Two large spherical stars with masses of M and 2M are positioned a distance D ap

Question

Two large spherical stars with masses of M and 2M are positioned a distance D apart (measured from to the center of one star to the center of the other star) as shown. A small spherical asteroid with mass m is located with its center exactly halfway between the two large stars. Find the magnitude and direction of the total gravitational force acting on the asteroid. A small probe of mass m is released from rest at a distance of 2R above the surface of a spherical planet of mass M and radius R. Find the speed probe will have when it hits the surface of the planet. Assume that the planet does not have an atmosphere and no forces other than gravity act on probe. Assume that the probe is released at the same distance from the planet but now with a velocity of v_0 = squareroot GM/4R. The probe has a rocket which fires to put it into a circular orbit at that height. Find the speed the probe would have once it is in the circular orbit and, using that, find the work done by the rocket. Assume that, all times, the probe is at the same distance from the planet and that there are no forces acting on the probe other than the planet's gravity and the rocket.

Explanation / Answer

force of mass M on asteroid= F1= GM m /(d/2)2=4GM m/d/2

this force f1 is towards M

force of mass 2M on asteroid= F1= G(2M) m /(d/2)2=8GM m/d/2

F2 is towards 2M

so F1 and F2 are in opposite directions,so the net force on asteroid= F=F2-F1

F=8GM m/d/2-4GM m/d

F=4GM m/d/2

the direction of net force is towards 2M

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