At a distance x from the centre of the Earth, the weight of an object is \(\frac{W}{4}\) where W is the weight of the object on the surface of the Earth. Which of the following is the expression of radius of the Earth in terms of x? A) \(\frac{x}{4}\) B) \(\frac{x}{2}\) C) \(\frac{3x}{4}\) D) 2x
Answer (1)
To solve this problem, we need to consider the gravitational force that acts on an object near the Earth. The weight of an object, which is the gravitational force acting on it, is given by Newton's law of universal gravitation:
F = r 2 GM m
Where:
F is the gravitational force (or weight) on the object.
G is the gravitational constant.
M is the mass of the Earth.
m is the mass of the object.
r is the distance from the center of the Earth to the object.
On the surface of the Earth, the weight W of the object is:
W = R 2 GM m
Where R is the radius of the Earth.
According to the problem, at a distance x from the center of the Earth, the weight of the object becomes 4 W . So we have:
4 W = x 2 GM m
Since W = R 2 GM m , we can substitute this into the equation:
4 R 2 GM m = x 2 GM m
Cancel out GM m from both sides:
4 R 2 1 = x 2 1
Cross-multiply to get:
4 x 2 = R 2
Taking the square root of both sides gives:
R = 2 x
Therefore, the expression for the radius of the Earth R in terms of x is 2 x .
The correct answer is option B: 2 x .
