Assume you are agile enough to run across a horizontal surface at 11.60 m/s on an airless spherical asteroid of uniform density 1.0x103 kg/m3, independently of the value of the gravitational field. To launch yourself into orbit by running, what would be (a) the radius? (b) the mass the asteroid? (c) What would be your period?
ma=F(grav)
mv²/R =GmM/R²
G •ρ•4πR³/3• m/R² =m(v²/R)
R=sqrt{3v²/4πρG}=...
M = ρ•4πR³/3= ….
T=2π/ω=2πR/v=....
To solve this problem, we need to use the principles of physics, specifically centripetal force and gravitational force.
(a) To find the radius of the asteroid, we can start by considering the centripetal force required to keep you in orbit while running. The centripetal force is provided by the gravitational attraction between you and the asteroid. The formula for centripetal force is:
F = m * v² / r
Where:
F = Centripetal force
m = Mass of the object (you)
v = Velocity (11.60 m/s)
r = Radius of the orbit
Since you want to launch yourself into orbit, the centripetal force must be equal to your weight. Your weight can be calculated using the formula:
Weight = mass * gravitational field strength
Since the value of gravitational field strength is independent of the value, we can ignore it in this calculation. Therefore, your weight is equal to:
Weight = m * g = m * 9.8 m/s²
Equating the centripetal force and weight, we get:
m * v² / r = m * 9.8
Canceling out the mass (m) from both sides, we have:
v² / r = 9.8
Rearranging the equation to solve for r:
r = v² / 9.8
Substituting the given velocity (v = 11.60 m/s) into the equation, we have:
r = (11.60 m/s)² / 9.8
Calculating this, we get:
r ≈ 13.74 meters
Therefore, the radius of the asteroid would be approximately 13.74 meters.
(b) To find the mass of the asteroid, we need to consider the density and the volume of the asteroid. The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r³
Substituting the radius we found previously (r ≈ 13.74 meters) into the equation, we get:
V = (4/3) * π * (13.74 meters)³
Calculating this, we get:
V ≈ 11733.88 cubic meters
The mass of the asteroid can then be found by multiplying the volume by the density:
mass = density * volume
Substituting the given density (density = 1.0x10³ kg/m³) and the calculated volume, we have:
mass = (1.0x10³ kg/m³) * 11733.88 cubic meters
Calculating this, we get:
mass ≈ 1.1734x10⁷ kilograms
Therefore, the mass of the asteroid would be approximately 1.1734x10⁷ kilograms.
(c) To find the period of your orbit, we can use the formula for the period of a satellite in circular motion:
T = 2π * r / v
Substituting the calculated radius (r ≈ 13.74 meters) and the given velocity (v = 11.60 m/s), we have:
T = 2π * 13.74 meters / 11.60 m/s
Calculating this, we get:
T ≈ 22.5 seconds
Therefore, the period of your orbit would be approximately 22.5 seconds.