Question

Two objects attract each other gravitationally with a force of 2.0×10⁻¹⁰ N

N when they are 0.22 m

m apart. Their total mass is 4.50 kg

kg . Find their individual masses.

m

mlarger =? msmaller?

Answer 1

Let the first mass be $m_1=m$. Then, the second one will be $m_2=4.5-m$. Let’s write the Law of Universal Gravitation:

$$F=\dfrac{Gm_1m_2}{d^2},$$ $$2.0\times10^{-10}=\dfrac{6.67\times10^{-11}\times(4.5m-m^2)}{(0.22\ m)^2},$$ $$\dfrac{(0.22\ m)^2\times2.0\times10^{-10}}{6.67\times10^{-11}}=4.5m-m^2,$$ $$m^2-4.5m+0.145=0.$$

This quadratic equation has two roots. So, solving the equation we can find the masses of the objects:

$$m_1=\dfrac{-b+\sqrt{b^2-4ac}}{2a}=\dfrac{-(-4.5)+\sqrt{(-4.5)^2-4\times1\times0.145}}{2\times1},$$ $$m_1=4.47\ kg,$$ $$m_2=\dfrac{-b-\sqrt{b^2-4ac}}{2a}=\dfrac{-(-4.5)-\sqrt{(-4.5)^2-4\times1\times0.145}}{2\times1},$$ $$m_2=0.032\ kg.$$

Answer:

$m_{larger}=4.47\ kg, m_{smaller}=0.032\ kg.$