Question

guys please answer this question. ​

Answer 1

Question :

If the average orbital radius of a planet around sun is changed from R to 1.01R then find out % change in time period of planet.

Solution :

♦ As per Kepler's third law of planetary motion, the squares of the time period of the planets are directly proportional to the cubes of radius of their orbital.

Mathematically, T² ∝ R³

$\sf:\implies\:\left(\dfrac{T_1}{T_2}\right)^2=\left(\dfrac{R_1}{R_2}\right)^3$

$\sf:\implies\:\left(\dfrac{T_1}{T_2}\right)^2=\left(\dfrac{R}{1.01R}\right)^3$

$\sf:\implies\:\dfrac{T_1}{T_2}=\left(\dfrac{1}{1.01}\right)^{\frac{3}{2}}$

$\sf:\implies\:\dfrac{T_1}{T_2}=\dfrac{1}{1.015}$

$\bf:\implies\:T_2=1.015T_1$

♦ % change in time period :

$\sf:\implies\:\%\:Change=\dfrac{T_2-T_1}{T_1}\times 100$

$\sf:\implies\:\%\:Change=\dfrac{1.015T_1-T_1}{T_1}\times100$

$\sf:\implies\:\%\:Change=\dfrac{0.015T_1}{T_1}\times 100$

$\sf:\implies\:\%\:Change=0.015\times 100$

$:\implies\:\underline{\boxed{\bf{\orange{\%\:Change=1.5\%}}}}$

Answer 2

Question :

If the average orbital radius of a planet around sun is changed from R to 1.01R then find out % change in time period of planet.

Solution :

♦ As per Kepler's third law of planetary motion, the squares of the time period of the planets are directly proportional to the cubes of radius of their orbital.

Mathematically, T² ∝ R³

$\sf:\implies\:\left(\dfrac{T_1}{T_2}\right)^2=\left(\dfrac{R_1}{R_2}\right)^3$

$\sf:\implies\:\left(\dfrac{T_1}{T_2}\right)^2=\left(\dfrac{R}{1.01R}\right)^3$

$\sf:\implies\:\dfrac{T_1}{T_2}=\left(\dfrac{1}{1.01}\right)^{\frac{3}{2}}$

$\sf:\implies\:\dfrac{T_1}{T_2}=\dfrac{1}{1.015}$

$\bf:\implies\:T_2=1.015T_1$

♦ % change in time period :

$\sf:\implies\:\%\:Change=\dfrac{T_2-T_1}{T_1}\times 100$

$\sf:\implies\:\%\:Change=\dfrac{1.015T_1-T_1}{T_1}\times100$

$\sf:\implies\:\%\:Change=\dfrac{0.015T_1}{T_1}\times 100$

$\sf:\implies\:\%\:Change=0.015\times 100$

$:\implies\:\underline{\boxed{\bf{\orange{\%\:Change=1.5\%}}}}$