Extended facts about orbits

In November when I *first* introduce Newton's Law of Gravitation and orbits, I keep things very simple.  I give the equation for gravitational force, define a gravitational field, and that's about it.  

We do a practice exercise where students must use circular motion facts to determine how the speed of a satellite in orbit depends on various parameters.  (We test predictions using an online simulation.)  You can see me do this exercise in AP classroom, on 3.8 Daily Video 1.

There's more to understanding orbits.  But students weren't ready for that "more" back in November.  Now they are.

In March as we begin our "tapering" review toward the May 12 exam, I want to revisit topics we've discussed previously, but in greater depth.  With regard to orbits, back in November we didn't know about energy or angular momentum!  Now we do.  So let's go back and discuss orbits again, this time with EVERYTHING we can think of.

Below are the NEW facts I'm bringing up, facts I ignored earlier in the course, alongside some commentary.

Two Types of mass:

Gravitational mass is measured using any relationship that involves a gravitational field or force.

Inertial mass is measured using any relationship that does NOT involve g, such as netF = ma or 

.

In all experiments ever performed, gravitational mass is equal to inertial mass.

Early on, I'm excited for students to simply understand that 100 kg is not a force, and that a 100 kg object on the moon has a mass of 100 kg.  There's no way I'm gonna try to explain the very subtle difference between gravitational and inertial mass!  Now, though, it's okay to add this subtlety on.

Gravitational potential energy

Near the surface of a planet, the potential energy of a planet-object system is mgh, with h = 0 at the lowest point of the motion.

Away from the surface, the potential energy of a planet-object system is treated differently:

·         PE is larger the farther from the planet’s center.

·         PE has a negative value (except when the object is way far away from the planet, in which case PE is zero).

·         The equation for potential energy is

Don’t use this equation unless you must derive an expression.  The negative sign is confusing.

That last italic piece is important.  Most AP1 orbits questions are not going to use this actual equation!  The first bullet is the big deal... planets getting farther away from one another increase their potential energy.  That means the potential energy gets closer to zero, gets less negative... but that subtlety is beyond many of my students.  Not to worry!  These facts will allow for a correct and clear answer to virtually any gravitational PE problem.

Orbits

In a circular orbit of a satellite around a planet, consider the planet-satellite system:

·         Kinetic energy is constant (same speed)

·         Gravitational potential energy is constant (same orbital radius)

·         Angular momentum mvr is constant (no external torques)

·         Total mechanical energy is constant (no external work, and no internal energy)

 

To find the speed of a circular orbit, set gravitational force

 equal to ma, with

In an elliptical orbit of a satellite around a planet, consider the planet-satellite system:

      ·         Kinetic energy is NOT constant (speed changes)

      ·         Gravitational potential energy is NOT constant (orbital radius changes)

      ·         Angular momentum mvr is constant (no external torques)

   ·         Total mechanical energy is constant (no external work, and no internal energy) 

Now that the class is well familiar with the concepts of energy and angular momentum, I want to be explicit about conservation in orbits.  This set of facts is also a good reminder/reinforcement of the statements elsewhere in my fact sheets about the conditions under which mechanical energy and angular momentum are conserved!