Question

The force of attraction between two objects of masses 2m amd m is F₁.When the object with mass 2m is replaced with another object of mass m/2 ,the new force of attraction is found to be F₂.The ratio of F₁:F₂ IS:-
(A)3:2 (B)2:3 (C)1:4 (D)=4:1​

Answer 1

Given :-

The force of attraction between two objects of masses 2m and m is F₁.

When the object of mass 2m is replaced with another object of mass m/2, the new force of attraction is found to be F₂ .

To Find :-

What is the ratio of F₁ : F₂.

Solution :-

$\clubsuit\: \: \sf\bold{\blue{In\: the\: first\: case\: :-}}$

As we know that :

$\bigstar\: \: \sf\boxed{\bold{\pink{F_g =\: \dfrac{Gm_1m_2}{r^2}}}}\: \: \bigstar$

Gravitational Force (\sf F_gFg ) = F₁

First Mass (m₁) = 2m

Second Mass (m₂) = m

Distance between the centre of masses (r²) = r

According to the question by using the formula we get,

$⟹ \: F </p><p>_ 1= \frac{g \: \times \: 2m \: \times \: m}{ {r}^{2} } </p><p>$

$\implies \sf F_1 =\: \dfrac{G \times 2m^2}{r^2}$

$\implies \sf\bold{\purple{F_1 =\: \dfrac{2Gm^2}{r^2}\: ------\: (Equation\: No\: 1)}}$

$</p><p>\clubsuit\: \: \sf\bold{\blue{In\: the\: second\: case\: :-}}</p><p>$

Given:-

Gravitational Force (\sf F_gFg ) = F₂

First Mass (m₁) = m/2

Second Mass (m₂) = m

Distance between the centre of masses (r²) = r

According to the question by using the formula we get,

$⟹F</p><p>_2= \: \frac{g \times \frac{ m }{2} \: \times m }{ {r}^{2} }$

$\implies \sf F_2 =\: \dfrac{G \times \dfrac{m^2}{2}}{r^2}$

$⟹F_2= \: \frac{ \frac{ {gm}^{2} }{2} }{ {r}^{2} }$

$\implies \sf F_2 =\: \dfrac{Gm^2}{2} \times \dfrac{1}{r^2}$

$\implies \sf\bold{\purple{F_2 =\: \dfrac{Gm^2}{2r^2}\: ------\: (Equation\: No\: 2)}}$

From the equation no 1 and the equation no 2 we get,

$\longrightarrow \sf \dfrac{F_1}{F_2} =\: \dfrac{\bigg\{\dfrac{2Gm^2}{r^2}\bigg\}}{\bigg\{\dfrac{Gm^2}{2r^2}\bigg\}}$

$</p><p></p><p></p><p>\longrightarrow \sf \dfrac{F_1}{F_2} =\: \bigg\{\dfrac{2\cancel{Gm^2}}{\cancel{r^2}}\bigg\} \times \bigg\{\dfrac{2\cancel{r^2}}{\cancel{Gm^2}}\bigg\}$

$</p><p></p><p></p><p>\longrightarrow \sf \dfrac{F_1}{F_2} =\: \dfrac{2}{1} \times \dfrac{2}{1}$

$</p><p></p><p></p><p>\longrightarrow \sf \dfrac{F_1}{F_2} =\: \dfrac{2 \times 2}{1 \times 1}$

$</p><p></p><p></p><p>\longrightarrow \sf \dfrac{F_1}{F_2} =\: \dfrac{4}{1}$

$\longrightarrow \sf\bold{\red{F_1 : F_2 =\: 4 : 1}}$

${\small{\bold{\underline{\therefore\: The\: ratio\: of\: F_1 : F_2\: is\: 4 : 1\: .}}}}$

Hence, the correct options is option no (D) 4 : 1.