The weight of an object is the same on two different planets. The mass of planet A is only fifteen percent that of planet B. Find rA/rB, which is the ratio of the radii of the planets.
0.15
Well, let's solve this cosmic conundrum, shall we? To figure out the ratio of the radii, we need to do some gravitational gymnastics.
Now, we know that weight is directly proportional to the mass and inversely proportional to the square of the distance. So, let's say the weight of the object on planet A is W and on planet B is also W (since you mentioned they're the same).
From W = m*g, where m is the mass and g is the acceleration due to gravity, we can deduce that:
W = m*A*gA
W = m*B*gB
Since weight is the same on both planets, W = W. This means:
m*A*gA = m*B*gB
Now here comes the funny part. We are given that the mass of planet A is only fifteen percent (0.15) that of planet B. So, we can rewrite the equation as:
0.15*m*B*gA = m*B*gB
The mass of the object on planet B cancels out, leaving us with:
0.15*gA = gB
Now, the acceleration due to gravity is inversely proportional to the square of the distance, so we can write:
gA/gB = (rB/rA)^2
Substituting our value for gB from earlier, we get:
0.15 = (rB/rA)^2
Taking the square root of both sides, we get:
√0.15 = rB/rA
So, rB/rA is approximately equal to the square root of 0.15. And there you have it! The cosmic clown math has been conquered.
To find the ratio of the radii of the planets, we need to consider the relationship between mass, weight, and the radius of a planet.
The weight of an object is the force exerted on it due to gravity, and it depends on the mass of the object and the gravitational pull on the planet. Weight is calculated using the formula:
Weight = mass * acceleration due to gravity
Now, let's address the given information. We know that the weight of the object is the same on both planets. This means that the mass of the object and the acceleration due to gravity on each planet must be inversely proportional. In other words, if the mass of planet A is 15% of planet B, the acceleration due to gravity on planet A must be 15 times that of planet B (since gravity is pulling 15 times harder on the object on planet B than on planet A, resulting in the same weight).
Let's denote the acceleration due to gravity on planet A as gA and on planet B as gB. We can write the following equation:
gA = 15 * gB
Now, the acceleration due to gravity on a planet is proportional to 1 divided by the square of its radius. So, we have:
gA ~ 1 / rA^2
gB ~ 1 / rB^2
Using the relationship between gA and gB, we can write:
1 / rA^2 = 15 * (1 / rB^2)
To find rA/rB, let's rearrange the equation:
rA/rB = √(15)
Therefore, the ratio of the radii of the planets, rA/rB, is approximately equal to the square root of 15.