Question

the gravitational force between two objects is F.how will this force change if distance between them is reduced to half and mass of each object is quadrupled​

Answer 1

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${\large{\bold{\rm{\underbrace{Requied \; Solution}}}}}$

~ As it's given that the gravitational force between two objects is F. How will this force change if distance between them is reduced to half and mass of each object is quadrupled?

${\large{\bold{\rm{\underline{Given \; that}}}}}$

★ The gravitational force between two objects is F.

★ Distance between them(two objects) is reduced to half.

★ Mass of each object is quadrupled.

${\large{\bold{\rm{\underline{To \; find}}}}}$

★ How will this force change if distance between them is reduced to half and mass of each object is quadrupled?

α) How will this force change if distance between them is reduced to half?

b) How will this force change if mass of each object is quadrupled?

${\large{\bold{\rm{\underline{Solution}}}}}$

α) At ${\sf{\red{4 \: F}}}$ force change when distance between them is reduced to half.

b) At ${\sf{\red{16 \: F}}}$ force change when mass of each object is quadrupled.

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${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ Answer for part α) - How will this force change if distance between them is reduced to half?

✯ To solve this question we have to work according to the universal law of gravitation.

✯ Assumptions are given below..!

◕ Let the gravitational force between the two given objects be M and M₁

◕ Let r be the distance between the two objects.

✯ As it's already given that the distance between them is reduced to half. Henceforth, r¹ = r/2 is the distance between them is reduced to half.

✯ Gravitational force = G × M × M₁ / r²

✯ According to the above information new force be the following,

⇝ F = G × M × M₁ / r²

⇝ F = G × M × M₁ / (r¹)²

⇝ F = G × M × M₁ / (r/2)²

⇝ F = 4 [G × M × M₁ / r²]

⇝ F = 4N

• 4F is the answer for part α)

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~ Answer for part b) - How will this force change if mass of each object is quadrupled?

⇝ M = 4M

⇝ M₁ = 4M₁

⇝ F = G(4M)(4M₁)/r²

⇝ F = 16MM₁/r²

⇝ F = 16F

• 16F is the answer for part b)

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${\large{\bold{\rm{\underline{Additional \; information}}}}}$

Gravitation - Every body attract other body by a force called force of gravitation.

Newton's law of gravitation - The force of gravitational attraction between two point bodies is directly proportional to the product of their masses and inversly proportional to the square of the distance between them. Consider two points bodies of masses m1 and m2 are placed at a distance r. The force of gravitational attraction between them, ${\bold{F \: = G \dfrac{m_{1} m_{2}}{r^{2}}}}$

Here, G is constant called gravitational constant.

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Value \: of \: G \: is \: 6.673 \times 10^{-11}Nm^{2}kg^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Dimensional \: formula \: for \: universal \: gravitational \: constant \: is \: M^{-1} L^{3} T^{-2}}}}$

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Answer 2

GIVEN :-

• Gravitational force between two objects = F.

TO FIND :-

• The change in force if distance between them is reduced to half and mass of each object is quadrupled.

SOLUTION :-

Let the distance between two objects be "d" and masses be "M" and "m".So by universal law of gravitation we have,

$\implies \displaystyle \sf \:F _1= \frac{GMm}{d ^{2} } \: \dashrightarrow \: \bigg \lgroup 1\bigg \rgroup$

Now According to the question,

$\implies \displaystyle \sf \:F_2 = \frac{G \times 4M \times 4m}{ \bigg(\dfrac{ d }{2} \bigg) ^{2} }$

$\implies \displaystyle \sf \:F _2=\frac{G \times 4(Mm)}{ \dfrac{d ^{2} }{4}}$

$\implies \displaystyle \sf \:F_2 =\frac{16(GMm)}{ {d ^{2} }} \dashrightarrow \: \bigg \lgroup 2\bigg \rgroup$

Now divide equation 2 by equation 1,

$\implies \displaystyle \sf \: \frac{F_2 }{F_1} = \frac{ \bigg(\dfrac{16(GMm) }{ {d ^{2} }} \bigg)} { \bigg(\dfrac{GMm}{ {d ^{2} } } \bigg)}$

$\implies \displaystyle \sf \: \frac{F_2 }{F_1} = 16$

$\implies \underline{ \boxed{\displaystyle \sf \: {F_2 } = 16F_1}}$

Hence we conclude that if distance between two objects is reduced to half and mass of each object is quadrupled then the change in force will be 16 times the initial force.