Consider a collection of identical boards, each with mass m, length L, and unifo

Question

Consider a collection of identical boards, each with mass m, length L, and uniform mass density. Clearly lacking other forms of entertainment, we decide place a board on a ledge, then a second board on top of the first board, then a third board on top of the second board, and so on. Let x_n represent the greatest possible horizontal distance between far edge of the uppermost board and the ledge, such that the assembly of boards may remain stationary. Prove the recursive relation for x_n, x_n= X_n-1 + L/2n Prove the explicit licit form of x_n Imagine Romeo is attempting to sneak into Juliet's nd story room. Lord Capulet has coated the ding wall with a material so slippery that is has no

Explanation / Answer

a) Length of the identical Blocks is L and Mass of of the Blocks is M

We have to calulate the center of mass of the structure . The distance between the centre of mass of the lower blockand right hand edge of the bottom blockwe'll define as L/2

From this we can write the equation

X2(2M)=(X1)M

Here X1=L/2

X2(2M)=(L/2)M

X2=L/4

at this point over hang of the block is X1+X2=L/2+L/4=3L/4

For third block

X3(3M)=(L/2)M

X3=L/6

at this point over hang of the block is

X1+X2+X3=L/2+L/4+L/6=11L/12

This value is close to L.

For fourth Block

X4(4M)=(L/2)M

X4=L/8

At this point over hang of the Block X1+X2+X3+X4=L/2+L/4+L/6+L/8=25L/24. This is greater than one

Xn(nM)=(L/2)M

The displacement for each block is L/2n

Hence we can write The recursive relation

Xn=Xn-1+L/2n

b) Over hang =L/2 (Summation of (k =1 to n))   

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