Consider a pile of books, identical in size and shape, that are stacked as that each extends further out from the one underneath. If each book is 10 inches in length and 10 books are stacked in the pile, how far can the stack extend without falling over?
2007-05-17 15:54:32 · 10 answers · asked by sportnwood32 1 in Science & Mathematics ➔ Mathematics
Essentially the center of gravity of the pile must be vertically above the base of the lowest book. If it is over the edge, the pile will topple.
Vertically, the center of gravity of the pile will be between the 5th and 6th book. So that point on top middle of the 5th book can be 5 inches to the side. Then just double that value to see how far the top of the pile can protrude.
Answer 10 inches. Each book is an inch further than the one below.
2007-05-17 16:00:45 · answer #1 · answered by Dr D 7 · 2⤊ 1⤋
Consider stack on the verge of falling. Bottom book won't fall
so assume stack above it is optimal so that it wont fail
internally. Then pivot point is where edge of bottom book meets stack above. The center of the mass on the left of the pivot point provides a torque equal and opposite to the torque
provided by the mass center on the right. This condition prevails for all the substacks on up with the new pivot points
for them. Intuitively speaking i think that the nth book slides
over n/(n+1) times the amount the book under it slid over.
If the books were infinitely thin i think we have a graph of lnx
if you turn your head to look at it . I think you can easily exceed the 20 inches, maybe go to 10pi. This is a great problem but i don't have the energy for it right now.
2007-05-18 00:01:42 · answer #2 · answered by knashha 5 · 0⤊ 0⤋
6 books high. or 5 inches over. you are looking at a physics problem, not an algebra one. If you try and extend the books so that the center of gravity is no longer over the bottom book, than the stack will topple
2007-05-17 22:59:59 · answer #3 · answered by evendims_keeper 1 · 2⤊ 1⤋
Start with the top book. You can move it over 5 in without it tipping over. Then find the center of mass of the top 2 books. where 0 is at the edge of the table. The top book is at 0 and the 2nd book is at 5. Their center of mass is at 5/2. So you can move the top two books over a distance of 5/2. Now find the center of mass of the top 3 books. This is 5/3. So you can move the top 3 books over 5/3. Continue this method for 10 books and you get
5/1 + 5/2 + 5/3 + 5/4 + ... + 5/10 = 7381/504 = 14.64 inches
2007-05-18 01:45:53 · answer #4 · answered by Demiurge42 7 · 0⤊ 1⤋
Calculate the center of mass on two books, then three and so on. See if a pattern emerges.
For example, with two books, the top book can extend by as much as 5 inches without toppling.
2007-05-17 23:00:34 · answer #5 · answered by Anonymous · 0⤊ 1⤋
I don't know the physics behind it but I tested this out using CD cases and was able to extend the top cd case out the entire length of a CD from the bottom CD. So I'd say 10 inches.
2007-05-17 23:08:07 · answer #6 · answered by Edgard L 2 · 1⤊ 1⤋
the stack will only fall over if the total combined density on the ascending side exceeds the density on the descending side, the volume of the is neglagable.
2007-05-17 23:21:07 · answer #7 · answered by ? 2 · 0⤊ 1⤋
I am not any good at these problems. There is a real good math site that may help you though. It is called Algebra Solutions. com. I hope you get the answer you need.
2007-05-17 22:59:57 · answer #8 · answered by phylobri 4 · 0⤊ 1⤋
thats hard to tell. you cant really find that out because there could be a sudden viberation... i guess u would have to try it out, ive never seen a problen like that before.
2007-05-17 22:57:34 · answer #9 · answered by Anonymous · 0⤊ 2⤋
no idea.
2007-05-17 22:58:55 · answer #10 · answered by laleela 1 · 0⤊ 2⤋