Suppose a planet with a mass of 2.14 ✕ 10^25 kg is orbiting a star with a mass of 4.05 ✕ 10^31 kg, and the mean distance between the planet and the star is 1.22 ✕ 10^12 m.
Using Newton's law of universal gravity, determine the speed of the planet when it is at the mean distance from the star.
how can you "drop" a wrench in zero gravity?
Asking for a friend ...
You just set it beside you and it stays there :)
To find the speed of the planet when it is at the mean distance from the star, we can use Newton's law of universal gravity equation:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force between the planet and the star,
G is the gravitational constant (6.67430 × 10^-11 N m^2/kg^2),
m1 is the mass of the planet,
m2 is the mass of the star, and
r is the distance between the planet and the star.
First, let's calculate the gravitational force:
F = G * (m1 * m2) / r^2
F = (6.67430 × 10^-11 N m^2/kg^2) * ((2.14 ✕ 10^25 kg) * (4.05 ✕ 10^31 kg)) / (1.22 ✕ 10^12 m)^2
Now, we know that the gravitational force is responsible for keeping the planet in orbit. This force provides the necessary centripetal force for circular motion. The centripetal force is given by:
F = m * v^2 / r
Where:
m is the mass of the planet,
v is the velocity or speed of the planet, and
r is the distance between the planet and the star.
We can equate the gravitational force to the centripetal force:
F = m * v^2 / r
Since the gravitational force is equal to the centripetal force, we can write:
G * (m1 * m2) / r^2 = m * v^2 / r
Rearranging the equation to solve for velocity (v):
v^2 = G * (m1 * m2) / r
Now, let's plug in the known values and solve for velocity:
v^2 = (6.67430 × 10^-11 N m^2/kg^2) * ((2.14 ✕ 10^25 kg) * (4.05 ✕ 10^31 kg)) / (1.22 ✕ 10^12 m)^2
Calculate the right-hand side of the equation and take the square root to find the velocity (v):
v = √[(6.67430 × 10^-11 N m^2/kg^2) * ((2.14 ✕ 10^25 kg) * (4.05 ✕ 10^31 kg)) / (1.22 ✕ 10^12 m)^2]
F = G m M /R^2 = m v^2/R
note m, the planet mass, is irrelevant. If you are on a project orbiting earth and drop a wrench, it orbits beside you :)
v^2 = G M/R