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
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is
Consider a cylindrical tank completely filled with water of height $1.6 \ m$ and cross-sectional area $0.5 \ m^2$. There is a hole in its side at a height of $90 \ cm$ from the bottom. Assume the cross-sectional area of the hole to be negligibly small compared to the cross-sectional area of the water tank. If a load of $50 \ kg$ is applied on the upper surface of water in the tank, then at the moment the hole is opened, the velocity of water coming out is:
($g = 10 \ m/s^2$)
Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length a. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length 2a and also the masses of the spheres are made 2m, then they will collide after __________ seconds.
A 4.0 cm long straight wire carrying a current of 8A is placed perpendicular to a uniform magnetic field of strength 0.15 T. The magnetic force on the wire is __________ mN.
- The equation for real gas is given by $\left(P+\frac{a}{V^2}\right) (V-b)=RT$, where P,V,T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $ab^{-2}$ is equivalent to that of :
Important Questions from Mechanics
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
