The crate on its small end slid no easier

# Self-authored for the EML case corpus (no external origin). Two movers must
# slide a 60 kg crate across a warehouse floor. Lying flat it touches the floor
# with 5000 square centimetres; one mover stands it on its small end, 1000
# square centimetres, and reasons that with a fifth of the surface rubbing, it
# will slide with a fifth of the push. What the push really is is computed
# below.
#
# The reasoning is careful. The small end really is a fifth of the flat face;
# the floor and the crate really are the same; rubbing really does happen
# where the surfaces touch; and the intent is exactly 'make the crate easier to
# slide'.
#
# Sliding friction is set by how hard the surfaces are pressed together - the
# crate's weight - times a factor for the two materials, and not by the area.
# On the small end the same weight presses on a fifth of the area, five times as
# hard on every square centimetre, and the two changes cancel: the push is 30 kg
# either way, not 6.

60 => crate_kg
50 => friction_factor_percent
5000 => flat_face_cm2
1000 => small_end_cm2

int(friction_factor_percent * crate_kg / 100) => push_needed_kg
int(push_needed_kg * small_end_cm2 / flat_face_cm2) => push_expected_on_the_small_end_kg
push_needed_kg - push_expected_on_the_small_end_kg => push_the_area_rule_misses_kg
int(crate_kg * 1000 / flat_face_cm2) => pressure_flat_grams_per_cm2
int(crate_kg * 1000 / small_end_cm2) => pressure_on_end_grams_per_cm2
int(flat_face_cm2 / small_end_cm2) => area_ratio

"crate                           : " + str(crate_kg) + " kg, friction factor " + str(friction_factor_percent) + " percent" ^0
"flat face                       : " + str(flat_face_cm2) + " cm2, pressing " + str(pressure_flat_grams_per_cm2) + " g on each cm2" ^0
"small end                       : " + str(small_end_cm2) + " cm2, a " + str(area_ratio) + "th of it, pressing " + str(pressure_on_end_grams_per_cm2) + " g on each cm2" ^0
"" ^0
"push needed, lying flat         : " + str(push_needed_kg) + " kg" ^0
"push needed, on its end         : " + str(push_needed_kg) + " kg" ^0
"push expected on its end        : " + str(push_expected_on_the_small_end_kg) + " kg" ^0
"push the area rule misses       : " + str(push_the_area_rule_misses_kg) + " kg" ^0
"" ^0

# ---- what the movers verified ----

"the less-surface reasoning" ^0
"  small end : a " + str(area_ratio) + "th of the flat face" ^0
"  floor and crate : the same" ^0
"  where rubbing happens : where the surfaces touch" ^0
"  intent : make the crate easier to slide" ^0
"  facts wrong : 0" ^0
"  verdict : ON ITS END IT SLIDES WITH A FIFTH OF THE PUSH" ^0
"" ^0
"  measuring both faces is the part done right here, and it is" ^0
"  why " + str(small_end_cm2) + " cm2 is exactly the area on the floor when it stands" ^0
"" ^0

# ---- what friction depends on ----

"pressed together, not spread out" ^0
"  what sets sliding friction : how hard the surfaces are pressed" ^0
"    together, times a factor for the two materials" ^0
"  how hard they are pressed : the crate's weight, " + str(crate_kg) + " kg, on any face" ^0
"  what the smaller face changes : the same weight on less area," ^0
"    " + str(pressure_on_end_grams_per_cm2) + " g on each cm2 instead of " + str(pressure_flat_grams_per_cm2) ^0
"  why it cancels : a fifth of the contact, each part pressed five" ^0
"    times as hard" ^0
"  the push : " + str(push_needed_kg) + " kg, flat or on end" ^0
"" ^0

# ---- what the movers got ----

"the push" ^0
"  believed : " + str(push_expected_on_the_small_end_kg) + " kg on its end" ^0
"  actual : " + str(push_needed_kg) + " kg, the same as flat, and a crate now tall enough" ^0
"    to tip" ^0
"  is the area wrong : no; the end is a " + str(area_ratio) + "th of the face" ^0
"  does friction follow the area : no; it follows the weight" ^0
"" ^0

