- #1

XanMan

- 14

- 1

**I am given the solution to the first part of the problem, however not the second part - would appreciate for someone to double check my work! Cheers.**

1. Homework Statement

1. Homework Statement

If a scale model of the solar system is made using materials of the same respective average density as the sun and planets, but reducing all linear dimension by a factor ##\alpha##, how will the velocity and period scale with ##\alpha##? Assume a circular orbit in your calculations.

## Homework Equations

[/B]

$$F = \cfrac{GMm}{r^2}

$$F = \cfrac{mv^2}{r}

## The Attempt at a Solution

Starting off with the basic relationship required:

$$\cfrac{GMm}{r^2} = \cfrac{mv^2}{r}$$

Then: ##v^2 = \cfrac{GM}{r} \propto \cfrac{G \rho R^3}{r}##, where ##R## is the radius of the Sun.

In general: $$v^2 \propto \cfrac{R^3}{r}$$

Re-scaling the solar system by ##\alpha##:

$$v^2 \propto \cfrac{\alpha^3 R^3}{\alpha r}$$

*Therefore ##v \propto \alpha##*

**i)**Now, ##v = \cfrac{2 \pi r}{T}##. Thus:

$$\cfrac{2 \pi r}{T} \propto \alpha$$

*Therefore ##T \propto \cfrac{1}{\alpha}##*

**ii)**