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## Sunday, 28 August 2011

### Physics GRE - #27

Suppose that there is a very small shaft in the Earth such that a point mass can be placed at a radius of $R/2$ where $R$ is the radius of the Earth. If $F(r)$ is the gravitational force of the Earth on a point mass at a distance $r$, what is: $\frac{F(R)}{F(2R)}?$

1. 32
2. 8
3. 4
4. 2
5. 1

Solution :

Recall that the gravitational force between two objects is given by $F(r)=\frac{GMm}{r^2}=\frac{k}{r^2}$ where $k$ is a constant.
Hence we have: $\frac{F(R)}{F(2R)}=\left.\frac{k}{R^2}\right/\frac{k}{(2R)^2}=4.$
On the same exam, they also asked for $\frac{F(R)}{F\left(\frac{R}{2}\right)}.$ Try it if you want some extra practice.
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