A satellite moving in a circular orbit with respect to the earth's center experiences a gravitational force. If the satellite is put into a new circular orbit of smaller radius, how will the gravitational force and the speed of the satellite change ,if at all?

To understand how the gravitational force and the speed of a satellite change when it is put into a new circular orbit of smaller radius, we need to consider the relationship between the force of gravity, orbital radius, and orbital velocity.

1. Gravitational Force: The force of gravity between two objects (in this case, the satellite and the Earth) is given by Newton's law of universal gravitation:

F = G * (m1 * m2) / r^2

Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.

When the satellite is put into a new orbit of smaller radius, the distance between the satellite and the Earth's center decreases. As a result, the gravitational force will increase because the denominator in the equation (r^2) will become smaller.

2. Orbital Velocity: The orbital velocity of a satellite is the speed required to maintain a stable circular orbit around the Earth. It is determined by the centripetal force, which is provided by the gravitational force:

F = (m * v^2) / r

Where F is the gravitational force, m is the mass of the satellite, v is the orbital velocity, and r is the orbital radius.

When the satellite is put into a new orbit of smaller radius, the orbital velocity needs to increase in order to maintain the same centripetal force. This is because the smaller radius results in a shorter distance for the satellite to cover in each orbit, hence requiring a higher speed.

So, to summarize:

- The gravitational force will increase when the satellite is put into a new circular orbit of smaller radius.
- The orbital velocity of the satellite will also increase when it is put into a new circular orbit of smaller radius.

It's important to note that these changes depend on the assumption that the mass of the satellite and the Earth remains constant.