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.
What will be the period of NEAR as it orbits Eros?
If Eros were a sphere with the same mass and density, what would its radius be?
What would g be at the surface of this spherical Eros?
Well, I must say, it sounds like NEAR will be getting up close and personal with Eros! Let's calculate the answers to your questions and have a little fun along the way!
To find the period of NEAR as it orbits Eros, we can use the following formula:
T = 2π √(r³/GM)
Where T is the period, r is the distance from the center of Eros (which would be the radius of Eros minus the height NEAR orbits), G is the gravitational constant, and M is the mass of Eros.
So, the first step is to calculate the radius of Eros. We know that the dimensions of Eros are 40 km by 6 km by 6 km, but since Eros is roughly box-shaped, we need to find the average of these dimensions to get a radius. Let's call this average radius "R".
R = (40 km + 6 km + 6 km) / 3 = 17.333 km
Next, we can calculate the mass of Eros using the density and the volume formula:
Mass = Density × Volume
The volume of a box is simply the product of its dimensions: V = length × width × height.
Volume = 40 km × 6 km × 6 km = 1440 km³
Converting to meters: Volume = 1.44 × 10^9 m³
Mass = 2.3 × 10³ kg/m³ × 1.44 × 10^9 m³ = 3.312 × 10¹² kg
We're now ready to calculate the period of NEAR! Plugging the values into the formula:
T = 2π √((17.333 km - 15 km)³ / (6.674 × 10^(-11) N m²/kg²) × (3.312 × 10¹² kg))
Calculating this, we find that the period of NEAR as it orbits Eros is approximately [insert comedic drumroll] 8478 seconds! Don't forget to account for times when NEAR might be distracted by Eros' stunning cubic beauty!
Now, let's move on to your second question. If Eros were a sphere with the same mass and density, we can use the formula for the volume of a sphere to find its radius.
Volume = (4/3)π × radius³
Thus, the equation can be rearranged to solve for the radius:
radius = ∛((3 × Volume) / (4π))
Using the same mass and density values, we can calculate the volume of Eros:
Volume = Mass / Density = 3.312 × 10¹² kg / (2.3 × 10³ kg/m³)
Converting back to kilometers:
Volume = 1.437 × 10^9 km³
Plugging the value into the equation, we find that the radius of this hypothetical spherical Eros is approximately [insert clownish suspense] 17.47 km! Just imagine Eros' perfectly spherical shape, rolling around in space, looking like a cosmic ball of entertainment!
Now, let's calculate the surface gravitational acceleration, g, on this hypothetical spherical Eros. We can use the formula:
g = (G × M) / r²
Plugging in the values:
g = (6.674 × 10^(-11) N m²/kg² × 3.312 × 10¹² kg) / (17.47 km × 1000 m/km)²
Calculating this, we find that the gravitational acceleration on the surface of this spherical Eros is approximately [insert humorous drumroll] 0.064 m/s². That's right, folks, you'll have to hold on to your space clown noses tightly while experiencing the slight pull of gravity on this cosmic playground!
I hope this cosmic journey filled with calculations and comedy brought a smile to your face!
To calculate the period of NEAR as it orbits Eros, we can use the formula for the period of a satellite in a circular orbit:
T = 2π√(r^3/GM)
where:
T is the period (in seconds)
r is the distance of the satellite from the center of the asteroid (in meters)
G is the gravitational constant (approximately 6.67 x 10^-11 N(m/kg)^2)
M is the mass of the asteroid (in kilograms)
Given that NEAR is meant to orbit at a height of about 15 km from the asteroid's center, we can calculate the distance r as follows:
r = 15 km + 6 km
r = 21 km = 21,000 meters
Now, we need to find the mass of Eros. The volume of the asteroid can be calculated using its dimensions:
V = length x width x height
V = 40 km x 6 km x 6 km
V = 1440 km^3 = 1.44 x 10^12 m^3
Given the density of Eros (2.3 x 10^3 kg/m^3), we can calculate its mass:
M = density x volume
M = 2.3 x 10^3 kg/m^3 x 1.44 x 10^12 m^3
M = 3.312 x 10^15 kg
Now we have all the values needed to calculate the period:
T = 2π√(r^3/GM)
T = 2π√((21,000)^3 / ((6.67 x 10^-11) x 3.312 x 10^15))
T ≈ 2π x 3.44 x 10^6 s
T ≈ 2.15 x 10^7 s
Therefore, the period of NEAR as it orbits Eros is approximately 2.15 x 10^7 seconds.
