The ratio of the weights of a body on Earth’s surface to that on the surface of a planet is 9:4 The mass of the planet is 19th of that of the Earth. If R is the radius of the Earth, what is the radius of the planet? (Take the planets to have the same mass density)

Sure, let's solve this step by step.

1. The weight of an object is given by the formula W = m*g, where m is the mass of the object and g is the acceleration due to gravity. On Earth, g is approximately 9.8 m/s^2.

2. The acceleration due to gravity on a planet is given by the formula g = GM/r^2, where G is the gravitational constant, M is the mass of the planet, and r is the radius of the planet.

3. Given that the ratio of the weights of a body on Earth to that on the planet is 9:4, we can write this as W_earth/W_planet = 9/4.

4. Substituting the formulas from steps 1 and 2 into this equation gives us (mg_earth)/(mg_planet) = 9/4.

5. The mass of the object cancels out, leaving us with g_earth/g_planet = 9/4.

6. Substituting the formula for g from step 2 into this equation gives us (GM_earth/R_earth^2)/(GM_planet/R_planet^2) = 9/4.

7. The gravitational constant G also cancels out, leaving us with (M_earth/R_earth^2)/(M_planet/R_planet^2) = 9/4.

8. Given that the mass of the planet is 1/19th of the mass of the Earth, we can substitute M_planet = M_earth/19 into this equation to get (M_earth/R_earth^2)/((M_earth/19)/R_planet^2) = 9/4.

9. Simplifying this equation gives us (19*R_earth^2)/R_planet^2 = 9/4.

10. Solving for R_planet, we find that R_planet = sqrt((19*R_earth^2)*4/9).

So, the radius of the planet is the square root of 76/9 times the radius of the Earth.