Answer: Weight will remain same as on the Earth
Explanation :
From Universal law of gravitation :
[tex]F=G\dfrac{M_em}{R^2}[/tex]......(1)
[tex]M_e[/tex] is the mass of earth
m is the mass of the person
R is the radius of the earth
We know that F= mg.....(2)
So, from equation (1) and (2)
[tex]mg=\dfrac{GM_em}{R^2}[/tex]
[tex]g=\dfrac{GM_e}{R^2}[/tex].....(3)
It is given that, If we stood on a planet having a mass four times that of earth mass and a radius two times of earth radius then from equation (3)
[tex]g=\dfrac{G\times 4M_e}{4R^2}[/tex]
The value of g is will not change. So, our weight would not change. It would remain the same as on the Earth.
The weight of the astronaut on the planet will be the same as on the Earth.
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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 !
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Given:
mass of Earth = M₁ = M
mass of Planet = M₂ = 4M
radius of Earth = R₁ = R
radius of Planet = R₂ = 2R
Asked:
weight of the astronaut on planet = w₂ = ?
Solution:
We will compare the weight of the astronaut on planet and earth as follows:
[tex]w_2 : w_1 = mg_2 : mg_1[/tex]
[tex]w_2 : w_1 = g_2 : g_1[/tex]
[tex]w_2 : w_1 = G\frac{M_2}{(R_2)^2} : G\frac{M_1}{(R_1)^2}[/tex]
[tex]w_2 : w_1 = G\frac{4M}{(2R)^2} : G\frac{M}{(R)^2}[/tex]
[tex]w_2 : w_1 = G\frac{4M}{4R^2} : G\frac{M}{R^2}[/tex]
[tex]w_2 : w_1 = G\frac{M}{R^2} : G\frac{M}{R^2}[/tex]
[tex]w_2 : w_1 = 1 : 1[/tex]
[tex]\large {\boxed {w_2 = w_1}}[/tex]
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From information above, we could conclude that the weight of the astronaut on the planet will be the same as on the Earth.
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Grade: High School
Subject: Physics
Chapter: Gravitational Fields
