Question

Two objects attract each other with a gravitational force of magnitude 1.00 x 10^(-8)N when separated by 20.0 cm. If the total mass of the 2 objects is 5.00 kg, what is the mass of each?

Answer 1

Let the first mass be $m_1=m$. Then, the second one will be $m_2=5.0\ kg -m.$

Let’s write the Law of Universal Gravitation:

$$F=\dfrac{Gm_1m_2}{r^2}.$$

Let's substitute the numbers:

$$1.0\cdot10^{-8} = \dfrac{6.67\cdot10^{-11}\cdot(5.0m-m^2)}{(0.2)^2},$$ $$\dfrac{1.0\cdot10^{-8}\cdot(0.2)^2}{6.67\cdot10^{-11}}=5.0m-m^2,$$ $$m^2-5m+6=0.$$

This uadratic eqqquation has two roots:

$$m_1=\dfrac{-b+\sqrt{b^2-4ac}}{2a}=\dfrac{5+\sqrt{5^2-4\cdot1\cdot6}}{2\cdot1}=3\ kg,$$ $$m_2=\dfrac{-b-\sqrt{b^2-4ac}}{2a}=\dfrac{5-\sqrt{5^2-4\cdot1\cdot6}}{2\cdot1}=2\ kg.$$

Answer:

$m_1=3\ kg, m_2=2\ kg.$