Kepler's Third Law for Planet Distance
This problem requires applying Kepler's Third Law of Planetary Motion, which relates a planet's orbital period ($T$) to the semi-major axis ($R$) of its orbit around the Sun. The law states that the square of the period is proportional to the cube of the semi-major axis, or $\frac{T^2}{R^3} = \text{constant}$.
Applying Kepler's Law
We can compare the Earth and planet X:
- Let $T_E$ be the Earth's period and $R_E$ be Earth's distance from the Sun.
- Let $T_X$ be planet X's period and $R_X$ be planet X's distance from the Sun.
According to Kepler's Third Law:
$ \frac{T_E^2}{R_E^3} = \frac{T_X^2}{R_X^3} $
Calculations
We are given:
- $T_X = 8$ years
- $T_E = 1$ year (standard orbital period for Earth)
- $R_E = 1.5 \times 10^{11}$ m
We need to find $R_X$. Rearranging the formula to solve for $R_X^3$: $ R_X^3 = R_E^3 \times \left(\frac{T_X}{T_E}\right)^2 $
Substitute the values:
$ R_X^3 = (1.5 \times 10^{11} \ m)^3 \times \left(\frac{8 \ years}{1 \ year}\right)^2 $
$ R_X^3 = (1.5^3 \times (10^{11})^3) \times 8^2 \ m^3 $
$ R_X^3 = (3.375 \times 10^{33}) \times 64 \ m^3 $
$ R_X^3 = 216 \times 10^{33} \ m^3 $
Now, take the cube root to find $R_X$: $ R_X = \sqrt[3]{216 \times 10^{33}} \ m $
$ R_X = \sqrt[3]{216} \times \sqrt[3]{10^{33}} \ m $
$ R_X = 6 \times 10^{11} \ m $
Therefore, the distance of planet X from the Sun is $6 \times 10^{11}$ m.
Similar Questions
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
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Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
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Important Questions from Mechanics
- When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
A uniform bar of length 12 cm and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
- A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is ______ N/m.
Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
