4A Homework Set 9
Univeral Law of Gravity
1. Tidal force: Two equal masses a distance L apart joined by a string are in free fall toward a planet.  The masses lie along a radial line from the planet's center.  Find the tension in the string.
2.  A particle is projected from the surface of the earth with a speed equal to twice the escape speed.  When it is very far away from the earth, what is its speed.  (Neglect air resistance.)
3.  Binary star - equal masses.  Two identical stars with mass M orbit around their center of mass.  Each orbit is circular and has radius R, so that the two stars are always on opposite sides of the circle.  (a)  Find the gravitational force of one star on the other.  (b) Find the orbital speed of each star and the period of the orbit.  (c) How much enery would be required to separate the two stars to infinity?
4.  Planet X rotates in the same manner as the earth, around an axis through its north and south poles, and is perfectly spherical.  An astronaut who weights 943.0N on the earth weights 915.0N at the north pole of Planet X and 850.0N at its equator.  The distance from the north pole to the equator is 18,850 km., measured along the surface of Planet X.  (a)  How long is the day on Planet X?  (b) if a 45, 000kg satellite is placed in a circular orbit 2000km above the surface of Planet X, what will be its orbital period.  
5.  Three identical bodies of mass M are located at the vertices of an equilateral triangle with side L.  At what speed must they move if they all revolve under the influence of one another's gravity in a circular orbit circumscribiing the triangel while still preserving the equilateral triangle?  
6. An exploding star: A planet of mass m is in a circular orbit about a star of mass M at an initial distance of r. The non-rotating star then explodes and ejects half of its mass radially outward in a symmetric fashion with none of its ejecta hitting the planet. After the star's explosion find the new radius of the planet about the star.
7. Get out of Dodge: An object of mass m has an orbital radius R about a planet of mass M. Find the additional speed it would need from its orbit to escape to infinity with a zero final kinetic energy.
8. Journey to the moon: Given is the mass of the earth and moon, the radius of the earth and moon, and the distance between their centers. Find the minimum speed required for an object launched from the Earth to just make it to the moon.


7.  A good place for an observatory: Although a single planet of given mass m in orbit about let's say, a star of mass M, with a known orbital radius can have only one period of its orbit for that given radius, if a third object were placed at certain special points within this system (i.e. the Lagrange points), then this third object object (with two forces of gravity acting on it from the star and then the planet) can have the same period of orbit as the planet has about the star. Find the radius of this third object from the star such that this could occur. One position would be in between the star and the planet and another position would be outside of the planet and star; both of those positions would be along a line joining the star and planet at any given instant. Find these two points. There are other Lagrange points but that's not this problem.