Question

a proposed gravity tractor has a mass of 3000 kg. at what distance should the gravity tractor be placed from an asteroid with a mass of 2.6 x 10^{10} kg in order to exert a gravitational force of 1.11 newtons on the asteroid?

Explanation:

Step1: Recall the formula for gravitational force

The formula for gravitational force is \(F = G\frac{m_1m_2}{r^2}\), where \(F\) is the gravitational force, \(G = 6.67\times10^{- 11}\space Nm^{2}/kg^{2}\) (universal gravitational constant), \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between them. We need to solve for \(r\). Rearranging the formula for \(r\) gives \(r=\sqrt{G\frac{m_1m_2}{F}}\).

Step2: Substitute the given values

We are given \(m_1 = 3000\space kg\), \(m_2=2.6\times 10^{10}\space kg\), \(F = 1.11\space N\), and \(G = 6.67\times10^{-11}\space Nm^{2}/kg^{2}\).
Substitute these values into the formula:

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First, calculate the numerator \(6.67\times3000\times2.6=6.67\times7800 = 52026\).
Then the denominator \(1.11\times10 = 11.1\).
So \(\frac{52026}{11.1}\approx4687\).
\(r=\sqrt{4687}\approx68.5\space m\)

Answer:

\(r\approx68.5\space m\)