3. Three lead balls of mass The region of the solar system between Mars and Jupi

Question

3.Three lead balls of mass

The region of the solar system between Mars and Jupiter contains many asteroids that orbit the Sun. Consider an asteroid in a circular orbit of radius 5.3 times 10 11 m. Find the period of the orbit. In recent years, a number of nearby stars have been found to possess planets. Suppose, the orbital radius of such a planet is found to be 4.2 times 10 11 m, with a period of 1140 days. Find the mass of the star. Three lead balls of mass m1 = 14 kg, m2 = 28 kg, and m3 = 8.9 kg are arranged as shown in the figure below. Find the total gravitational force exerted by balls 1 and 2 on ball 3. Be sure to give the magnitude and the direction of this force.

Explanation / Answer

1. P = 2*pi*sqrt( a^3 / (GM) )

Where:

a is the semi-major axis (or radius, when speaking loosely)

G is the gravitational constant

M is the mass of the central body

= 2*pi*sqrt( (5.3*10^11)^3 / (6.67*10^(-11)*1.99*10^30) )

= 6*10^7 seconds

2. P = 2*pi*sqrt( a^3 / (GM) )

P = 9.85*10*7 seconds

So, M = 4.52*10^30 kg

Mass of star = 4.52*10^30 kg

3. Gravitational force = GM1M2/r^2

For M1 and M3,

8.86*10^-10 i - 1.478*10^-10 j

For M2 and M3,

2.61*10^-9 i + 2.61*10^-9 j

Total force = 3.5*10^-9i + 2.46*10^-9 j

Magnitude = 4.3*10^-9 N

Direction 35.1 deg

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