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What Do a Falling Apple and an Orbiting Moon Have in Common?
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science
May 2023
If you drop an object, it will fall. It's a motion that we've all seen hundreds of times. We've also all seen plenty of the moon, which makes one complete orbit around our planet every 27.3 days (as seen from the Earth). Falling and orbiting may seem like radically different types of motion, but they're not! The same physics explains them both.
There is a famous story about Isaac Newton making the connection thanks to a falling apple. (It's probably not true--but it might be.) Still, his realization is kind of amazing, so I'm going to walk you through the whole process. It includes some concepts that people living today might take for granted, but the building of knowledge like this isn't trivial, and Newton didn't figure everything out on his own. He built on ideas from Galileo, who studied the motion of falling objects, Robert Hooke, who explored the effects of things moving in circles, and Johannes Kepler, who produced ideas about the motions of the planets and moon.
Let's start with what happens to an object as it falls. In the third century BC, Aristotle asserted that a massive object will fall faster than a low-mass one. Sounds reasonable, right? That seems to fit with what we see--imagine dropping a rock and a feather at the same time. But Aristotle wasn't big on testing his theories with experiments. It just seemed to make sense that a heavier object falls faster. Like most of his philosopher peers, he preferred to come to conclusions based on armchair logic.
Aristotle also reasoned that objects fall at a constant speed, meaning they don't slow down or speed up as they go. He probably arrived at this conclusion because dropped objects fall quickly, and it's really hard to spot changes in speed with the naked eye.
But much later, Galileo Galilei (who went by his first name because he thought that was cool) came up with a way to slow things down. His solution was to roll a ball down a ramp instead of dropping it. Rolling the ball at a very slight angle makes it much easier to tell what's going on. It might look something like this:
Now we can see that as the ball rolls down the track, it increases in speed. Galileo suggested that during the first second of motion, the ball will increase in speed a certain amount. It will also increase by the same amount of speed in the next second of motion. That means that during the time interval between 1 and 2 seconds, the ball will travel a farther distance than it did in the first second.
He then suggested that the same thing happens as you increase the steepness of the angle, as it would produce a greater increase in speed. That must mean that an object on a completely vertical ramp (which would be the same as a falling object) would also increase in speed. Boom--Aristotle was wrong! Falling objects don't fall at a constant speed, but instead change speed. The rate at which the speed changes is called acceleration. On the surface of the Earth, a dropped object will accelerate downward at 9.8 meters per second per second.
We can write the acceleration mathematically as a change in velocity divided by the change in time (where the Greek symbol ? indicates a change).
OK, now let's see if Aristotle was also wrong about heavier objects falling faster.
What happens if you roll a more massive ball down the ramp? If the incline stays at the same angle, then it will roll and increase in speed, just like a ball with a smaller mass does. In fact, Galileo's setup shows that both balls--no matter their mass--take the same time to get to the end of the ramp, and both have the same acceleration as they roll down the ramp.
The same turns out to be true if you drop two objects of different masses from the same height. They will fall with the same downward acceleration and hit the ground at the same time.
In fact, on the surface of the Earth most dropped objects will hit the ground at the same time. For a simple experiment, try dropping a tennis ball and a basketball from the same height. Even though the basketball is many times the mass of the tennis ball, they will pretty much hit the ground at the same time. If you don't believe it, use the slow motion video feature on your phone.
So it looks like Aristotle is wrong again--but why? After all, this seems counterintuitive. If you hold these two objects at the same time, one feels heavier to you. It seems clear that the gravitational force pulls down more on the heavier object. Then why do they fall with the same acceleration?
People often assume that objects on the surface of the Earth fall the same because gravity itself is the same. Not quite. Newton's answer to this problem was to say that the acceleration of an object depends on both the total gravitational force and the object's mass. And the gravitational force on the object increases with the mass of the object (mass x g). From this we get Newton's second law, which we can write like this:
If the only force on a falling object is gravity, and that force depends on the mass, then we get the following equation:
In this equation, G is a constant with a value of 9.8 meters per second per second--the free fall acceleration of an object on the surface of the Earth.
OK, so remember how I said "most dropped objects" hit the ground at "pretty much" the same time? There's a reason why their landing times might be slightly different, and it has nothing to do with acceleration. It has to do with a force called air drag.
If you put your hand out of the window of a moving car, you can feel this force as your hand collides with molecules of air. It's a backwards-pushing force that increases as the speed of an object increases. So when you drop objects on Earth, there are actually two forces acting on them during the fall. Gravity pulls down, while air drag pushes up. An object's mass-to-drag ratio affects how fast it falls.
Both the tennis ball and basketball are heavy relative to their size. So while they both experience air drag, it's small compared to their weight. In the end, the relative air drag force pushing up on each is insignificant compared to the gravitational force pushing them downward. It doesn't make much difference in how fast they fall.
But if you compare the tennis ball to something like a feather, the feather is very light relative to its size, and so air drag makes more difference. The air drag on the feather can counter the downward push of gravity enough that the feather won't accelerate as it falls, which means it would land after the tennis ball.
In other words: Objects fall with the same acceleration regardless of mass--but only if there is no air resistance.
In 1971, during the Apollo 15 mission, astronaut David Scott decided to perform an awesome experiment to demonstrate this idea. The moon has gravity, but no air--and therefore no air drag. While standing on the surface of the moon, he dropped a hammer and a feather at the same time. Both hit the ground simultaneously. This showed that Aristotle was wrong, and Newton and Galileo were right: If you get rid of air drag, all objects fall at the same speed.
