The ratio of the weights of a body on the Earth's surface to that on the surface of a planet is 9:4. The mass of the planet is 1/9th that of the Earth. If 'R' is the radius of the Earth, what is the radius of the planet?

Correct option is B. R2
Since mass of the object remains same
∴Weight of object will be proportional to 'g'(acceleration due to gravity)
Given
Wearth/Wplanet = 9/4
gearth/gplanet
Also, gsurface = GM/R² (M is mass planet, G is universal gravitational constant, R is radius of planet)
∴9/4 = (GMearth/Rplanet²)/(GMplanet/Rearth²)= (Mearth/Mplanet) × (Rplanet²/Rearth²)= 9(Rplanet²/Rearth²)
∴Rplanet/Rearth = √(1/9) = 1/3
∴Rplanet = Rearth/3 = R/3

There appears to be a mistake in the provided solution. The equation correctly derives that the ratio of radii is 1/3, not 1/2. Let's re-examine the steps:

1. Weight Ratio: The ratio of weights is directly proportional to the ratio of accelerations due to gravity: Wearth/Wplanet = gearth/gplanet = 9/4
2. Gravity Equation: The acceleration due to gravity on the surface of a planet is given by: g = GM/R², where G is the gravitational constant, M is the planet's mass, and R is its radius.
3. Ratio of Gravities: Substituting the gravity equation into the weight ratio gives:
   (GMearth/Rearth²)/(GMplanet/Rplanet²) = 9/4
4. Simplifying: This simplifies to:
   (Mearth/Mplanet) * (Rplanet²/Rearth²) = 9/4
5. Given Mass Ratio: We are given that Mplanet = (1/9)Mearth. Substituting this into the equation gives:
   (Mearth/((1/9)Mearth)) * (Rplanet²/Rearth²) = 9/4
6. Solving for Radius Ratio: This simplifies to:
   9 * (Rplanet²/Rearth²) = 9/4
   (Rplanet²/Rearth²) = 1/4
   Rplanet/Rearth = 1/2
7. Radius of Planet: Therefore, Rplanet = (1/2)Rearth = R/2

Therefore, the radius of the planet is R/2. The provided solution contains a calculation error. The correct answer is not present in the options. The correct option should be R/2