# ---- null control ----

# The same crate pushed with friction worked out from its weight instead of from
# the area in contact.
6 => nc_push_read_from_the_area_kg
30 => nc_push_read_from_the_weight_kg
24 => nc_kg_the_weight_rule_restores

"null control - work friction out from the weight" ^0
"  push, read from the area : " + str(nc_push_read_from_the_area_kg) + " kg" ^0
"  push, read from the weight : " + str(nc_push_read_from_the_weight_kg) + " kg" ^0
"  kg the weight rule restores : " + str(nc_kg_the_weight_rule_restores) ^0
"  no crate and no floor changed; the friction was computed from" ^0
"  what presses the surfaces together" ^0
"" ^0

# ---- the rule ----

"what standing the crate on a smaller face guarantees" ^0
"  less of it touches the floor : exactly, a " + str(area_ratio) + "th" ^0
"  it slides with less push : not addressed; friction follows the" ^0
"    weight pressing the surfaces together, " + str(crate_kg) + " kg on any face, and" ^0
"    the push stays " + str(push_needed_kg) + " kg" ^0
"" ^0

"friction is a bargain between weight and contact, and the weight always wins" ^0
"it back; shrink the contact and every part of it bears down harder" ^0
"" ^0

"The small end is a " + str(area_ratio) + "th of the flat face - measured. But sliding friction follows" ^0
"the weight pressing the surfaces together, not the area, so on its end the" ^0
"crate still needs " + str(push_needed_kg) + " kg of push, not " + str(push_expected_on_the_small_end_kg) + ", until friction is worked out from the" ^0
"weight rather than from the surface in contact." ^0

crate_kg = 60
friction_factor_percent = 50
flat_face_cm2 = 5000
small_end_cm2 = 1000
push_needed_kg = int(friction_factor_percent * crate_kg / 100)
push_expected_on_the_small_end_kg = int(push_needed_kg * small_end_cm2 / flat_face_cm2)
push_the_area_rule_misses_kg = push_needed_kg - push_expected_on_the_small_end_kg
pressure_flat_grams_per_cm2 = int(crate_kg * 1000 / flat_face_cm2)
pressure_on_end_grams_per_cm2 = int(crate_kg * 1000 / small_end_cm2)
area_ratio = int(flat_face_cm2 / small_end_cm2)
print("crate                           : " + str(crate_kg) + " kg, friction factor " + str(friction_factor_percent) + " percent")
print("flat face                       : " + str(flat_face_cm2) + " cm2, pressing " + str(pressure_flat_grams_per_cm2) + " g on each cm2")
print("small end                       : " + str(small_end_cm2) + " cm2, a " + str(area_ratio) + "th of it, pressing " + str(pressure_on_end_grams_per_cm2) + " g on each cm2")
print("")
print("push needed, lying flat         : " + str(push_needed_kg) + " kg")
print("push needed, on its end         : " + str(push_needed_kg) + " kg")
print("push expected on its end        : " + str(push_expected_on_the_small_end_kg) + " kg")
print("push the area rule misses       : " + str(push_the_area_rule_misses_kg) + " kg")
print("")
print("the less-surface reasoning")
print("  small end : a " + str(area_ratio) + "th of the flat face")
print("  floor and crate : the same")
print("  where rubbing happens : where the surfaces touch")
print("  intent : make the crate easier to slide")
print("  facts wrong : 0")
print("  verdict : ON ITS END IT SLIDES WITH A FIFTH OF THE PUSH")
print("")
print("  measuring both faces is the part done right here, and it is")
print("  why " + str(small_end_cm2) + " cm2 is exactly the area on the floor when it stands")
print("")
print("pressed together, not spread out")
print("  what sets sliding friction : how hard the surfaces are pressed")
print("    together, times a factor for the two materials")
print("  how hard they are pressed : the crate's weight, " + str(crate_kg) + " kg, on any face")
print("  what the smaller face changes : the same weight on less area,")
print("    " + str(pressure_on_end_grams_per_cm2) + " g on each cm2 instead of " + str(pressure_flat_grams_per_cm2))
print("  why it cancels : a fifth of the contact, each part pressed five")
print("    times as hard")
print("  the push : " + str(push_needed_kg) + " kg, flat or on end")
print("")
print("the push")
print("  believed : " + str(push_expected_on_the_small_end_kg) + " kg on its end")
print("  actual : " + str(push_needed_kg) + " kg, the same as flat, and a crate now tall enough")
print("    to tip")
print("  is the area wrong : no; the end is a " + str(area_ratio) + "th of the face")
print("  does friction follow the area : no; it follows the weight")
print("")
nc_push_read_from_the_area_kg = 6
nc_push_read_from_the_weight_kg = 30
nc_kg_the_weight_rule_restores = 24
print("null control - work friction out from the weight")
print("  push, read from the area : " + str(nc_push_read_from_the_area_kg) + " kg")
print("  push, read from the weight : " + str(nc_push_read_from_the_weight_kg) + " kg")
print("  kg the weight rule restores : " + str(nc_kg_the_weight_rule_restores))
print("  no crate and no floor changed; the friction was computed from")
print("  what presses the surfaces together")
print("")
print("what standing the crate on a smaller face guarantees")
print("  less of it touches the floor : exactly, a " + str(area_ratio) + "th")
print("  it slides with less push : not addressed; friction follows the")
print("    weight pressing the surfaces together, " + str(crate_kg) + " kg on any face, and")
print("    the push stays " + str(push_needed_kg) + " kg")
print("")
print("friction is a bargain between weight and contact, and the weight always wins")
print("it back; shrink the contact and every part of it bears down harder")
print("")
print("The small end is a " + str(area_ratio) + "th of the flat face - measured. But sliding friction follows")
print("the weight pressing the surfaces together, not the area, so on its end the")
print("crate still needs " + str(push_needed_kg) + " kg of push, not " + str(push_expected_on_the_small_end_kg) + ", until friction is worked out from the")
print("weight rather than from the surface in contact.")

