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

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 out? One book book center of mass ? overhang Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.3 lec 8F.4 1
Book Stacking Book Stacking book center balances if of mass One book One book book center center of mass of mass over table 1 2 1-book overhang = 1 2 Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.5 lec 8F.6 n books n books 1 1 2 2 n n center of mass Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.7 lec 8F.8 2
n books n books 1 1 2 2 center of mass of the whole stack n n balances if center of mass over table Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.9 lec 8F.10 n+1 books Δ- overhang 1 Δ-overhang ::= 2 horizontal distance from center of mass of all n+1 books at table edge n-book to (n+1)-book center of mass of n top n books at centers of mass edge of book n+1 n+1 � Δoverhang Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.11 lec 8F.12 3
Δ- overhang Δ- overhang table edge Δ n n 1 1 1/2 1/2 1 1 2 Δ = n+1 = 2(n+1) Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.15 lec 8F.16 � � n+1 books n+1 books 1 1 2 2 center of mass center of mass of all n+1 books of all n+1 books at table edge center of mass of center of mass of n n top n books at top n books edge of book n+1 n+1 n+1 � � 1 Δoverhang 2(n +1) Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.17 lec 8F.18 4
Book stacking summary Harmonic Sums H n ::=1+ 2 + 1 1 3 + ⋯ + 1 B n ::= overhang of n books n B 1 = 1/2 1 n th Harmonic number B n+1 = B n + 2(n +1) B n = H n /2 ⎛ ⎞ 2 1+ 1 1 2 + 1 3 + ⋯ + 1 B n = ⎜  ⎝ ⎠ n Albert R Meyer, April 6, 2012 Albert R Meyer, April 6, 2012 lec 8F.19 lec 8F.20 5
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