Question

Two objects attract each other with a gravitational force of a magnitude$1.00\times {10}^{-8}\text{N}$when separated by 20.0 cm. If the total mass of the two objects is 5.00 kg, what is the mass of each?

Short answer

The masses of objects are $m_1=3.00kg$and$m_2=2.00kg$ .

Step-by-step solution

Step 1: Newton’s Law

Newton’s law of universal gravitation states that the gravitational force of attraction between any two particles of masses $M_1$and $M_2$separated by a distance r has the magnitude:

$F_g=G\frac{m_1.m_2}{r^2}$

Where, $G=6.674\times {10}^{-11}\text{N}·{\text{m}}^{\text{2}}{\text{/kg}}^{\text{2}}$is the universal gravitational constant. This equation enables us to calculate the force of attraction between masses under many circumstances.

Given

Gravitational force:$F_g=1.00\times {10}^{-8}\text{N}$

Distance between two objects:$r=20\text{cm}=0.2\text{m}$

Find the masses

Wefindthereasonfortheabovequestionbyfindingthemassofsphere.

Let suppose the first mass $m_1,$is and the other mass will be$m_2=5-m_1,$

According to Newton’s law of universal gravitation, we have

$$\begin{gathered} F_g=G\frac{m_1.m_2}{r^2} \\ 1\times {10}^{-8}=6.67\times {10}^{-11}\times \frac{m_1.\left(5-m_1\right)}{{0.2}^2}..............\left(\because m_2=5-m_1\right) \\ {m}_1.\left(5-m_1\right)=\frac{0.04\times {10}^{-8}}{{10}^{-11}\times 6.67} \\ {m}_1.\left(5-m_1\right)=5.99\approx 6 \\ {m_1}^2-5m_1+6=0 \end{gathered}$$

On simplifying and solving, we get,

$$\begin{gathered} m_1=3.00\text{kg} \\ {m}_2=5-m_1=5-3.00=2.00\text{kg} \end{gathered}$$

Hence, the masses of objects are$m_1=3.00kg$ and $m_2=2.00kg$.