Explaining acceleration due to gravity without math

[Nightowl]

Hi,

I teach a physics class to a group of homeschooled students. Many of them aren't yet at the level of algebra.

Last week we went over Newton's laws and did some experiments to show objects fall at the same rate regardless of mass. One student though, has asked me to explain why the mass doesn't matter. He believes that the acceleration is different even if it is so small we can't detect it to be so (because of the different masses of the objects).

I assured him that the mass of objects doesn't matter when it comes to the acceleration of gravity, but that the force each mass feels is different. The trouble I'm having is that my own brain can't get past the math. I can prove it using math, but he isn't at the level of understanding that yet. He's a very smart boy, and I think asking excellent questions so I want to explain in a way to have his lightbulb go off. But so far, everything I've come across to help me get past my own teaching limitations (as in I can't show him the math to prove it and I'm stumped) also uses math, or just says it doesn't depend on it without explanation. I'm not a fan of "that's the way it is" type of answers. I really want to help him "get it".

Any suggestions on how to help him understand the mass of the Earth matters but the mass of the objects don't? Without using math. ; )

Thanks in advance!

 

[Drakkith]

You can do experiments to show that the acceleration an object feels under gravity is independent of mass, and you can explain that although the force increases with mass, the acceleration stays the same because the force is counterbalanced by the increase in the mass. You can also explain that many, many experiments have been done over the last few centuries and all support the idea that acceleration under gravity is independent of mass.

If none of those convince him, then unfortunately sometimes the only real answer to someone's question is for them to wait and learn more and ask again later.

He believes that the acceleration is different even if it is so small we can't detect it to be so (because of the different masses of the objects).

Explain that if we cannot detect it, then we cannot include it in our laws of nature. Science only explains what we can actually detect and measure. He can believe that there is a small difference, but unless someone detects this difference then it will never be included in our laws.

 

[andrewkirk]

How about this?

The acceleration of an object is the net effect of two inputs: the force applied and the inertia of the object.

For a falling object - ignoring air resistance - the force is the weight of the object, which is about ten times its mass. Imagine holding up a litre milk carton, whose mass is 1kg, and weight is about ten Newtons. Now imagine holding up two milk cartons. We have to push twice as hard to hold it up. So on with three, four milk cartons and so on.

Now for the inertia part. Imagine pushing a well-oiled trolley across a level floor, where that trolley is initially stationary. Inertia depends on mass (I'm avoiding saying 'is proportional to') so, if the trolley's mass is 200kg, we have to push twice as hard to get it moving at a given rate as we do if it is 100kg.

For a falling object, the acceleration increases with the force applied and decreases with the inertia. If we double the mass then the force doubles, but so does the inertia, so the rate of acceleration is unchanged. It's like an evenly balanced tug-of-war. If you add one more person to each end of the rope, and they both have the same strength, you won't change the outcome. The object's weight is pushing it down and its inertia is resisting the push ('pushing it up').

An aside: In my experience, the reason people feel that heavier objects fall faster is that the heavy objects they have experienced - like bowling balls - are denser than the light objects - like pieces of paper. They are also often shaped with a higher ratio of volume to surface area. The reason those less dense, flatter, objects fall more slowly is air resistance. So people's intuitions are formed by air resistance, and those intuitions are correct: paper sheets do fall more slowly than bowling balls in air. What is amazing, and cannot be witnessed unless one has access to a high-tech lab with a vacuum chamber, is that a feather falls at the same rate as a bowling ball in a vacuum.

 

[kuruman]

A simple experiment that I have used as an introduction to the acceleration of gravity is this. Hold a large-sized and relatively heavy textbook in one hand and a letter-size sheet of paper in the other. Ask the question, "If I let go simultaneously, which object will hit the floor first?" If anyone says, "Yes, but there is air resistance", say "please just answer the question." At this point the answer should be unanimous. Drop the objects to verify that the book hits the floor first. Now place the paper on top of the textbook, say "Observe" and release the book. Point out that when the book is used to push the air away, the speed of the much lighter sheet changes exactly the same way as the book's speed and so they stay together. If the speed of the sheet changed at a slower rate, then it would lag behind - not what happens. This is a cheap version of the "feather and farthing" demo but just as convincing and it always works.

 

[andrewkirk]

@kuruman That's a really ingenious experiment. Like the experiment with yo-yo 1 in your rolling motion Insights, it immediately had me doing lots of practical trials to see this in action (my intuition was completely wrong in that yo-yo experiment). The trials confirmed your prediction.

I wonder whether this one might be reaching the right conclusion for - at least partly - the wrong reason, because there will be a downwards suction effect on the paper when the book it is sitting on is dropped. To test this, I again sat the paper on the book and then, without touching the paper, pulled the book down with an acceleration greater than gravity. The paper stayed flat against the book all the way down. I wasn't able to time it (no assistants nearby to operate the stopwatch), so I can't be certain I was accelerating the book at more than 1G, but I think I was.

 

[robphy]

Experiment:
https://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html
(YouTube version)

and


Galileo's argument:
https://en.wikipedia.org/wiki/Galileo's_Leaning_Tower_of_Pisa_experiment

 

[kuruman]

... because there will be a downwards suction effect on the paper when the book it is sitting on is dropped.

