Angle of Instability The above shows how to calculate the angle of instability f

Question

Angle of Instability

The above shows how to calculate the angle of instability for 4 blocks stack, and the angle of instability was 14 degrees for 4 blocks stack. So can someone show me how to calculate the angle of instability of the system for 3 blocks stack. The 3-block system should have a lesser angle of instability since I already did an experiment for this and 3 blocks stack has a lower center of mass.

My experiment result for 3 and 4 blocks set:

3 block stuck has a lower center of mass, and tipped over at about 15 degrees.

4 block stuch has a higher center of mass, and tipped over at avout 10 degrees.

Here is the 4 block stack. The center of mass will be at the middle of the stack between the 2nd and 3rd block. The base of the object is the width of the bottom block. See how a vertical line drawn from the center of mass is exactly at the edge of the base? This is what they mean by "vertical projection". You can calculate Angle of instability = 90 degrees - Inner Angle We can find the inner angle with trigonometry Inner Angle = arctan((1/2)side / (2*side)) side = the length of a side of the cube See how the side cancels out in the top and the bottom so we get Inner Angle = arctan((1/2) / (2)) = rctan(1/4) = 14 degrees Angle of instability = 90-14 degrees = Now for the 3 block stack, you do the same thing. You draw three blocks on top of each other, the center of the mass will be in the middle of the 2nd block. Calculate the inner angle based on your drawing and the calculate the angle of instability.

Explanation / Answer

for a 3 blocks stack, the centre of mass will be inside the second block.
so ,
angle of instability = 90 - inner angle,

and .
inner angle = arctan((1/2)side/1.5*side)
inner angle = arc tan(0.5/1.5) = 18.4349

so
angle of instability = 90 - 18.4349= 71.565 degrees (for 3 blocks stack)

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