Book Stacking Harmonic Sums table Albert R Meyer, - - PowerPoint PPT Presentation

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Book Stacking Harmonic Sums table Albert R Meyer, - - PowerPoint PPT Presentation

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Book Stacking Mathematics for Computer Science MIT 6.042J/18.062J Book Stacking Harmonic Sums table Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.1 lec 8F.2 Book Stacking Book Stacking How far

slide-1
SLIDE 1

1

lec 8F.1 Albert R Meyer, April 6, 2012

Mathematics for Computer Science MIT 6.042J/18.062J

Book Stacking Harmonic Sums

lec Albert R Meyer, April 6, 2012

table

8F.2

Book Stacking

lec 8F.3 Albert R Meyer, April 6, 2012

?

  • verhang

Book Stacking

How far out?

lec 8F.4 Albert R Meyer, April 6, 2012

Book Stacking

One book

book center

  • f mass
slide-2
SLIDE 2

2

lec 8F.5 Albert R Meyer, April 6, 2012

Book Stacking

One book

book center

  • f mass

lec 8F.6 Albert R Meyer, April 6, 2012

Book Stacking

1 2

One book

book center

  • f mass

balances if center of mass

  • ver table

1-book overhang

= 1 2

lec 8F.7 Albert R Meyer, April 6, 2012

n books

1 2 n

lec 8F.8 Albert R Meyer, April 6, 2012

n books

center

  • f mass

1 2 n

slide-3
SLIDE 3

3

lec 8F.9 Albert R Meyer, April 6, 2012

n books

1 2 n

lec 8F.10 Albert R Meyer, April 6, 2012

n books

center of mass

  • f the whole stack

1 2 n

balances if center of mass

  • ver table

lec 8F.11 Albert R Meyer, April 6, 2012

n+1 books

center of mass

  • f all n+1 books

at table edge center of mass of top n books at edge of book n+1

Δoverhang

1 2 n n+1

  • lec 8F.12

Albert R Meyer, April 6, 2012

Δ-overhang

Δ-overhang ::= horizontal distance from n-book to (n+1)-book centers of mass

slide-4
SLIDE 4

4

lec 8F.15 Albert R Meyer, April 6, 2012

Δ-overhang

1

n

1/2

lec 8F.16 Albert R Meyer, April 6, 2012

Δ = 1 2 n+1 = 1 2(n+1) Δ-overhang

1

n

1/2

Δ

table edge

lec 8F.17 Albert R Meyer, April 6, 2012

n+1 books

center of mass

  • f all n+1 books

at table edge center of mass of top n books at edge of book n+1

Δoverhang

1 2 n n+1

  • lec 8F.18

Albert R Meyer, April 6, 2012

n+1 books

center of mass

  • f all n+1 books

center of mass of top n books

  • 1

2 n n+1

1 2(n +1)

slide-5
SLIDE 5

5

lec 8F.19 Albert R Meyer, April 6, 2012

Bn ::= overhang of n books B1 = 1/2 Bn+1 = Bn + Bn =

Book stacking summary

1 2(n +1)

1 2 1+ 1 2 + 1 3 +⋯+ 1 n ⎛ ⎝ ⎜ ⎞ ⎠ 

lec 8F.20 Albert R Meyer, April 6, 2012

1 Hn ::=1+ 2 + 1 3 +⋯+ 1 n

Harmonic Sums

nth Harmonic number Bn = Hn/2

slide-6
SLIDE 6

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