Question

1. This problem explores the "cosmology" of the 2-dimensional surface of a spherical balloon, presently of radius R. Two "galaxies" are separated by angle $\theta$ as shown, and the distance between two galaxies is measured by the arc length $l$. a. If the balloon inflates at rate $dR/dt$, show that galaxies obey an equation similar to Hubble's Law. What is the analogy to the Hubble's Law equation on the balloon surface "universe"? What is the Hubble's constant of this balloon universe? b. Suppose that instead of a sphere, the 2-dimensional universe actually has the shape of a squashed beachball, as shown. The squashed beachball universe expands, always keeping the same shape. Does this universe obey Hubble's Law? Carefully explain your answer. [You need to think carefully about how distances on the surface scale with the size of the balloon in this more complicated geometry.]

          1. This problem explores the "cosmology" of the 2-dimensional surface of a spherical balloon,
presently of radius R. Two "galaxies" are separated by angle $\theta$ as shown, and the distance
between two galaxies is measured by the arc length $l$.
a. If the balloon inflates at rate $dR/dt$, show that galaxies obey an equation similar to
Hubble's Law. What is the analogy to the Hubble's Law equation on the balloon
surface "universe"? What is the Hubble's constant of this balloon universe?
b. Suppose that instead of a sphere, the 2-dimensional universe actually has the shape
of a squashed beachball, as shown. The squashed beachball universe expands,
always keeping the same shape. Does this universe obey Hubble's Law? Carefully
explain your answer. [You need to think carefully about how distances on the
surface scale with the size of the balloon in this more complicated geometry.]

1. This problem explores the "cosmology" of the 2-dimensional surface of a spherical balloon,
presently of radius R. Two "galaxies" are separated by angle θ as shown, and the distance
between two galaxies is measured by the arc length l.
a. If the balloon inflates at rate dR/dt, show that galaxies obey an equation similar to
Hubble's Law. What is the analogy to the Hubble's Law equation on the balloon
surface "universe"? What is the Hubble's constant of this balloon universe?
b. Suppose that instead of a sphere, the 2-dimensional universe actually has the shape
of a squashed beachball, as shown. The squashed beachball universe expands,
always keeping the same shape. Does this universe obey Hubble's Law? Carefully
explain your answer. [You need to think carefully about how distances on the
surface scale with the size of the balloon in this more complicated geometry.]

Added by Jennifer B.

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