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)
$\frac{T_B^2}{T_A^2}
=\frac{a_B^3}{ a_A^3}$
$\frac{T_B^2}{T_A^2}
=\frac{(4X)^3}{(X)^3}$
$\frac{T_B^2}{T_A^2}
=\frac{64X^3}{X^3}$
$\frac{T_B^2}{T_A^2} =64$
$\frac{T_B^2}{T_A^2} =64$
${T_B^2} =64{T_A^2}$
${T_B} =8{T_A}$
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