Question

The mass of a ball is four times the mass of another ball. When these balls are separated by a distance of 10 cm, the force of gravitation between them is 6.67 × 10⁻⁷ N. Find the masses of the two balls.

Answer 1

according to Newton's law of gravitation, force is directly proportional to product of masses and inversely proportional to square of separation between masses.
e.g., $F=\frac{Gm_1m_2}{r^2}$
here, G is gravitational constant.

a/c to question,
$m_1=4m_2$ , r = 10cm = 0.1m
and F = 6.67 × 10^-7 N

we know, G = 6.67 × 10^-11 Nm²/kg²

so, F = G$(4m_2)(m_2)$/r²

or, 6.67 × 10^-7 = 6.67 × 10^-11 × 4 × $m_2$²/(0.1)²

or, 10⁴ = 4$m_2$²/0.01

or, $m_2$² = 25

or, $m_2$ = 5 kg

and then, $m_1$ = 4 × 5 = 20kg

Answer 2

$\sf{\boxed{\bold{\tiny{Heya \: mate.\: Solution \: below.}}}}$
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$<b><u><font color = "red">ᎪNᏚᏔᎬᎡ - ↓ </font color></u>$




⏩$\mathfrak{\blue{\underline{Gravitation}}}$



• Gravitation is the force which attracts any two bodies in the universe.

• Gravitation was first noticed by Newton (the apple fell and then bla bla bla...). Around then, he gave the famous law for gravitation and this came to be known as Newton's Law of Gravitation.

• According to Newton's Gravitational Law, the gravitational force between any two bodies is directly proportional to the mass of the two bodies and inversely proportional to the square of distance between them. Combining these, and adding a Universal Gravitational Constant, we can find the gravitational force between two objects as -

$\sf{F = \frac{Gm_1m_2}{{d}^{2}}}$

where,

F = the Gravitational Force

G = Universal Gravitational Constant

$\sf{m_1}$= mass of one object

$\sf{m_2}$= mass of second object

d = Distance between the two objects





♠ So, coming back to your question,



⏩ Let the mass of one object (m_1) be "m"

And, mass of second object (m_2) be "4m"

Also, given distance between them $\sf{= 10 cm = 0.01 m}$

And, we know that G $\sf{= 6.67 \times {10}^{-11}}$

Also, given Gravitation Force between them =$\sf{6.67 \times {10}^{-7}}$



⏩ Then, according to question,


$\sf{=> F = \frac{G \times m_1 \times m_2}{{d}^{2}}}$


$\sf{=>6.67 \times {10}^{-7} = \frac{6.67 \times {10}^{-11} \times m \times 4m}{{0.1}^{2}}}$


$\sf{=>6.67 \times {10}^{-7} = \frac{6.67 \times {10}^{-11} \times m \times 4m}{0.01}}$


$\sf{=>6.67 \times {10}^{-7} \times 0.01 = 6.67 \times {10}^{-11} \times m \times 4m}$


$\sf{=>6.67 \times {10}^{-7} \times {10}^{-2} = 6.67 \times {10}^{-11} \times m \times 4m}$


$\sf{=>6.67 \times {10}^{-9} = 6.67 \times {10}^{-11} \times m \times 4m}$


$\sf{=>6.67 \times {10}^{-9} = 6.67 \times {10}^{-11} \times {4m}^{2}}$


$\sf{=> {4m}^{2} = \frac{6.67 \times {10}^{-9}}{6.67 \times {10}^{-11}}}$


$\sf{=> {4m}^{2} = {10}^{2}}$


$\sf{=> {4m}^{2} = 100}$


$\sf{=> {m}^{2} = \frac{100}{4}}$


$\sf{=> m = \sqrt{25}}$


$\sf{=> m = 5}$



•°• $\sf{m = 5}$

$\sf{4m = 4(5) = 20}$




⏩⏩⏩ •°• The mass of the smaller object is 5 kg and that of bigger object is 20 kg.
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