Question 16: Gravitation

Question

If the distance between the Sun and Earth is doubled, then the duration of the year on earth will be :
[Given actual duration of the year $=\mathbf{T}$ ]

Accepted Answer

According to KeplerÃ¢ÂÂs third law, the square of the orbital period is proportional to the cube of the semi-major axis:
$$T^{2}\propto a^{3}.$$
If the distance (i.e. $a$) is doubled, $aÃ¢ÂÂ=2a$, then
Compute the new period $TÃ¢ÂÂ$:
$$\bigl(TÃ¢ÂÂ\bigr)^{2}\propto (2a)^{3}=2^{3}\,a^{3}=8\,a^{3}.$$
Compare to the original:
$$\frac{(TÃ¢ÂÂ)^{2}}{T^{2}}=\frac{8\,a^{3}}{a^{3}}=8      \quad\Longrightarrow\quad      TÃ¢ÂÂ=T\sqrt{8}=2\sqrt{2}\;T.$$
So the new year would last
$$\boxed{2\sqrt{2}\;T}$$
(Option A).

Answer

$2 \sqrt{2} T$

Answer

$\frac{T}{2}$

Answer

$\sqrt{\mathbf{2}} \boldsymbol{T}$

Answer

$\frac{T}{\sqrt{2}}$

Gravitation

If the distance between the Sun and Earth is doubled, then the duration of the year on earth will be : [Given actual duration of the year =\mathbf{T} ]

Choose the correct answer: