Paper Ball and Bean Bag Dimensions

Background

In these experiments we investigate the dimension of approximately spherical objects. We assume that there is a power law relationship between the mass and radius of the object:

M = k r^d,

where M is the mass, r is the radius, k is a constant of proportionality, and d is the dimension.

Examples of this relation are presented in the discussion of mass dimension.

To find the dimension of the object, we need to solve this equation for d. Taking logs of both sides we obtain

log(M) = log(k) + dlog(r).

Solving for d gives

d = (log(M) - log(k))/log(r).

For mathematical objects of infinite extent, we evaluate d by taking the limit as r approaches infinity. Because k is a constant, log(k)/log(r) -> 0 as r -> infinity. So in this case we find

d = lim_{r → ∞} log(M)/log(r).

Physical objects are of finite extent, so instead of taking this limit we go back to the first log equation

log(M) = d log(r) + log(k),

and note it is a linear equation in the form y = mx + b. From experimental data, d is the slope of the best fit line through the data points

(log(r₁),log(M(r₁))), (log(r₂),log(M(r₂))), ..., (log(r_n),log(M(r_n))).

Example 1 Suppose these values of (r, M(r)) are measured.

(1, 1.01812), (2, 1.36625), (3, 1.54131), (4, 1.8103), (5, 1.97791), (6, 2.04471), (7, 2.18991), (8, 2.20176), (9, 2.29044), (10, 2.464), (11, 2.64258), (12, 2.62917), (13, 2.77048), (14, 2.94974), (15, 3.04551), (16, 3.02433), (17, 3.15033), (18, 3.19124), (19, 3.26849), (20, 3.21769)

Plotting (log(r), log(M(r))) for these values gives this graph

We see the points are reasonably well-fit with a line of slope 0.4, so we can say the data support the claim this object has mass dimension dm ≈ 0.4.

Example 2 Suppose these values of (r, M(r)) are measured.

(1, 2.7809), (2, 3.07854), (3, 2.76506), (4, 1.94689), (5, 2.14993), (6, 2.613), (7, 3.64626), (8, 3.53464), (9, 3.63159), (10, 2.99401), (11, 3.91546), (12, 4.66868), (13, 3.64654), (14, 2.55153), (15, 2.94254), (16, 5.05635), (17, 4.44356), (18, 5.21844), (19, 3.77781), (20, 3.35247))

Plotting (log(r), log(M(r))) for these values gives this graph

We see the points are not reasonably well-fit with any line, so we can say the data do not support the claim this object has a well-defined mass dimension.

Finally, though the point shouldn't need to be mentioned, we must point out that no deduction at all can be drawn from a graph having only two points. For example, suppose the underlying data are the red points in this graph, but only the two blue points have been measured. Given these two points, certainly one can draw a line passing through them. Does it follow that all subsequently measured data points will lie near to this line? Not at all.

In general, the larger the range of r-values over which a power law scaling is observed, the better the claim that the underling process is fractal.

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