The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km from the asteroid's center. Eros is roughly box-shaped, with dimensions 40 km 6 km 6 km. Assume Eros has a density (mass/volume) of about 2.3 103 kg/m3.
(a) What will be the period of NEAR as it orbits Eros?
(b) If Eros were a sphere with the same mass and density, what would its radius be?
(c) What would g be at the surface of this spherical Eros?
what is 15E3 and how do you know the mass of Near?
To answer the questions, we need to apply the principles of celestial mechanics and gravitation. Let's go through each question step by step:
(a) To calculate the period of NEAR as it orbits Eros, we can use Kepler's third law, which states that the square of the orbital period is proportional to the cube of the average distance between the objects.
1. Calculate the average distance between NEAR and the center of Eros:
Average distance = height + (length/2) = 15 km + (40 km/2) = 35 km
2. Convert the average distance to meters:
Average distance = 35 km = 35,000 meters
3. Now, we can use Kepler's third law to find the period of the orbit:
T^2 = (4π^2 * r^3) / (G * M)
Where T is the period, r is the average distance, G is the gravitational constant, and M is the mass of Eros.
Gravitational Constant (G) = 6.67430 * 10^-11 m^3 kg^-1 s^-2
Mass of Eros = Density * Volume
Volume of Eros = width * length * height = 6 km * 40 km * 6 km = 1440 km^3
Mass of Eros = Density * Volume = 2.3 * 10^3 kg/m^3 * 1440 km^3 * (10^3 m/km)^3
Plug in the values into the equation to find the period:
T^2 = (4π^2 * (35,000)^3) / (6.67430 * 10^-11 * 2.3 * 10^3 * 1440 * (10^3)^4)
Calculate T by taking the square root of the right-hand side of the equation.
(b) If Eros were a sphere with the same mass and density, we can calculate its radius by assuming it has a uniform density. The formula for the average density of a sphere is:
Density = Mass / (4/3 * π * r^3)
In this case, we have the density and the mass (calculated in part (a)) of Eros. We can rearrange the equation to solve for the radius (r):
r = (3 * Mass / (4 * π * Density))^(1/3)
(c) To find the acceleration due to gravity at the surface of the spherical Eros, we can use the formula for gravitational acceleration:
g = (G * M) / r^2
Where G is the gravitational constant, M is the mass of Eros (calculated in part (a)), and r is the radius of Eros (calculated in part (b)).
Now, using the formulas and calculations explained above, you can find the answers to parts (a), (b), and (c) of the question.
a. find the volume of the box, then knowing density, find the mass.
So now, the period.
centripetal force= gravitational force
massNear*v^2/15E3=GMassEros*massNEAR*/(15E3)^2
you have all that , solve for V. then,
V= 15E3*2PI/Period
solve for period.
15E3 meters = 15km= 15*10^3 meters
Mass Near= volume*density=XxYxZ*density
you are given those dimensions.
billy millie
I would say m=v2-Fc
So, r2xFc3=M2g