a space traveller determines the radius of a planet to be half of earth . After landing on its surface , he finds the acceleration due to gravity to be twice of that of surface of earth. Find the ratio of mass of planet to the earth
1:2
acceleration on earth: GM/r^2
acceleration on planet X: GM'/(r/2)^2
GM'/(r/2)^2 = 2GM/r^2
4M'/r^2 = 2M/r^2
M' = M/2
makes sense, since with the same mass and half the radius, the force would be 4 times as much.
1:2
Well, it seems like that planet is really pulling some extra weight! To find the ratio of the mass of the planet to Earth, we can utilize Newton's law of universal gravitation.
Now, let's label some variables. Let's call the radius of the Earth "R" and the radius of the planet "r" (which we are given is half of the Earth's radius). Let's also call the acceleration due to gravity on Earth "g" and the acceleration due to gravity on the planet "g'".
According to Newton's law of universal gravitation, the gravitational force is directly proportional to the masses of the objects and inversely proportional to the square of the distance between their centers. We can express this as:
F = G * (m1 * m2) / (r^2)
Where F is the force of gravity, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Since we are comparing the acceleration due to gravity, we know that the force of gravity experienced by the space traveler on the planet is twice the force of gravity experienced on Earth. Therefore:
g' = 2g
Dividing the two equations, we get:
g' / g = (G * (m1 * m2) / (r^2)) / (G * (M1 * M2) / (R^2))
Now, since the traveler is on the planet, m1 (mass of the traveler) and m2 (mass of the planet) can be considered constant. Similarly, M1 (mass of the traveler) and M2 (mass of the Earth) can be considered constant. Therefore, we can simplify the equation to:
g' / g = (m1 * m2) / (M1 * M2) * (R^2 / r^2)
We are given that the radius of the planet is half of the Earth's radius, so r = R/2. Substituting this value into the equation, we get:
g' / g = (m1 * m2) / (M1 * M2) * (R^2 / (R/2)^2)
Simplifying further, we find:
g' / g = (m1 * m2) / (M1 * M2) * (4 / 1)
Given that g' = 2g, we can substitute that in the equation:
2g / g = (m1 * m2) / (M1 * M2) * 4
Now, as g cancels on both sides:
2 = (m1 * m2) / (M1 * M2) * 4
Rearranging the equation, we get:
(m1 * m2) / (M1 * M2) = 2 / 4
Simplifying the right side, we get:
(m1 * m2) / (M1 * M2) = 1/2
So, the ratio of the mass of the planet to Earth is 1/2 or 0.5.
In simpler terms, we could say that the planet weighs half as much as Earth. Talk about being on a cosmic diet!
To find the ratio of the mass of the planet to that of the Earth, we need to use the information given about the radius and acceleration due to gravity.
Let's denote the radius of the planet as R and the radius of the Earth as RE.
From the given information, the radius of the planet is half that of the Earth, so we can write:
R = (1/2) * RE
Now, let's denote the acceleration due to gravity on the planet as g and the acceleration due to gravity on Earth as gE.
From the given information, the acceleration due to gravity on the planet is twice that of Earth, so we can write:
g = 2 * gE
The acceleration due to gravity on Earth is a constant, known as the gravitational constant, denoted by G. It is approximately equal to 9.8 m/s^2.
Since the acceleration due to gravity on Earth is given by the formula:
gE = G * (ME / RE^2)
where ME is the mass of the Earth, we can substitute this into the equation for the planet's acceleration:
g = 2 * G * (ME / RE^2)
Next, we need to express the acceleration due to gravity on the planet in terms of its mass and radius. The acceleration due to gravity on the planet is given by the formula:
g = G * (M / R^2)
where M is the mass of the planet.
Substituting the expressions for g and gE, we have:
2 * G * (ME / RE^2) = G * (M / R^2)
Now, let's simplify this equation and solve for the ratio of the mass of the planet to that of the Earth.
2 * (ME / RE^2) = M / R^2
Since R = (1/2) * RE, we can substitute this expression:
2 * (ME / (1/2)^2 * RE^2) = M / ((1/2) * RE)^2
Simplifying the denominators and rearranging, we have:
2 * (ME / (1/4) * RE^2) = M / (1/4) * RE^2
Now, cross-multiply and solve for M:
2 * ME = M
ME = M/2
Therefore, the ratio of the mass of the planet to that of the Earth is 1:2.