Assignment:
Problem 1.
A) Sketch the path of an object in circular orbit with constant speed around the earth. At some point in the circular orbit indicate the direction of the velocity and acceleration of the object.
B) Why doesn't a satellite in circular orbit around the earth fall and crash into the earth's surface due to the gravitational pull of the earth?
Problem 2.
A) Find the weight of an astronaut of mass 60 kg when she is in a circular orbit 6.37 x 106 m above the surface of the earth. (That is she is two earth radii from the center of the earth.)
B) Why does this astronaut feel weightless?
C) Find the speed of the astronaut in this orbit.
Problem 3.
A) Find the radius of a geosynchronous orbit of a satellite around Mars. You will have to look up the relevant Mars data.
B) Find the radius of the circular orbit of a satellite that completes two orbits around the earth each day.
Problem 4. Look up the mass and radius of the moon.
A) Derive an expression for the acceleration due to gravity on the surface of the moon.
B) Find the escape speed from the surface of the moon.
C) Find the speed for a circular orbit of around the moon, if the orbital path is to have a radius of twice the radius of the moon.
Problem 5. Ignore the eects of air resistance.
A) Find the speed that a rocket must have, when the rocket is a distance of 3 earth radii from the center of the earth if it is to escape from the earth.
B) Find the maximum height reached by a rocket launched with a speed of 9,000 m/s vertically from the surface of the earth. Ignore the eects of air resistance.
C) A rocket is launched from the surface of the earth and has a speed of 2000 m/s when it is very far from the earth. With what speed was the rocket launched?
Problem 6. Ignore the eect of air resistance. Consider an object dropped from a height of 2 x 104 meters above the surface of the earth. (That's about twice the height of the tallest mountain above sea-level.)
A) Find the speed with which the object would strike the surface of the earth using the full gravitational potential energy UG = -G m ME/r.
B) Find an approximate speed with which the object would strike the surface of the earth using the approximate gravitational potential energy formula valid for motion near the surface of the earth, UG = m gEy.
C) Find the percent error introduced by using the approximate method of part B)
Problem 7. Repeat Problem 6 if the object is dropped from a height of 2 x 106 meters above the surface of the earth.
Consider an object of mass m that is a distance h above the surface of the earth.
A) Show that the size of the gravitational force on this object can be written as
FG = m gE (1+h/RE)-2
B) Recalling that for |x| «1, (1+x)n = 1 +nx + n(n-1)x2/2! + O(x3), show that, for h/RE « 1
FG = m gE (1-2h/RE+ O(h/RE)2)
C) Let h = 10, 000 m. Compare the gravitational force obtained using
1. the exact result (Equation (1)).
2. The approximate result (Equation (2)).
3. The result used in earlier chapters FG ≈ m gE.
(Keep enough signi cant digits to see the dierences in these results.)