We can use the conservation of momentum to solve. Since the satellites stick together, there is only one final velocity term.

We know the masses for both satellites are equal, and the second satellite is initially stationary.

Now we need to find the velocity of the first satellite. Since the satellite is in orbit (circular motion), we need to find the tangential velocity. We can do this by finding the centripetal acceleration from the centripetal force.

Recognize that the force due to gravity of the Earth on the satellite is the same as the centripetal force acting on the satellite. That means .

Solve for  for the satellite. To do this, use the law of universal gravitation.

Remember that  is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

Now that we know the force, we can find the acceleration. Remember that centripetal force is . Set our two forces equal and solve for the centripetal acceleration.

Now we can find the tangential velocity, using the equation for centripetal acceleration. Again, remember that the radius is equal to the sum of the radius of the Earth and the height of the satellite!

This value is the tangential velocity, or the initial velocity of the first satellite. We can plug this into the equation for conversation of momentum to solve for the final velocity of the two satellites.

The period describes how long it takes the satellite to make one full orbit. If you go back to the definition of velocity, , we can apply that to our new circular orbit, in which the distance is equal to the circumference of the circle and the time is equal to the period: . The circumference divided by the period will give us the average velocity.

The problem gives us the radius, but we need to find the tangential velocity. We can do this by first solving for the centripetal acceleration from the centripetal force.

Recognize that the force due to gravity of the Earth on the satellite is the same as the centripetal force acting on the satellite. That means .

Solve for  for the satellite. To do this, use the law of universal gravitation.

Remember that  is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

Now that we know the force, we can find the acceleration. Remember that centripetal force is . Set our two forces equal and solve for the centripetal acceleration.

Now we can find the tangential velocity, using the equation for centripetal acceleration. Again, remember that the radius is equal to the sum of the radius of the Earth and the height of the satellite!

We now have a value for the tangential velocity, which we can use in the equation for velocity from the beginning to find the period.

To solve, use Newton's law of universal gravitation:

We are given the values for the mass of each planet, as well as the distance (radius) between them. Using these values and the gravitational constant, we can solve for the force of gravity.

Remember that Newton's second law states that . The force acting upon the planet in question will be the force due to gravity. Once we find that, we can find the acceleration.

To solve, use Newton's law of universal gravitation:

Remember that the weight of the astronaut is the same as the gravitational force acting between the planet and the astronaut.

We are given the gravitational constant, the radius of the planet, the mass of the astronaut, and the magnitude of the force generated. Using these values in the universal gravitation equation, we can solve for the mass of the planet.

To solve this problem, use Newton's law of universal gravitation:

We are given the value of the force, the distance (radius), and the gravitational constant. We are also told that the masses of the two satellites are equal. Since the masses are equal, we can reduce the numerator of the law of gravitation to a single variable.

Now we can use our give values to solve for the mass.

There are two forces acting on the falling skydivers.  The first is the force of gravity (their weight) which we will denote as .  The second is the force of air resistance pushing up against them which we will denote as .

Newton’s 2nd Law tells us that the net force acting on these two objects is equal to their mass times their acceleration.

The net force is the sum of both forces acting together.

We know that the air resistance acts opposite to gravity and is equal to ¼ their total weight.

Substitute this value into our net force equation.

This simplifies to

The force of gravity is calculated by multiplying the mass by the acceleration due to gravity.

Substitute in our known variables and solve 

The first thing is to identify the forces involved in this situation.  There is the force of gravity (or her weight) which is pulling down on the rope.  We can calculate this by

The other forces are the force of Tension on each side of the wire as she stands in the midpoint.  These two Tension forces are what hold up Sally and keep her from falling.  However, these two Tensions forces are at an  angle below the horizontal.  This means that we need to analyze the components of the Tension force.  The -components of each Tension force are equal in magnitude and opposite in direction as this is what keeps the rope connected to both buildings.  The -components of each Tension force are equal in magnitude and in the same direction as they both are keeping Sally up.  So we can sum up the forces acting in the -direction as:

Which can be simplified to

Since Sally is not accelerating, the forces are balanced and the net force must equal .

Earlier we calculated the force of gravity so we can substitute this in to find the Tension in the  Direction.

This is the Tension in the -direction.  However, the problem is asking for the overall Tension in the wire.  At this point, we must use trigonometric functions to determine the hypotenuse (the overall Tension) in the wire.  Since the -component of the Tension is the opposite side of the triangle from the  angle, we can use cosine to find our hypotenuse.

 will be the weight of the object. Weight is a very specific force: it is the mass times gravity. As it turns out, the angle is irrelevant in finding weight.

Using Newton's second law and the given values for mass and gravity, we can solve for .

Note that the weight is negative, because it is acting in the downward direction.