Scaling Laws

Suppose that a special effects studio wants to film a scene in which the Leaning Tower of Pisa topples to the ground. In order to achieve this goal, the studio might make a scale model of the tower, which is (say) 1 m tall, and then film the model falling over. The only problem is that the resulting footage would look completely unrealistic because the model tower would fall over too quickly. The studio could easily fix this problem by slowing the film down. But, by what factor should the film be slowed down in order to make it look realistic?

Although, at this stage, we do not know how to apply the laws of physics to the problem of a tower falling over, we can, at least, make some educated guesses as to the factors upon which the time, $t_f$, required for this process to occur depends. In fact, it seems reasonable to suppose that $t_f$ depends principally on the mass of the tower, $m$, the height of the tower, $h$, and the acceleration due to gravity, $g$. In other words,

$$t_f = C\,m^{\,x}\,h^{\,y}\,g^{\,z},$$ (1.11)

where $C$ is a dimensionless constant, and $x$, $y$, and $z$ are unknown exponents. The exponents $x$, $y$, and $z$ can be determined by the requirement that the previous equation be dimensionally consistent. Incidentally, the dimensions of an acceleration are $[L]/[T]^2$. Hence, equating the dimensions of both sides of Equation (1.11), we obtain

$$[T] = [M]^{\,x}\,[L]^{\,y}\,\left(\frac{[L]}{[T]^2}\right)^{\,z}.$$ (1.12)

We can now compare the exponents of $[L]$, $[M]$, and $[T]$ on either side of the previous expression. These exponents must all match in order for Equation (1.11) to be dimensionally consistent. Thus,

$$= y + z,$$ 0 (1.13)

$$= x,$$ 0 (1.14)

$$1 = -2\,z.$$ (1.15)

It immediately follows that $x=0$, $y=1/2$, and $z=-1/2$. Hence,

$$t_f = C\sqrt{\frac{h}{g}}.$$ (1.16)

Now, the actual tower of Pisa is approximately $100\,$m tall. It follows that because $t_f\propto \sqrt{h}$ ($g$ is the same for both the real and the model tower) the 1 m high model tower would fall over a factor of $\sqrt{100/1}=10$ times faster than the real tower. Thus, the film must be slowed down by a factor of 10 in order to make it look realistic.