Question

The values of acceleration due to gravity on planets A and B are in the ratio 3 : 5. If the planets have equal radii, calculate the ratio of their masses. ​

Answer 1

Answer:

3 : 5

Explanation:

$\underline{\bigstar\:\textsf{Given:}}$

• Ratio of acceleration due to gravity on planet A and B ($\sf{g_A\::\:g_B}$) = 3 : 5
• Radius of A = Radius of B

$\underline{\bigstar\:\textsf{To\:find:}}$

• Ratio of their masses ($\sf{m_{A}\::\:m_{B}}$) = ?

$\underline{\bigstar\:\textsf{Solution:}}$

We know, acceleration due to gravity or $\green{\sf{g\:=\:\dfrac{GM}{R^2}}}$

Here, two planets are given A and B whose g are in the ratio of 3 : 5 and radii means $\sf{R_{A}\:=\:R_{B}}$

$\:\:\:\::\implies\:\displaystyle\sf\dfrac{g_{_A}}{g_{_B}} = \dfrac{\: \: \frac{GM_A}{R_A^2}}{\: \: \: \frac{GM_B}{R_B^2} \: \:}$

$\:$

$\sf{\because\:R_{A}\:=\:R_{B}}$ (They will be canceled)

$\:$

$\:\:\:\::\implies\:\sf{\dfrac{3}{5}\:=\:\dfrac{G\:\times\:M_{A}}{G\:\times\:M_{B}}}$

$\:$

$\sf{\because\:G\:is\:a\:constant\:it\:is\:also\:canceled}$

$\:$

$\:\:\:\:\small:\implies\underline{\boxed{\sf\purple{\dfrac{3}{5}\:=\:\dfrac{M_{A}}{M_{B}}}}}$

Hence, ratio of their masses will also be 3 : 5.

Answer 2

$\red{\mid{\underline{\overline{\textbf{Answer}}}\mid}}$

Ratio of masses = 3 : 5

$\red{\mid{\underline{\overline{\textbf{explanation}}}\mid}}$

According Law of universal gravitation the force of attraction between two bodies is directly proportional to product of their masses and inversely proportional to square of distance between them. This law is defined by the expression.

$\bigstar \: \orange{\mid{\underline{\overline{\textbf{f = GMm ÷ d2}}}\mid}}$

We also use this equation to find acceleration due to gravity of a planet. The equation used for it is :

$g = \frac{Gm}{ {r}^{?} }$

In this equation :

• G = Universal gravitational constant
• M = Mass of first body
• m = mass of second body
• d = distance between them.

For planets or spherical bodies we take the distance between them as their radii.

Value of Universal Gravitational constant is :

$\mapsto \: 6.673 \times {10}^{ - 11}$

Now :-

$\blue{\mid{\underline{\overline{\textbf{Coming to question:-}}}\mid}}$

In this question we have been given that ratio of

their Gravitational force is in ratio 3 : 5 their radii are equal but their masses are not known, we have to calculate the ratio of their masses.

Now let's take the mass of planet A be :

$\mapsto \: m_a$

And the mass of planet B be :

$\mapsto \: m_b$

Radius of planet A = r_a

Radius of planet B = r_b

Acceleration due to gravity of planet A is :

$g_a = \frac{Gm_a}{ {r_a}^{2} }$

Acceleration of gravity due to planet B is :

$g_b = \frac{Gm_b}{ {r_b}^{2} }$

According to the question g_a : g_b = 3:5

Therefore we can say that:

$\frac{g_a}{g_b} = \frac{3}{5} = \frac{ \frac{Gm_a}{ {r_a}^{2} } }{ \frac{Gm_b}{ {r_b}^{2} } }$

Since radii of both planets are equal so they will cancel each other like:

$\frac{3}{5} = \frac{ \frac{Gm_a}{\cancel r_a2} }{ \frac{Gm_b}{\cancel r_b2} }$

$\mapsto \: \frac{3}{5} = \frac{Gm_a}{Gm_b}$

$\mapsto \: \frac{3}{5} = \frac{6.673 \times {1 0 }^{ - 11} \times \: m_a }{6.673 \times {10}^{ - 11} \times m_b}$

G will also be canceled since it's a constant.

$\mapsto \: \frac{3}{5} = \frac{m_a}{m_b}$

Therefore we got that ma : mb = 3 : 5.

Thus ratio of their masses is also 3:5.

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BY SCIVIBHANSHU

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