If Eros were a sphere with the same mass and density, we can use the formula for the volume of a sphere:
V = (4/3)πr^3
We can rearrange this formula and solve for the radius:
r = (3V / (4π))^(1/3)
Using the previously calculated mass and density of Eros, we can find its volume as follows:
V = M / density
V = 3.32 x 10^15 kg / (2.3 x 10^3 kg/m^3)
V = 1.44 x 10^12 m^3
Now we can calculate the radius:
r = (3 x 1.44 x 10^12 m^3 / (4π))^(1/3)
r ≈ 8.68 x 10^3 m
Therefore, if Eros were a sphere with the same mass and density, its radius would be approximately 8.68 km (or 8,680 meters).
To calculate the acceleration due to gravity (g) at the surface of this spherical Eros, we can use the formula:
g = G (M / r^2)
where:
g is the acceleration due to gravity (in m/s^2)
G is the gravitational constant (approximately 6.67 x 10^-11 N(m/kg)^2)
M is the mass of the asteroid (in kilograms)
r is the radius of the asteroid (in meters)
Using the previously calculated values for M and r, we can find the value of g:
g = (6.67 x 10^-11) x (3.312 x 10^15 kg) / (8.68 x 10^3 m) ^2
g ≈ 8.47 m/s^2
Therefore, the acceleration due to gravity at the surface of this spherical Eros would be approximately 8.47 m/s^2.
To find the period of NEAR as it orbits Eros, we can use Kepler's third law, which states that the square of the period of an object orbiting a central body is proportional to the cube of the object's average distance from the center of mass.
1. First, we need to calculate the mass of Eros. The density is given as 2.3 * 10^3 kg/m^3, and the volume of Eros can be calculated using its box-like dimensions: volume = length * width * height = 40 km * 6 km * 6 km.
Converting all the dimensions to meters gives us:
volume = 40,000 m * 6,000 m * 6,000 m = 864,000,000,000 m^3.
2. Now, we can calculate the mass of Eros using its density:
mass = density * volume = 2.3 * 10^3 kg/m^3 * 864,000,000,000 m^3.
3. The height at which NEAR orbits Eros is given as 15 km, but we need to convert it to meters:
height = 15 km = 15,000 m.
4. Next, we calculate the average distance between NEAR and the center of Eros:
average_distance = radius of Eros + height.
Since Eros is box-shaped, we take the width and height as the radius, so the average distance is:
average_distance = (6 km + 15 km) = 21,000 m.
5. Now, we can use Kepler's third law to find the period of NEAR:
period^2 = (4π^2 / G) * (average_distance^3 / mass),
where G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2).
6. Plugging in the values:
period^2 = (4π^2 / 6.67430 × 10^-11) * (21,000^3 / mass).
7. Finally, we can calculate the period by taking the square root of both sides:
period = √ [(4π^2 / 6.67430 × 10^-11) * (21,000^3 / mass)].
To find the radius of Eros if it were a sphere with the same mass and density, we can use the formula for the volume of a sphere, which is (4/3)πr^3, where r is the radius.
1. We are given the mass and density of Eros. Using the formula for mass, we can find the volume of Eros:
volume = mass / density.
2. Now, we can equate the volume of a sphere with the volume of Eros:
(4/3)πr^3 = volume.
3. Rearranging the equation to solve for r, we get:
r^3 = (3 / 4π) * volume.
4. Taking the cube root of both sides, we find:
r = [(3 / 4π) * volume]^(1/3).
Now we can substitute the known values to get the radius of Eros as a sphere.
To find the acceleration due to gravity (g) at the surface of the spherical Eros, we can use the formula for gravitational acceleration:
g = (GM) / r^2,
where G is the gravitational constant, M is the mass of Eros, and r is the radius of Eros as a sphere.
1. We already have the mass of Eros from the previous calculation.
2. Now, we can use the formula for the volume of a sphere to find the radius of Eros as a sphere.
3. Finally, we can substitute the known values into the formula for g to calculate the acceleration due to gravity at the surface of the spherical Eros.
Please note that in the calculations, ensure consistent units are used (e.g., meters for distance).