Calculation of true thickness of the bed

45. Two planets A and B orbit around their Sun, B being four times farther away than A from their Sun. Then the length of the year on B, compared to that A, would be

1) The same

2) Twice

3) Four times

4) Eight times

Solution:

From Kepler’s third law,

$T^2 =\frac{4\pi^2}{GM} a^3$

T-Planet's Period

a-semi major axis of the orbit

M- mass

G-Universal gravitational constant

$T_A^2 =\frac{4\pi^2}{GM} a_A^3$ ---(1)

$T_B^2 =\frac{4\pi^2}{GM} a_B^3$ ---(2)

Divide eq(2) by eq(1)

$\frac{T_B^2}{T_A^2} =\frac{a_B^3}{ a_A^3}$

$\frac{T_B^2}{T_A^2} =\frac{(4X)^3}{(X)^3}$

$\frac{T_B^2}{T_A^2} =\frac{64X^3}{X^3}$

$\frac{T_B^2}{T_A^2} =64$

$\frac{T_B^2}{T_A^2} =64$

${T_B^2} =64{T_A^2}$

${T_B} =8{T_A}$