Newton's second law and universal gravitation law we can the answers are:
a) r = 7.3 10⁴ m
b) T = 1.53 10⁴ s
Newton's second law states that force is proportional to slats and the acceleration of the body,
F = m a
The universal gravitational law says that the force between two bodies is proportional to their plus and inversely proportional to the distance between them
F = [tex]G \frac{Mm}{r^2}[/tex]
Where the bold letters indicate vectors and F is the force, G the universal gravitation constant (G = 6.67 10⁻¹¹ N m² /kg²), M the most of the asteroid, m the most of the ball and r the separation between the center of the asteroid and the ball
a) The acceleration is centripetal since the ball rotates around the asteroid in a circular orbit
a = v² / r
We substitute
[tex]G \frac{Mm}{r^2} = m \frac{v^2}{r}[/tex]G Mm / r² = m v² / r
[tex]G \frac{M}{r}[/tex] = v²
Density is defined by the relationship between the mass and the volume of a body
ρ = [tex]\frac{M}{V}[/tex]M / V
M = ρ V
In this case it indicates that the asteroid is circular, so the volume (V) of a sphere is
V = [tex]\frac{4}{3} \pi r^3[/tex]
M = [tex]\rho \ \frac{4}{3} \pi r^3[/tex]
In this case we have assumed that the distance to the ball and the radius of the asteroid are equal
we substitute
[tex]\frac{4}{3} \pi \ G\ \rho r^2 = v^2[/tex]
r² = [tex]\frac{3}{4} \ \frac{v^2} {\pi \ G \ \rho }[/tex]i
Let's calculate
r² = ¾ [tex]\frac{30^2}{ \pi \ 6.67 \ 10^{-11} \ 2400}[/tex]
r = [tex]\sqrt{ 0.536879 \ 10^{10}}[/tex]
r = 0.7327 10⁵ m
r = 7.3 10⁴ m
b) As the motion of the ball is circulating, the modulus of its velocity is constant, for which we can use the kinematic relations of the uniform motion
v = d / T
T = d / v
Where d is the distance and T the period of the orbit
The distance of a circle is
d = 2π r
d = 2π 7.3 10⁴
d = 4.60 10⁵ m
we calculate
T = 46.0 10⁴ / 30
T = 1.53 10⁴ s
In conclusion using Newton's second law and the universal law of gravitation we can the answers are:
a) r = 7.3 10⁴ m
b) T = 1.53 10⁴ s
Learn more about the universal gravitation law here:
https://brainly.com/question/858421
