[2 Mark Questions]

Question

State Newton’s law of gravitation. Write an expression for acceleration due to gravity on the surface of the earth. If the ratio of acceleration due to gravity of two heavenly bodies is 1 : 4 and the ratio of their radii is 1 : 3, what will be the ratio of their masses?

Vidyadip · Accepted Answer

Newton’s law of gravitation states that every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
$F =\frac{G m_1 m_2}{d^2}$
Acceleration due to gravity $g =\frac{G M}{R^2}$
Where $G$ is gravitational constant
$M$ is the mass of the earth
$R$ is radius of the earth
Ratio of acceleration due to gravity $=1: 4$
Ratio of radii of two bodies $=1: 3$
Acceleration due to gravity is $g$
$=\frac{ GM }{ R ^2}$
$g_1 =\frac{ GM _1}{ R _1^2}$
$\therefore M _1 =\frac{g_1 R _1^2}{ G \text { SamacherKKalvi.Guide }}$
$M _2 =\frac{g_2 R _2^2}{ G }$
Dividing Equation (1) by equation (2) we get
$\frac{ M _1}{ M _2}=\frac{g_1 R_1^2}{ G } \times \frac{ G }{g_2 R _2^2}$
$\frac{ M _1}{ M _2}=\frac{g_1 R _1^2}{g_2 R _2^2}$
$\quad \frac{g_1}{g_2}=\frac{1}{4} \text { and } \frac{ R _1}{ R _2}=\frac{1}{3}$
$\therefore \frac{ M _1}{ M _2}=\frac{1}{4} \times \frac{(1)^2}{(3)^2}$
$\quad \quad=\frac{1 \times 1}{4 \times 9}=\frac{1}{36}$
$\text { Samacheerkalvi.Gude }$
$\therefore M _1: M _2=1: 36$
$\therefore \text { Ratio of their masses }=1: 36$

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