## Modern mathematical comments on Two Falling GlobesSome of the notation below relies on the HTML superscript tag <sup>. a^{b} should appear as ``a superscript b''. Newton used lower case
letters in the diagram, whereas I have used capitals (to avoid ambiguity
with the constant g.)
The
experiment
begins with the release of globe First, let us analyse the problem assuming uniform acceleration, with value g. This corresponds to no air resistance. In this case, t, the time taken to fall a distance x is given by t = sqrt ( 2 x / g ) [1] For the Two Falling Globes experiment,
t
Substituting for t sqrt ( EH ) = sqrt ( EF ) + sqrt ( GH ) [5]
By setting up this experiment and adjusting the length of the string FG so that
In this way, one can demonstrate the presence of air resistance etc
t Now take the case where the globes do not accelerate uniformly, but with acceleration given by
d so that the acceleration is reduced in proportion to the speed. Newton asserts the effect of air resistance takes this form (and we might assume that the reduced weight of moving bodies in his mechanical theory of gravity could too.) For air resistance, the constant k depends on the mass and size of the globes. For a body falling from rest at time t=0, the solution of [6] is
x = ( g / k This can be verified by substituting into [6] (or by solving [6] by the change of variable v = dx / dt, to obtain v(t), and then integrating this.) The uniform acceleration case with k=0 can be recovered by expanding the exponential as a series and cancelling. Only the term
(1/2) . g t is independent of k and remains if k=0. This is equivalent to [1] above.
For the Two Falling Globes experiment with k not zero and two identical globes,
t Interestingly, Newton records an observation by Galileo which would provide a fourth equation and allows us to solve for the times and the two constants. This requires that the experiment is performed with a globe of the same size and mass as Galileo (so that k is the same.) Since Newton carefully notes the mass and material of the globe Galileo used, this would be possible. Newton's intentions are discussed on another page. © 1994-2001 Andrew McNab. Back to isaacnewton.org.uk |