1J11.20 Leaning Tower of Lire
Abstract
Given a sufficiently large number of identical rectangular blocks (or books, or coins), it is possible to stack
them in such a way that the top block completely clears the base of the stack. A simple geometrical pattern
describes an efficient method for achieving this with just a few blocks.
Picture
Setup
Setup is 5 minutes
Safety Concerns
None.
Equipment
• Set of blocks (5 or more)
1
Procedure
At least five blocks are needed in order to comfortably achieve an overhang such that the top block clears
the table. More blocks can be used in a single stack, making the demonstration both more impressive and more
convincing.
The maximum overhang is most easily achieved when the blocks are balanced from the top down. Begin by
lining up all of the blocks in a vertical stack near (but not protruding from) the edge of a table. Use your fingers
to keep all the lower blocks steady. Slowly push the top block to one side until its centre of mass is almost at the
edge of the block immediately below it. Once this point is reached the block will begin to teeter. At this point it
is wise to move the block slightly back to a more stable position. This will help keep the blocks parallel with the
table.
Once the top block is positioned, proceed to the block immediately below. Again keeping the lower blocks
secure, move the second-highest block in the same direction the top block was moved until the tipping point is
just reached. In this case both blocks should begin to teeter simultaneously. This will occur at about half the
distance that the top block was moved.
Proceed to the next block and continue in this way, halving the distance each time, remembering to keep the
base steady as each new tipping point is approached. The last step is to balance the entire stack at the table
edge, following the same half-distance rule as the rest of the stack.
Theory
Common experience with stackable objects shows that there is a certain maximal distance that items in the
stack can be shifted before they fall off the stack. An old problem in physics and mathematics asks what is
the greatest achievable shift of the top item, with respect to the base, such that the stack remains at static
equilibrium. The result is perhaps unintuitive.
Some assumptions about the blocks are necessary to derive a result. First, assume that the blocks are all of
equal length l, and all have equal, uniform mass density. In order to apply the theory to the real world, assume
that the corners are slightly tapered. As labelled in Figure 1, n blocks will be used, with numbering starting at
the top block and proceeding down the stack, .
l
1
d1
2
d2
3
n
dn
Support Structure
Dn
Figure 1: Diagram of a leaning tower of Lire using n blocks of equal length l.
It is of interest to find an expression for the overhang distance Dn between the far end of the top block and
the edge of the support, shown in Figure 1. To do so, it is instructive to first find the local overhang di of the
ith block, which is the distance between its overhanging end and the corner of the block immediately below it.
2
From Figure 1, it is apparent that for n blocks, the overhang distance is
Dn =
n
X
di .
(1)
i=1
There are two conditions that must be met to ensure that a rigid body remains at rest. First, the forces acting
on the object must sum to zero, by Newton’s second law. Secondly, the object must be rotationally stable. In the
case of this demonstration, the forces acting on a given block are gravity and the normal forces exerted by any
objects in contact with the block. These are the only forces acting on a block, and therefore as long as a block’s
centre of mass is located above some supporting structure, the necessary conditions are met.
Consider the case n = 1. A simple consideration of geometry leads to the conclusion that the greatest
overhang for a single block is achieved when the long side (length l) extends outward from the table at 90◦ , and
the centre of mass of the block is as close to the edge of the table as possible before it starts to tip. The block
has uniform density, so its centre of mass is located at its geometrical centre. The maximum theoretical local
overhang, in ideal conditions, is therefore half of the block’s width (d1 = 21 l).
Extending this model to n > 1, a simple computation of the centre of mass of the stack shows that di is
halved each time i is increased by one. Then Dn is given by
n
(Dn )ideal =
l X1
,
2 i=1 i
(2)
a harmonic series. Note that as n approaches infinity, so too does Dn . That is, there is no theoretical limit on
the maximum overhang of ideal blocks, which may be surprising.
In reality, blocks are not ideal, nor is the structure that supports the stack. In particular, the corners are not
perfectly sharp. This implies that the apparent overhang distance is slightly smaller than the actual overhang
distance, since the corner of the supporting body is tapered. The “effective” corner of the block is actually slightly
inset. Let this discrepancy be denoted ∆x, as in Figure 2. Assume that ∆x, is roughly the same for all blocks.
Consider the ith block in a stack of n blocks. To find the
local overhang di of this block, first define an x-axis parallel
to the length of the block, where x = 0 at the end of the
block that is closer to the base of the stack, as in Figure
i
3. Denote xc,i as the x-coordinate of the collective centre
of mass of the first i blocks. Each block is balanced at the
(i + 1)
tipping point, where the centre of mass is at the “effective
corner“ of the block below, so xc,i is located at the corner of
the block underneath it. The edge of the supporting block
∆x
has the x-coordinate (xc,i + ∆x). It follows that the local
overhang distance is given by
Figure 2: Diagram of the imperfect “corners” of the
di = l − (xc,i + ∆x) .
(3)
blocks used to construct a leaning tower of Lire.
Similarly, xc,(i−1) shall refer to the collective centre of
mass of the first (i − 1) blocks. This point is located at the top corner of the ith block, with an x-coordinate of
xc,(i−1) = l − ∆x.
