Question

The distance of Venus from the sun is $ 0.72\;AU $. The orbital period of the Venus is:
\[\begin{align}
  & A.200days \\
 & B.320days \\
 & C.225days \\
 & D.325days \\
\end{align}\]

Answer 1

Hint: From Keplar’s law of planetary motion we know that the time period of revolution of the planets is proportional to the distance of the planet from the sun. Here we are given the details of Venus, we can find the time period of Venus by comparing the same with earth.

Formula used:
 $ T^{2}\propto R^{3} $

Complete step-by-step answer:
From newton’s law of gravitation we know that the gravitational force between the planets is given as $ F=\dfrac{Gm_{1}m_{2}}{R^{2}} $ where, $ G $ is the gravitational constant, $ m_{1} $ and $ m_{2} $ are the masses of the planets which are separated at a distance $ R $ from each other.
Since the planets are in constantly moving on the circular path, we can say that they experience a centripetal force towards the centre or the sun, this force is given as $ F_{C}=mR\omega^{2} $ where $ \omega $ is their angular velocity.
For the planet to remain in its orbital the two forces must be equal.
 $ \implies F=F_{C} $
 $ \implies mR\omega^{2}=\dfrac{Gm_{1}m_{2}}{R^{2}} $
 $ \implies mR\left(\dfrac{2\pi}{T}\right)^{2}=\dfrac{Gm_{1}m_{2}}{R^{2}} $
 $ \implies T^{2}\propto R^{3} $
Here, given that $ r_{v}=0.72AU $, we must find the time period of Venus $ T_{v} $ .
We know that radius of earth is $ r_{e}=1AU $ and the time period of earth $ T_{e}=1\;yr=365\;days $
Taking the ratio between the two, we get,
 $ \left(\dfrac{T_{v}}{T_{e}}\right)^{2}=\left(\dfrac{r_{v}}{r_{e}}\right)^{3} $
 $ \implies \left(\dfrac{T_{ v}}{365}\right)^{2}=\left(\dfrac{0.72}{1}\right)^{3} $
 $ \implies T_{v}=365\times (0.72)^{\dfrac{3}{2}} $
 $ \implies T_{v}=222.93\;days. $
Since this value is not there in the given options, we can take the nearness possible value.
Thus the answer is option \[C.225days\]

So, the correct answer is “Option C”.

Note: We know that Kepler’s law of planetary motion assumes that the planets revolve around the sun in elliptical orbits. He also gave three laws called the Kepler’s laws of planetary motion, which as the name suggests described the motion of planets. According to Kepler, the area swept by the planet during an equal time interval is equal. He also says that the square of the orbital time period is proportional to the third power of the length of the semi-major axis of the elliptical orbit, which is nothing but the distance of the planet from the sun during its revolution around the sun.