Orbits

Newton's second law, the principle of inertia states that an object tends to remain in the state of rest or uniform speed. This is known as the inertia of the body.

We can experience the effects of inertia when we are traveling forward and we brake or try to stop. Our body, due to inertia, tends to continue going forward. In the same way, when we accelerate, we tend to "sink" in our seats, again due to inertia.

The same happens, but now with more dramatic effects, when we take a curve. The object tends to continue in its straight course, and when we start turning we feel a force that pulls us away from the center of the curve. This is an inertial force, called the centrifugal force, which always points in a direction opposite to the center of the curve.

We can feel this force when riding a bicycle. We are forced to tilt the bicycle in order not to be overturned by the centrifugal force.

The magnitude of this centrifugal force is related to the mass of the object, its speed and the radius of the curve. The expression is the following :

where Fc : the centrifugal force

m : the mass of the object

V : the speed of the object

R : the radius of the curve

Analyzing the formula, several conclusions can be formulated. The higher the speed, the bigger the centrifugal force will be. We know from our own experience, from driving a car or riding a bicycle, that in order to negotiate a curve successfully the speed must be reduced.

We can also see that if the radius is small, the force will increase. A curve with a small radius is what we call a sharp turn, while a curve with a large radius is a smooth one. The sharper the turn, the greater the force will be.

When spacecraft go into outer space, they frequently circumnavigate the Earth in what are called orbits. An orbit is a circular or an elliptical path around the Earth.

In an orbit, two basic forces act over the object : its weight, pulling it down towards the center of the Earth, and the centrifugal force, pointing away from the center of the orbit, in this case, the center of the Earth. If the weight is bigger than the centrifugal force, the object will fall back into the Earth and so reduce the radius of its orbit. If the centrifugal force is greater than the weight, the craft will tend to escape from the Earth and so the radius of its orbit will grow. If the weight is equal to the centrifugal force, the object will remain in equilibrium and, as there is no friction in space, maintain its orbit endlessly.

Velocity needed to achieve orbit

If we consider any given body of mass m and the Earth, the formula for the attractive force would be :

W is the weight (the weight is obviously the force with which the Earth attracts any body or object) and R the radius of the Earth, supposing that all the Earth's mass is concentrated at its center.

According to Newton's first law :

The acceleration can be determined by dividing force by mass:

In the case of weight, the acceleration would correspond to gravity :

For an orbit to remain stationary, the centrifugal force must equal the gravitational attraction of the Earth (weight) :

where R is the radius of the Earth and r the radius of the orbit above Earth's surface (see figure)

Weight was :

(a person's weight would descend in a higher orbit, because g will decrease with height)

Equaling the centrifugal force with weight :

Finding out V :

which is finally the expression for the velocity V needed to obtain an orbit of radius r

The following graph shows the relationship between the height of the desired orbit and the velocity needed
:

Applying the above expression, try to determine the velocity needed to achieve a certain orbit and, inversely, the height obtained with a certain velocity for Low Earth Orbit

Velocity given the radius :

Radius of the orbit given the velocity :