A hypothetical spherical planet consists entirely of iron (p=7860 kg/m^3). Calclate the period of a satellite that orbits just above its surface.
I started out with the pressure formula P=M/V, but i kept subbing in other formulas, but i get stuck..can someone help me? thanks.
You need to know the radius of the planet to answer this.
I have no idea what your "pressure formula" means. It is incorrect, and pressure has nothing to do with this problem.
Im sorry! I meant density! It was represented by a small p, so i thought pressure >_<
Anyways, density = mass/volumne. My teacher gave me a hint about radius cancelling each other off in the formula manipulation. Hope this helps!
To calculate the period of a satellite that orbits just above the surface of a planet, you need to consider the gravitational force acting on the satellite. The period of an orbiting object is defined as the time it takes to complete one full revolution around the planet.
In this case, you are given that the planet is spherical and consists entirely of iron. The density of iron is given as p = 7860 kg/m^3.
To start, you can use the formula for the gravitational force between two objects:
F = G * (M1 * M2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.674 x 10^-11 N(m/kg)^2)
M1 and M2 are the masses of the two objects involved (in this case, the planet and the satellite)
r is the distance between the centers of the two objects (in this case, the radius of the planet)
Since the satellite orbits just above the surface of the planet, the distance between their centers is equal to the radius of the planet. Therefore, r is the radius of the planet.
To determine the mass of the planet, you can multiply the volume of the planet by its density:
M1 = (4/3) * π * r^3 * p
Next, you can relate the gravitational force to the centripetal force acting on the satellite. The centripetal force is given by the formula:
F = (M2 * v^2) / r
Where:
v is the orbital velocity of the satellite
Equating the gravitational force to the centripetal force, we can solve for the orbital velocity, v:
G * (M1 * M2) / r^2 = (M2 * v^2) / r
Simplifying the equation and solving for v:
v^2 = (G * M1) / r
Now that you have the orbital velocity, you can calculate the period of the satellite's orbit. The orbital period, T, is given by the formula:
T = (2π * r) / v
Substituting the value of v obtained above, we get:
T = (2π * r) / sqrt((G * M1) / r)
Finally, substitute the value of M1 (calculated earlier) and the given radius of the planet to obtain the final answer.
Remember to use consistent units throughout the calculations.
Time required for 1 revolution,
T= (2πr^(3/2))/√(G∙M) (p. 146 of 7th ed Cutnell & Johnson)
Volume of a sphere,
V=4πr^3 (book inside cover)
Formula for mass density,
M=ρ∙V (p.321)
Where
T: the time period
G: universal gravitational constant, 6.673 E-11 (N∙m^2)/kg^2
note: N=(kg∙m)/s
r: distance from center of planet to satellite, aka planet’s radius
ρ: mass density of planet.
Iron density = 7860 kg/m^3
V: Volume of planet
Substitute formula for volume of a sphere into equation for mass:
M = ρ∙ 4πr^3
Then substitute this into the equation for time period
T= (2πr^(3/2))/√(G∙ρ∙ 4πr^3)
The r^(3/2) in the numerator cancels the √(r^3) in the denominator,
So the equation simplifies to
T= 2π/√(G∙ρ∙4π)
Plug in the known value of G and given value of ρ.
For a planet made of iron, satellite period
T ≈ 2447.39 seconds