This real time strobed animation, of an experiment done elsewhere in our solar system, illustrates the butter side up version of an angular impulse problem.

Practical Problem

At what minimum height h above the edge of a height-H table should you hold a square (side 2R) piece of toast butter side up, so that if its edge clips the table edge when you drop it the toast will land butter side down?

• Does this change if the toast is circular, or if a corner rather than an edge is clipped?

• What if you use margarine instead of butter, or cashew butter with strawberry jam?

• How high would you hold it if you want it to land butter side up after one turn?

• How big must H/R be to allow a double somersault butter side up?

• Do any of these answers depend on the size of g?

• Should this be an event in the kitchen olympics?

Discussion of Data

If the table is 1 meter high and this strobed animation is in real time, how accurately can you determine:

(i) the acceleration due to gravity on the experiment site,

(ii) where in the solar system this data might have been taken,

(iii) the size of the impulse FΔt/m the toast experiences, and

(iv) the dimensionless ratio ξ=I/mR² for the toast?

Discussion of Models

If the toast is height h above the table, 
the speed when it hits will be v = Sqrt[2gh].
But how might we model the table's impact?
Suppose after collision that the toast center 
has a downward speed v' and angular velocity 
of ω=v'/R where R is the edge distance.  Then
collision impulse FΔt=m(v-v') and angular 
impulse RFΔt=Iω are linked.  Solving for 
v' gives v'=v/(1+I/mR2).  Time to the floor 
and the amount of rotation can now both be 
calculated as a function of h, and then h 
adjusted so that the butter side lands down.  
Will that work? 

See Also

Footnotes

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