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 density formula P=M/V, but i kept subbing in other formulas, but i get stuck..can someone help me? thanks.
I posted this before, but i made a mistake in the question and I am now fearing that whoever helped me will not go 3 pages into the history just to find this question and help me again. Thanks =P
Also, teacher gave me a hint that "r(radius)" was suppose to cancel during the formula manipulation process..
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
~Marianne
To calculate the period of a satellite orbiting just above the surface of a spherical planet, you can use the formula for the period of an orbiting object:
T = 2π√(r^3 / GM)
Where:
T - Period of the satellite (in seconds)
π - Pi, approximately 3.14159
r - Radius of the planet (in meters)
G - Gravitational constant, approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2
M - Mass of the planet (in kilograms)
To solve the problem, you need to gather the necessary information:
1. Obtain the density of the planet (p = 7860 kg/m^3) - This information is not directly needed to calculate the period.
2. Determine the radius of the planet (r) - This information is missing from the question. You need to know the radius in order to proceed with the calculation. Once you have the radius, be sure to convert it to meters if it's given in a different unit, such as kilometers.
3. Determine the mass of the planet (M) - Using the density and the formula for the volume of a sphere, you can find the mass of the planet.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius. Since the density is given as p = M/V, you can rearrange the equation to solve for M:
M = p * V
Substituting the formula for the volume of a sphere, you get:
M = p * (4/3)πr^3
Once you have the mass, you can proceed with calculating the period of the satellite using the formula mentioned earlier.