Just crumple up the sheet of paper into a ball, put it back on the book and drop. As best as I can tell, it doesn't separate although turbulence blew the paper sideways on a couple of occasions.

 

[brainpushups]

I think there are some good examples for observational evidence given so far, but these don't address the explanation of why. To get there you'll need at least the concept of proportion. I think that Hewitt's Conceptual Physics book does a pretty good job of explaining this with diagrams and 'cartoon' equations. You may want to check that out.

 

[kuruman]

I think there are some good examples for observational evidence given so far, but these don't address the explanation of why.

To every explanation there is another why. This particular case will unavoidably end up at "Why does the Earth attract objects?" to which the only reply I know is "Because it does." The observational evidence supports the explanation, "because the Earth attracts objects of different masses in a way that their speed changes in exactly the same way."

The belief that less massive objects move faster if dropped from the same height is a preconception that stems from the following observation. If I drop a paperback from a height of a couple of feet on my hand resting against a table, my hand will experience no pain. However, if I drop a lead brick from the same height, I will suffer a lot of pain if not a broken hand. (Don't try this at home.) The preconception lies in the faulty logic, "Everybody knows that the faster an object moves, the more damage it does. Therefore, if the paperback causes no damage but the lead brick does, it follows that the lead brick must be moving faster." So before producing any further explanation, I would take steps to root out the preconception and then explain. In some cases, the explanation becomes obvious once the preconception has been removed.

 

[brainpushups]

To every explanation there is another why. This particular case will unavoidably end up at "Why does the Earth attract objects?" to which the only reply I know is "Because it does." The observational evidence supports the explanation, "because the Earth attracts objects of different masses in a way that their speed changes in exactly the same way."

The OP was asking for a way to explain that objects fall at the same rate regardless of mass and wasn't satisfied with a 'that's the way it is' explanation which is all the experiments provide. Of course why can always continue to be asked and we'll eventually reach a limit, but I don't think this situation has even started down that road yet.

 

[vela]

Why completely avoid the math? It seems like a good opportunity to teach the utility of math, finding the relationship between quantities, and perhaps introducing simple algebra.

 

[kuruman]

He believes that the acceleration is different even if it is so small we can't detect it to be so (because of the different masses of the objects).

This is an important piece of information; the student believes that the acceleration is different. Does the student have any evidence to back his claim? If he does, then it's false. As I mentioned above, there is a preconception here that has to be removed. It is not a matter of explanation, it is a matter of shattering a false belief. I believe that the best way to do that is by showing how nature actually works. Then comes the math as @vela suggested.

 

[FactChecker]

If the comparison of a crumpled sheet of paper to an entire book doesn't convince him, then I'm afraid that he will never be convinced. (A small wad of paper would be better.) I would just encourage him to think about that experiment more carefully. Remind him that there is more force on the book but also more inertia in the book. The spectacular fact that the two effects are exactly equal might encourage him to be curious about learning more.

 

[CWatters]

Any suggestions on how to help him understand the mass of the Earth matters but the mass of the objects don't? Without using math. ; )

Perhaps get them doing an experiment to demonstrate that it's harder to accelerate a massive object than a light one.

Then explain that a more massive object does experiences a greater force due to gravity but it's also harder to accelerate because it's more massive. So these two effects cancel each other out. and everything accelerates at the same rate. .

 

[kuruman]

Then explain that a more massive object does experiences a greater force due to gravity but it's also harder to accelerate because it's more massive. So these two effects cancel each other out. and everything accelerates at the same rate.

It's a good approach, but it will probably not work in this case. According to OP, the student

... believes that the acceleration is different even if it is so small we can't detect it to be so (because of the different masses of the objects).

To show the exact cancellation of the two effects, math is needed and the constraint is "no math."

 

[FactChecker]

To show the exact cancellation of the two effects, math is needed and the constraint is "no math."

I'm under the impression that the identical 'mass' involved in both inertial and gravitational forces can only be completely explained by General Relativity (that the gravitational acceleration is really inertial motion along a time-space geodesic). In that case, the math required would be very advanced.

 

[brainpushups]

I'm under the impression that the identical 'mass' involved in both inertial and gravitational forces can only be completely explained by General Relativity (that the gravitational acceleration is really inertial motion along a time-space geodesic). In that case, the math required would be very advanced.

I doubt that it will occur to the student that there is a conceptual difference between gravitational and inertial mass and that simply canceling the appropriate factors in a simple proportion, in conjunction with experimental evidence, will be the most convincing way to approach the problem. If the student resists without articulating a reason then they probably just need a little while to mature.

 

[FactChecker]

I doubt that it will occur to the student that there is a conceptual difference between gravitational and inertial mass

Ha! Ok. I'll buy that. (Although pointing out that coincidence might stimulate a curious mind.)

 

[gmax137]

Galileo's argument:
https://en.wikipedia.org/wiki/Galileo's_Leaning_Tower_of_Pisa_experiment

I like this approach. The two balls & string thing.

We know "pure reason" can be a very misleading way to explore the workings of the universe, but without experiment or math, what is left?

Making the two balls & string thing and then tossing it out of a second floor window might be interesting/fun as well.