Let us consider two bodies having masses m and m' respectively.
Let they are separated by a distance of r from each other.
As per the Newtons law of gravitation ,the gravitational force between two bodies is given as - [tex]F = G\frac{mm'}{r^{2} }[/tex] where G is the gravitational force constant.
From the above we see that F ∝ mm' and [tex]F\alpha \frac{1}{r^{2} }[/tex]
Let the orbital radius of planet A is [tex]r_{1}[/tex] = r and mass of planet is [tex]m_{1}[/tex].
Let the mass of central star is m .
Hence the gravitational force for planet A is [tex]f_{1} =G \frac{m_{1}*m }{r^{2} }[/tex]
For planet B the orbital radius [tex]r_{2} =2r_{1}[/tex] and mass [tex]m_{2} = 3 m_{1}[/tex]
Hence the gravitational force [tex]f_{2} =G\frac{m m_{2} }{r^{2} }[/tex]
[tex]f_{2} =G\frac{m*3m_{1} }{[2r_{1}] ^{2} }[/tex]
[tex]= \frac{3}{4} G\frac{mm_{1} }{r_{1} ^{2} }[/tex]
Hence the ratio is [tex]\frac{f_{2} }{f_{1} } = \frac{\frac{3}{4}G mm_{1/r_{1} ^2} }{Gmm_{1}/r_{1} ^2 }[/tex]
[tex]=\frac{3}{4}[/tex] [ ans]
The ratio F₂ : F₁ = 3 : 4
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Newton's gravitational law states that the force of attraction between two objects can be formulated as follows:
[tex]\large {\boxed {F = G \frac{m_1 ~ m_2}{R^2}} }[/tex]
F = Gravitational Force ( Newton )
G = Gravitational Constant ( 6.67 × 10⁻¹¹ Nm² / kg² )
m = Object's Mass ( kg )
R = Distance Between Objects ( m )
Let us now tackle the problem !
[tex]\texttt{ }[/tex]
Given:
Gravitational force on planet 1 = F₁
Gravitational force on planet 2 = F₂
mass of planet 1 = m₁
mass of planet 2 = m₂ = 3m₁
distance between planet 1 and star = R₁
distance between planet 2 and star = R₂ = 2R₁
Asked:
ratio of force = F₂ : F₁ = ?
Solution:
[tex]F_2 : F_1 = G \frac{ M m_2} { (R_2)^2 } : G \frac{ M m_1} { (R_1)^2 }[/tex]
[tex]F_2 : F_1 = \frac{m_2} { (R_2)^2 } : \frac{ m_1} { (R_1)^2 }[/tex]
[tex]F_2 : F_1 = \frac{3m_1} { (2R_1)^2 } : \frac{ m_1} { (R_1)^2 }[/tex]
[tex]F_2 : F_1 = \frac{3} { 4 } : 1[/tex]
[tex]\boxed{F_2 : F_1 = 3 : 4}[/tex]
[tex]\texttt{ }[/tex]
[tex]\texttt{ }[/tex]
Grade: High School
Subject: Physics
Chapter: Gravitational Fields