(4)
Next, the x-coordinate of the center of mass xcm of a system with N parts is defined by
xcm =
N
1 X
m j xj ,
M i=1
(5)
P
where mj is the mass of the j th part, xj is the x-coordinate of that part, and M =
mj is the total mass of
the system. For the block stack of Figure 3, the system shall be defined as the top i blocks, so that in Equation
5, N = i. If these blocks all have equal mass m, then M = im. Divide the system into two parts, one being the
3
y
(i − 1)
i
di
x=0
xc,i
x
xc,(i−1)
l
(xc,i + ∆x)
Figure 3: Diagram of the imperfect “corners” of the blocks used to construct a leaning tower of Lire.
ith block alone, and the other the combination of the first (i − 1) blocks. Using Equation 5, the x-coordinate of
the centre of mass of the first i blocks is
xc,i =
1
(m1 x1 + m2 x2 ) ,
im
(6)
where m1 = m is the mass of the ith block alone, x1 is the x-coordinate of the centre of mass of the ith block
alone, m2 is the combined mass of the first i − 1 blocks, and x2 is the x-coordinate of the combined centre
of mass of the first i − 1 blocks. x1 = 21 l, since the centre of mass of a single uniform block is located at its
centre. By definition, the first (i − 1) blocks have combined centre of mass x2 = xc,(i−1) . Their combined mass
is m2 = (i − 1) m. When these values are used in Equation 6, the m factors cancel out, and the x-coordinate of
the centre of mass of the first i blocks is given by
xc,i =
l
2
+ (i − 1) xc,(i−1)
.
i
Substituting xc,(i−1) from Equation 4 into Equation 7 yields, with simplification,
1 l
xc,i = (l − ∆x) −
− ∆x ,
i 2
(7)
(8)
which can be substituted into Equation 3. Simplified, the result is the local overhang distance for the ith block,
1 l
di =
− ∆x .
(9)
i 2
Combining Equations 1 and 9 gives the total overhang of n block,
n
X
1 l
Dn =
− ∆x .
i 2
i=1
(10)
Like Equation 2, the summation of Equation 10 is a harmonic series - note that if l and ∆x are independent
of i, the expression in parentheses may be factored out of the summation. As with the ideal case, Dn approaches
infinity as n does. In practice, there are other factors that must be considered. For instance, most blocks are not
perfectly rigid, and the bending of flexible blocks causes the stack to ”droop” at the top, placing a limit on the
maximum achievable Dn .
Observe that if the edge of a table is used as a base, the error (∆x)s of the nth block’s overhang is a property
of the table edge, not the blocks. To reflect this, Equation 10 may be modified to give
" n
#
X 1 l
l
− ∆x
+
− (∆x)s ,
(11)
Dn =
i
2
2n
i=1
4
where (∆x)s is the distance between the centre of mass of the block stack and the apparent edge of the supporting
table(similar to the ∆x shown in Figure 3). Equation 11 is valid for n > 1. For the n = 1 case, the summation
term is omitted in the calculation.
Using Equation 11, it is possible to calculate the minimum number of blocks necessary to create a stack whose
topmost block clears the edge of the support structure at the base. The blocks used in this demonstration have
a typical length of l = 19.15 cm. The typical offset due to corner taper is measured to be approximately ∆x
= 0.3 cm, and the error due to corner taper of the table used for testing is (∆x)s = 0.4 cm. Table 1 gives a
series of values for various n, calculated from Equation 11, as well as measured from empirical trials using the
demonstration blocks. For comparison, theoretical values without consideration for error are also calculated.
n
Experimental Dn (cm)
1
2
3
4
5
8.80
13.45
16.50
18.80
20.50
Theoretical Dn (cm), with
Error [Equation 11]
9.18
13.66
16.70
19.00
20.84
Theoretical Dn (cm), without
Error [Equation 2]
9.58
14.36
17.55
19.95
21.86
Table 1: Theoretical and Experimental values of Dn for n = 1, 2, 3, 4, 5.
Observing Table 1, it is clear that the theoretical values from the model with error are much closer to empirical
values than those from the model without. The difference between the theoretically-calculated Dn with error and
the experimental Dn has a sample average of 0.27 cm. Measurement misalignments, random error in the process
of estimating the balance point of each block, and the imperfect construction of the blocks would all contribute
this discrepancy.
In Table 1, also note that for the given parameters, the top block of the stack clears the table (i.e. Dn > l =
19.15 cm) when n = 5. In the “ideal” case, the minimum number of blocks required is 4, which is at odds with
the empirical results.
Finally, it may be of interest to note that if any possible stacking orientation is allowed, this method of stacking
does not achieve the maximum possible overhang distance for a given number of blocks. Allowing for some blocks
to act as counterweights, it is possible to obtain significantly greater values for Dn . This case and others are
explored by Hall[1].
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References
[1] J. F. Hall. ”Fun with stacking blocks”. American Journal of Physics, 73(12), December 2005. pg1107.
[2] R. M. Sutton. ”A Problem of Balancing”. American Journal of Physics, 23(8), November 1955. pg 547.
[3] P. B. Johnson. ”Leaning Tower of Lire”. American Journal of Physics, 23(4), April 1955. pg 240.
[4] L. Eisner. ”Leaning Tower of The Physical Reviews”. American Journal of Physics, 27(2), February 1959. pg
121.
[5] R. P. Boas. ”Cantilevered Books”. American Journal of Physics, 41(5), May 1973. pg 715.
[6] D. J. Steck. ”An Experiment in Discovery”. The Physics Teacher, 18(9), December 1980. pg 672.
[7] G. D. Freier and F . J. Anderson. ”Mp-11. Shifting Center of Gravity”. American Association of Physics
Teachers, 2002. pg M-37.
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