onto its side.
Example 2C
If Mike lets go of the box from the following position, what will happen?

Because he lets go there will again only be the weight force acting on the box. However, this time the line of action of the weight is directly above the pivot. Thus the weight will produce no turning moments on the box and the box will remain in this position.

The box is in a state of unstable equilibrium.
Maximum Angle of Tilt
The position in which the box is in unstable equilibrium is the maximum angle that the box can be tilted before toppling over.

It can be easily shown that this angle is the same as that made between the line of action of the weight and the diagonal to the pivot as shown below.

 

Question
What is the maximum angle of tilt the uniform cuboid can undergo before toppling over?

 
Angle of Tilt and Weight
Note that the weight doesn’t come into finding the maximum angle of tilt.

An empty water bottle and a completely full water bottle will both topple over when rotated by exactly the same angle.

However, of course it will still require less force to push the empty water bottle over to the tipping point as it has less weight and so will produce less turning moments.
Increasing Stability
An object can be made more stable by:

1. lowering its centre of gravity

2. Increasing the area of its base

You can see in both of the above examples that the angle of maximum tilt has increased.

e.g for the lowering of CoG the object will now be able to tilt this far without toppling over onto its side:

 

Question
Which method of setting up this sign would be best? Why?

The final diagram is the best way as the base area is largest making the sign the most stable against toppling over – e.g if blown by wind.

Suspended Objects

In the above examples we have been considering objects resting on a flat surface. There is also the case of objects suspended.

Consider the following heart-shape cut out of card. We will consider this to be uniform (density) and rigid.

It’s centre of gravity has been marked with a cross and a small hole cut out of it to allow it to be suspended.

Making Holes
We assume that the small hole has only removed an insignificant amount of the card and so the position of the centre of gravity hasn’t been altered.

In practice, the hole would be just large enough to allow the card to rotate freely with negligible friction.

If we suspend the card from a pin the object’s own weight will cause it to swing.

The turning moment about the pin can be calculated if we know the weight and measure the shortest distance (perpendicular) from the pivot (pin) to the line of action of the weight.

moment = force x distance

moment = weight x d

Of course these moments will be clockwise and so the heart will begin to rotate clockwise about the pin.

Where will the shape stop?

The above card may oscillate back and forth for a while but will always stop swinging in the following position:

i.e with the centre of gravity directly below the pivot.

As weight acts vertically (straight down), the line of action of the weight passes through the pivot. Thus there is no resultant moment produced at this position.

This object is now in a stable equilibrium. If tilted slightly from this position it will always return to this position.

 

There is only one position that the object can be released from that will not return end up in the above position. That’s when the centre of gravity is directly above the pivot.

Because the centre of gravity is directly above the pivot, the line of action of the weight will pass directly through the pivot:

 

Thus, again there will be no resultant moment in this position.

This object is now in an unstable equilibrium. If tilted slightly from this position it will always move away from this position.

 

Stacked Boxes Question
Two identical blocks of length L are stacked on top of each other.

A force starts to push the top block to the right.

How far can the top block be pushed until it topples off?

The top box will topple off if its centre of gravity no longer lies above the bottom box.

Thus the furthest point the box can be pushed is this:

So the largest distance is L/2.
Balance Toy Question
A toy (blue) is balanced on a a triangular base (grey).

The toy has a body built from hollow plastic (light), but is given heavy solid metal (heavy) hands (blue circles).

(a) Estimate the position of the centre of gravity of the toy

Its position of its CoG would be along the vertical line of symmetry.


If the only mass was just the two ‘hands’, the combined CoG would be the midpoint between these two. As the body has a little mass it will be raised slightly above this point as indicated by the red cross in the diagram above.

 

(b) Explain why the toy balances at this position

As the CoG is directly below the pivot there is no turning moment produced around the pivot.

The line of action of the weight force passes directly through the pivot.

This position is a stable equilibrium.

 

The toy is now tipped (rotated clockwise) as shown below. 

(c) What will happen when it is released?

When the toy is rotated its centre of gravity shifts position (although it is staying at the same position on the toy itself (along the centre of symmetry and slightly up from the midpoint between the ‘hands’).

The new position of the CoG  is now to the left of the pivot.

This toy’s own weight now provides a clockwise turning moment about the pivot. This is because the line of action of the weight force lies to the left of the pivot.

The clockwise moments will make the toy swing back to its original position.

 

The ‘hands’ have been removed from the toy

 

(d) Will the toy still behave the same way as before?

The new position of the CoG will be much higher now that the heavy ‘hands’ have been removed. A likely position of the new CoG is indicated by the red cross below.


As the weight is directly above the pivot, there is no turning moment produced and so the toy will remain in this position.

It is an unstable equilibrium.

 

Consider the object again being rotated slightly in the same direction as before.

Its CoG has now moved to the right of the pivot.

This will result in the weight providing a clockwise moment around the pivot.

The result is that the toy will continue to rotate further to the right and will no longer be able to balance.
<< Back Turning Effects of a Force

 

Links
Stability – OPENSTAXCOLLEGE online text
Video – Concentration & Balance
Great PDF notes covering the whole chapter
Carnival Scams – Rope Bridge

 

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