To make a relationship between a falling apple and the moon, let's start with the fact that the moon circles the Earth over a period of close to 27 days. (It's not a perfectly circular orbit, but pretty close.)
Early Greek astronomers had a fairly accurate value for the radius of the moon's orbit. Their basic idea was to look at the shadow of the Earth on the moon during a lunar eclipse. With some simple measurements of the size of the shadow compared to the size of the moon, they found that the distance to the moon was 60 times the radius of the Earth. Remember that: That number is going to be important. (The Greeks' value for the size of the Earth was pretty good too.)
But how is an object moving in a circle similar to an object falling on Earth? That's a tough connection, so let's start with a demonstration. You could do this yourself if you are brave enough. Take a bucket and add some water. Now take the bucket by the handle and swing it around in a circle over your head. If you do this fast enough, the water stays in the bucket. Why doesn't it fall out?
To show why not, here's another fun demo: Put a cup of water on a rotating platform like a lazy Susan and spin it. The surface of the water won't stay flat. Instead, it will create a parabola, like the shape of a sagging string. Here's a picture of what that looks like--I added blue dye to the water so you can see it better:
Why does the surface of the water make this shape? We can assume that all of the water is rotating with the same angular velocity. This means that in one revolution, water near the edge of the cup has to travel a greater distance (in a larger circular path) than water near the center of the cup. So it's going faster.
Now let's focus on two blobs of water: one near the center and one near the edge. On the surface, the rest of the water can only push on these blobs in a direction perpendicular to the surface. As the surface curves up, the water below the outer blob pushes it towards the center. Here's a diagram:
But if there is a force pushing that water towards the center of the cup, why doesn't it move towards the center? (If it did, the water should form a dome, not a sagging parabola.) Before Newton, the common explanation, from 17th-century scientist Robert Hooke, was that the water blob was in a state of balance, meaning that if one force was pushing water towards the center, another must be pushing it away. Hooke called this a centrifugal force. But what Hooke didn't know is that water moving in a circle is actually accelerating towards the center of the circle. That acceleration is just like a ball rolling down an inclined ramp. The magnitude of this acceleration depends on both the speed of the object (or water) and the distance from the center of the circle.
The faster (v) something moves in a circle, the greater the acceleration. Also, the smaller the radius of the circle (r), the greater the acceleration.
If the moon is moving around the Earth in a circle, that means it is accelerating. We can even calculate this acceleration knowing just the size of the moon's orbit and its speed. The Greeks had a reasonable value for the radius of the moon's orbit at about 1/60th the radius of the Earth. Since it takes the moon 27.3 days to orbit, then we can find the moon's speed. It's the distance around the circle divided by the time. This gives us a value of about 1,000 meters per second, or 2,280 miles per hour. Plugging this into our equation for the acceleration of an object moving in a circle gives a value of 0.0027 meters per second squared.
Now for the real connection. What if this acceleration of the moon and the acceleration of a falling object on the surface of the Earth are both due to the same interaction? Why would there be such a different acceleration for the moon's orbit--0.0027 m/s2 compared to 9.8 m/s2 for a falling object on the surface of the Earth?
Newton's solution to this problem was to let the gravitational force on an object decrease with distance. Suppose that the gravitational force still depends on the mass of the object and the mass of the Earth. This was really difficult to measure back in Newton's day, but it is inversely proportional to the square of the distance between the center of the Earth and the object. We call this distance r. We can write this as the following equation:
In this expression, G is a gravitational constant and ME is the mass of the Earth. Newton didn't know the value of either of these. But if you have an object with a mass of m, then it should have an acceleration of:
Now we can do something. Let's compare the acceleration of a falling object to the acceleration of the moon as a ratio.
You see how nice it is to work with ratios? We don't need to know the value of G or the mass of the Earth (ME). Heck, we don't even need to know the radius of the Earth (RE). In the end, this says that the acceleration of an object on Earth should be 602 times larger than the acceleration of the moon.
Let's try it. Using the calculated value of the moon's acceleration, this is what we get:
Well--that's pretty darn close to 3,600. (I rounded the numbers a little bit.) But this does indeed suggest that the gravitational force decreases with distance. It's kind of a big deal. It shows that the physics that work on the surface of the Earth are the same physics that work in the heavens. That's why it's called Newton's law of universal gravitation.
Before Newton's gravitational force model, there were already some ways to predict the motion of objects in the solar system. Johannes Kepler used existing data on the motions of planets to develop the following three laws of planetary motion:
The orbit of a planet creates a path in the shape of an ellipse. (And a circle is technically an ellipse.)
As a planet moves around the sun, it sweeps out equal areas in equal times, so a planet will increase in speed as it gets closer to the sun.
There is a relationship between the orbital period (T) and the orbital distance (technically the semi-major axis of the orbit--a) such that T2 is proportional to a3.
Newton was able to show that his universal law agreed with these three laws. His gravity could explain a falling apple, the motion of the moon, and the rest of the objects in the solar system. And remember, he didn't even know the value of G, the gravitational constant.
It was a huge win. Without it, we never would have been able to solve the big questions posed by astronomy and eventually space exploration. We wouldn't be able to use the orbital period of a moon to calculate the mass of a planet. We wouldn't be able to calculate the trajectory for a spacecraft going to the moon. In the end, we would have never sent people to the moon--and David Scott would never have gotten a chance to drop the hammer there.
