The Massive Problem of Inertia

June and July are tough observing months, for amateurs at least.  With the sky not turning dark until well into the night hours, not to mention the bugs and generally cloudy or hazy skies, this is the time of year that I turn into more of an armchair astronomer.  My last post brought us to consider our dizzy motion through the Universe, with the realization that nothing stands still, and there are no really “fixed stars” against which to measure motion.  This time around, I want to lead you to a deeper understanding of how everything is truly interconnected, by considering nothing more abstract than an apple held in your hand.

Hold an apple in your hand.  (Of course, any object of reasonable size will do – I suppose the Apple could be an I-Pod).  The apple rests on the palm of your hand, and its weight can easily be felt as your muscles need to exert force to keep your hand and its apple from falling to your side.  If you can’t feel the weight, try keeping that apple in your outstretched hand for 5 minutes.  What creates the weight is the force of Earth’s gravity pulling the apple toward Earth’s center. 

The amount of the force is proportional to the mass of the apple, times the mass of the Earth.  Put a second apple in your hand, and the force is doubled, as you now are holding twice the mass.  The force also varies inversely as the distance between the center of the apple and the center of the Earth squared.  (A slight confusion is possible here, as the mass of the Earth needs to stay closer to its center than the apple for this to be true).  So, if you take your apple up about 4000 miles – about twice your normal distance from Earth’s center – it will weigh a quarter what it weighs now.

“But wait a minute, the astronauts on the space station are weightless, and they’re only 230 miles up, how can you be right?”  Well, here comes a detour, but it’s an important one. 

If I take the apple and throw it across my yard, it will go up maybe 30 feet, turn around and fall back to the ground about 100 feet away from me.  If I put the apple in a cannon (um, inside a hollow cannon ball, let’s say) and fire the cannon, the apple might go 300 feet in the air, and fall back to Earth 2000 feet away.  Put the apple in a small rocket, and it may go a mile high in the air, falling back to Earth 10 miles away.  In a larger missile, 50 miles high, falling thousands of miles away.  Finally, if I put it in a spacecraft and launch it “into space”, it will reach 100’s of miles in height, and will fall… off the “end” of the Earth!  The spacecraft will then go into orbit around the Earth, which is the same as constantly “falling”.

The term “weightless” is very misleading.  The proper term is “free fall” – meaning a state in which an object is falling and is completely unsupported.  You can experience free fall – or a good approximation – at almost any amusement park on a roller coaster, or even on the “Free fall” ride, where you are dropped vertically through a couple dozen feet.  At no time do you lose your “weight”, your weight is the reason you fall.  A spider crawling on the apple that I threw across the yard experienced free fall during his trip.

Ok, before we get nauseous, let’s get back to our nice apple sitting in the palm of our hand.  Its weight is caused by its mass and Earth’s mass pulling toward each other.  But what exactly is this thing called “mass”?  Well, to be honest, I don’t think anyone can answer that question.  However, there is another way of trying to define it. 

All objects resist motion.  Take your apple and swing it back and forth in your hand in front of you.  Notice that you need to apply a very small amount of force to change the direction of motion each time you switch from left to right to right to left.  As before, if you don’t feel that force, try swinging it back and forth for 5 minutes and see if you get tired.  Since you are not raising and lowering the apple, it is not gravity that you are feeling each time you change the apple’s direction.  What you are feeling is the apple’s inertia. 

Inertia is the resistance of an object to a change in its motion, and in particular, to the start of motion.  Inertia is again measured by a property named “mass” – the larger the mass, the larger the force needed to overcome its inertia and start (or change) its motion.

But this inertia is not related to “weight”.  In free fall, when objects appear to be “weightless”, they still have inertia.  Otherwise, the Space Shuttle would not have needed a crane to lift objects into and out of its cargo bay – the astronauts could have done it, or even an ant.  In fact, if an object had no inertia, any force at all would cause it to fly away at “infinite” speed.  An object will still have the same inertia no matter where in the universe it is located – whether on the surface of huge planet, or in the deepest stretches of empty space between the galaxies.

So we have gravitational force which depends on an object’s mass, and inertia, which depends on something we also call “mass”.  And we see that even when an object is not near a large object – and therefore has no “weight” – it will still have inertia.  To make the distinction clear, we call the mass that creates weight the “gravitational mass”, and the mass that causes inertia “inertial mass”.  But are these two masses the same?

This was a – very – big question in the physics of the 1800s, and even well into the 20th century.  Using very delicate equipment that allowed the measurement of gravitational attraction between masses in a lab, and highly accurate measurements of the inertia of objects, the equivalence of gravitational and inertial mass was measured to higher and higher degrees of accuracy under a host of variable conditions.  Objects were heated and cooled to extreme temperatures, subjected to intense magnetic and electric fields, crushed under immense pressures, and in every case the inertial and gravitational masses were found to be the same, to the limits of measurement.  And yet there was no explanation as to why this equivalence should hold.

And now we return to astronomy.  Throughout the 20th century, we have become increasingly aware of the vastness of the universe.  Our solar system is one of hundreds of billions of solar systems in the Milky Way, the Milky Way is one of dozens of galaxies in the Local Group of galaxies, the Local Group is one of hundreds of galaxy groups within the Virgo Supercluster of galaxies, and there are millions of superclusters. 

Though the matter in the universe is distributed in clumps on a “local” scale (the galaxies, groups and superclusters), on a very large scale we can say that wherever in the universe we look, the structure appears about the same.  In the science of cosmology – the study of the structure and behavior of the universe – this fact is known as the cosmological principle.  That principle states that – on a large enough scale – the matter in the universe is evenly distributed in all directions.

Let’s come back to our simple apple yet again.  As it sits in your hand, what forces act upon it?  As we said earlier, the gravity of Earth pulls it toward Earth’s center.  But that is not the only force of gravity that is acting.  Any two objects located anywhere in the universe pull upon each other with a gravitational force.  The sun pulls upon the Earth to keep Earth in its orbit.  The Moon pulls upon the Earth and affects the height of our tides.  As you read this, I pull upon you and you upon me with a slightest of gravitational attraction.

Everything in the universe – all matter, in all solar systems, in all galaxies of the universe – pulls upon everything else through the force of gravity.  The individual force exerted by any distant star upon our apple is vanishingly small – but it is not zero.  And there is an immense amount of matter in the universe.

So if our apple is being pulled upon by the distant planets, stars and galaxies, why isn’t it constantly in motion – why doesn’t it fly off in some particular direction?  Because of the cosmological principle.  There is an equal amount of matter surrounding it in all directions, pulling upon it equally from all sides.  When you move your apple from left to right, you pull it against the gravity of all of the matter of the universe that lies to your left. 

And therein is the answer to the great puzzle of inertia.  The resistance of an object to a change in motion is caused by none other than the gravity of all of the mass of the universe pulling in the direction opposite to the change.  This astounding realization quickly explains why inertial mass and gravitational mass are the same value – they are the same phenomenon.  There is no such thing as “inertial” mass – there is only gravitational mass.

And so as you relax in your recliner on a Sunday afternoon, uninspired to rise to wash the dishes or mow the lawn, remember that it is not just your own inertia you must overcome, you need to fight against the power of the entire universe to accomplish your chores.  I won’t blame you if you need to sit there a bit longer to let that sink in.