The weight of an object is the same on two different planets. The mass of planet A is only sixty percent that of planet B. Find the ratio of the radii of the planets.

force=kM/r^2

the force is the same, so

M1/r^2=M2/R^2

r/R= sqrt(M1/M2)

To find the ratio of the radii of the planets, we can use the equation for the weight of an object, which is given by:

Weight = Mass * Gravitational Acceleration

Let's denote the weight on planet A as WA, the weight on planet B as WB, the mass of the object as m, the gravitational acceleration on planet A as gA, and the gravitational acceleration on planet B as gB.

Since the weight of the object is the same on both planets, we have WA = WB. Therefore, we can write the equation as:

m * gA = m * gB

Since the mass of the object cancels out, we have:

gA = gB

Now we need to find the ratio of the gravitational accelerations, which can be expressed using the equation:

g = (G * M) / r^2

where G is the gravitational constant, M is the mass of the planet, and r is its radius.

Let's denote the ratio of the radii of the planets as RA/RB, where RA is the radius of planet A and RB is the radius of planet B.

Using the equation for gravitational acceleration, we can write:

gA = (G * MA) / (RA^2)
gB = (G * MB) / (RB^2)

Since gA = gB, we have:

(G * MA) / (RA^2) = (G * MB) / (RB^2)

Canceling out the gravitational constant and rearranging the equation, we get:

(MA / MB) * (RB^2 / RA^2) = 1

Given that the mass of planet A is only sixty percent that of planet B, we have MA / MB = 0.6.

Substituting this value into the equation, we get:

0.6 * (RB^2 / RA^2) = 1

Simplifying the equation, we find:

RB^2 / RA^2 = 1 / 0.6

RB^2 / RA^2 = 10/6

Cross multiplying, we get:

6 * RB^2 = 10 * RA^2

Simplifying further, we have:

RB^2 = (10/6) * RA^2

Taking the square root of both sides, we find:

RB = √(10/6) * RA

Therefore, the ratio of the radii of the planets is:

RA / RB = 1 / √(10/6) or √(6/10) / 1

Simplifying this, we get:

RA / RB = √(3/5) or approximately 0.774, which means the radius of planet A is approximately 0.774 times the radius of planet B.