File:ButterSideUp.gif

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[edit]

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[edit]

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[edit]

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[edit]

Footnotes[edit]

Summary[edit]

Licensing[edit]

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