X and Y are planets.Accelaration due to gravity at a height half the radius from surface of planet X is equal to that of at the surface of planet Y. If mass of X is four times that of Y, ratio of radii of planets X and Y is
A) 4:3 B) 3:4 C) 2:1 D) 1:2
To find the ratio of the radii of planets X and Y, we need to use the concept of gravitational acceleration and the formula for gravitational force.
Let's assume the radius of planet X is denoted by rX and the radius of planet Y is denoted by rY.
The gravitational acceleration, denoted by g, is given by the formula:
g = (G * M) / R^2
where G is the universal gravitational constant, M is the mass of the planet, and R is the distance from the center of the planet.
Given that the acceleration due to gravity at a height of half the radius from the surface of planet X is equal to the acceleration at the surface of planet Y, we can write the following equation:
(gX/2) = gY
First, let's consider planet X. The gravitational acceleration at a height of half the radius from the surface of planet X is given by:
gX/2 = (G * MX) / (rX/2)^2
Simplifying this equation, we get:
gX/2 = (4 * G * MX) / rX^2
where MX is the mass of planet X.
Next, let's consider planet Y. The gravitational acceleration at the surface of planet Y is given by:
gY = (G * MY) / rY^2
where MY is the mass of planet Y.
Since gX/2 = gY, we can equate the two equations:
(4 * G * MX) / rX^2 = (G * MY) / rY^2
Canceling out the G and simplifying the equation, we get:
(4 * MX) / rX^2 = MY / rY^2
Since we are given that MX = 4 * MY, we can substitute this value into the equation:
(4 * (4 * MY)) / rX^2 = MY / rY^2
Simplifying further, we get:
16 * MY / rX^2 = MY / rY^2
Canceling out MY and simplifying, we get:
16 / rX^2 = 1 / rY^2
Cross-multiplying and simplifying, we get:
16 * rY^2 = rX^2
Taking the square root of both sides gives:
4 * rY = rX
Thus, the ratio of the radii of planets X and Y is 4:1.
Therefore, the correct answer is option C) 2:1.