crate                           : 60 kg, friction factor 50 percent
flat face                       : 5000 cm2, pressing 12 g on each cm2
small end                       : 1000 cm2, a 5th of it, pressing 60 g on each cm2

push needed, lying flat         : 30 kg
push needed, on its end         : 30 kg
push expected on its end        : 6 kg
push the area rule misses       : 24 kg

the less-surface reasoning
  small end : a 5th of the flat face
  floor and crate : the same
  where rubbing happens : where the surfaces touch
  intent : make the crate easier to slide
  facts wrong : 0
  verdict : ON ITS END IT SLIDES WITH A FIFTH OF THE PUSH

  measuring both faces is the part done right here, and it is
  why 1000 cm2 is exactly the area on the floor when it stands

pressed together, not spread out
  what sets sliding friction : how hard the surfaces are pressed
    together, times a factor for the two materials
  how hard they are pressed : the crate's weight, 60 kg, on any face
  what the smaller face changes : the same weight on less area,
    60 g on each cm2 instead of 12
  why it cancels : a fifth of the contact, each part pressed five
    times as hard
  the push : 30 kg, flat or on end

the push
  believed : 6 kg on its end
  actual : 30 kg, the same as flat, and a crate now tall enough
    to tip
  is the area wrong : no; the end is a 5th of the face
  does friction follow the area : no; it follows the weight

null control - work friction out from the weight
  push, read from the area : 6 kg
  push, read from the weight : 30 kg
  kg the weight rule restores : 24
  no crate and no floor changed; the friction was computed from
  what presses the surfaces together

what standing the crate on a smaller face guarantees
  less of it touches the floor : exactly, a 5th
  it slides with less push : not addressed; friction follows the
    weight pressing the surfaces together, 60 kg on any face, and
    the push stays 30 kg

friction is a bargain between weight and contact, and the weight always wins
it back; shrink the contact and every part of it bears down harder

The small end is a 5th of the flat face - measured. But sliding friction follows
the weight pressing the surfaces together, not the area, so on its end the
crate still needs 30 kg of push, not 6, until friction is worked out from the
weight rather than from the surface in contact.

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