The Physics of Weight Loss

Today I want to talk about weight loss. Or getting into shape. Or both. This is something I’ve been thinking about lately, and while I’m sure probably everyone reading this has had to lose a few pounds at one point or another. Everyone knows the deal: To get in shape you need to watch what you eat, hit the gym, take up running/jogging/biking, so on and so forth.

Time to hit the gym

But why? Why does any of this stuff work? Everyone knows that running can help you lose a few pounds, but most people don’t actually know why, or more importantly where that weight goes. So it’s time for me to do what I do best: overanalyze things with physics.

LOSING WEIGHT VS. GETTING IN SHAPE

We often use these two terms interchangeably, but in fact they are completely different—albeit related—things. Your weight is essentially your mass—how much stuff you’re carrying around with you. It’s how hard you’re pulled toward the center of the earth. Every single part of you—fat, muscles, bones, skin, organs, everything—contributes to your weight.

Your shape, on the other hand, is really about how that weight is arranged. In American culture, you’re generally considered to be in shape if you have low amounts of body fat and at least some muscle. Obviously that’s a broad generalization but we have to define our terms somehow.

When talking about either recreational fitness or overall health and wellbeing, your shape is far more important than your weight. Especially since muscle weighs more than fat. It’s not your weight that reduces your lung capacity or puts you at a higher risk of stroke and heart attack, but the amount and distribution of fat in your body. This is why the Body Mass Index (BMI) which compares only your weight to your height, is worse than useless as a measure of health. Just going by their BMIs, most professional football players would be considered obese.

Just look at this fat slob

But there’s another reason for distinguishing between weight and shape, and that’s because the two of them change somewhat independently of each other. Anyone who’s ever started lifting weights to get into shape knows this. As you replace the fat with muscle your shape will improve while your weight increases. And people who starve themselves will see their weight fall while their shape deteriorate as their bodies pack on stores of fat. There is somewhat of a correlation between the two, but what exactly that is differs from person to person depending on body type and genetics.

HOW TO LOSE WEIGHT

Physically speaking, losing weight is incredibly easy. One of the most fundamental rules of our universe is the conservation of mass. That is, mass can neither be spontaneously created nor can it spontaneously disappear. And since weight depends on mass, this gives us an equation for the change in a person’s weight:

The amount by which your weight will change over any given time is exactly the difference between how much weight you added to your body (by eating) and how much weight you removed from your body (through various excretions). Notice that this depends only on the amount of food you eat, not the quality of it. In terms of your weight, a pound of celery is just as bad as a pound of birthday cake. Also notice that this does not depend on how much or how little you exercise.  It’s simply the difference between what’s going into your body and what’s coming out.

HOW TO GET IN SHAPE

This is where it gets complicated, and I’ll admit I’m by no means an expert in biochemistry. But I’m not too sure how many people would be interested in those details anyway so I’ll just summarize. Where your weight is all about the mass in your body, your shape is all about energy.

All of our energy comes from what we eat. It’s stored in the chemical bonds of food and we measure it in calories. In our stomachs we use acid to dissolve the food, storing that energy in glucose molecules. That glucose is then either used for energy immediately or stored in the body as glycogen in fat deposits. As our bodies consume energy throughout the day we dip into these fat reserves, so the key to staying shape, or at least the key to not gaining any fat, is to use more energy than you consume.

What makes this difficult is that it’s very hard to measure exactly how much energy your body is using. We are literally using energy all of the time, even when we’re sleeping. It takes energy just to be alive. We’re constantly spending energy to heat ourselves up, and an incredible twenty percent of all of the energy we use throughout the day goes to powering just our brains. We call the amount of energy it takes to sustain your resting body your Resting Metabolic Rate (RMR, although when I learned it we called it the Basal Metabolic Rate), or more commonly known as your metabolism. Since muscle tissue requires more energy to sustain than fat, the more muscles you have the higher your RMR will be. Any extra energy your body uses due to exercise adds to this metabolic rate to determine your body’s total energy output. If you’re consuming less energy than you’re using, your body will naturally take the difference from its stores of fat, leading to you being in better shape.

MYSTERIOUS NIGHTLY WEIGHT LOSS

I’m sure you’ve heard before that you weigh less in the morning than you do at night, and it’s true. It’s not just because of all the food you ate during the day, either. So a big question is: where does that weight go? It’s tempting to say that it’s due to the energy our bodies use while we sleep, as mentioned earlier, but that’s not right. We’re not stars, so we can’t convert mass into energy. Honestly, I think the real answer is even stranger than that.

We breathe all that extra mass away. That might sound ridiculous, but think about it. When we breathe, we inhale air with a density of around 1.2 grams per liter and exhale carbon dioxide which is considerably heavier with a density of 1.977 grams per liter. It’s not enough to notice a difference with a single breath, but multiplied over the course of an entire night it can add up to over a pound. And if this still seems a bit farfetched, consider that the exact opposite process—inhaling carbon dioxide and exhaling lighter oxygen—is where trees get all their mass.

EXERCISING TO LOSE WEIGHT

According to everything I’ve said so far, it would seem that there’s no link between exercise and weight loss. But anyone who’s ever exercised knows that isn’t true. I’ve come back from 10 mile rollerblading trips a full three pounds lighter than when I left. So where is that weight going?

As you can probably guess, a significant amount of this is water weight which we lose through sweat, but not all of it can be explained this way. The process of the body using fat to power itself is called ketosis, and through a lengthy series of interactions the glycogen mentioned earlier is converted into ATP, the form of energy which can be used by our cells. The biggest byproducts of this reaction are water and carbon dioxide. So as your body undergoes ketosis, the fat cells shrink as the glycogen is removed from them and you sweat and breathe out the byproducts, causing you to both lose weight and get into better shape.

FINAL THOUGHTS

You’re probably expecting me to have some final wisdom about the best way to get into shape or lose those pesky pounds, and I really wish I did, but ultimately it boils down to knowing your body and using common sense. One of the strangest things working against us in the battle of the bulge is our sense of hunger. Don’t trust it. Studies suggest that for people in developed countries (with ready access to stocked refrigerators and grocery stores) hunger is more of a social and conditioned response than a physiological one. One thing I’ve found lately is that I was just eating way more than I needed to, even though my eating habits before hadn’t been particularly unhealthy. What I’ve come to realize is that what I considered my resting state before was actually my body’s way of saying it was full. I was essentially filling the tank every time it dropped below three-quarters, and in doing so I was keeping my body from using its stored energy. If you find yourself consistently feeling stuffed after meals that might be why. Try to really pay attention to what your stomach is telling you, and just because it’s growling doesn’t necessarily mean it’s empty.

Blue Skies Ahead

Question: Why is the sky blue?

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Answer: It isn’t. Not the same way that bluebirds, sapphires, and the members of Blue Man Group are, anyway. To see why, we need to first talk about what exactly we mean when we say that something is ‘blue’ or ‘red’ or any color at all.

To talk about color, we must first talk about light. Light, at the smallest scale, is broken up into photonsindividual particles or pieces of light. Those photons each have a certain amount of energy, and a corresponding frequency, that determines their color. In physics, there are only six colors in the visible spectrum (red, orange, yellow, green, blue, and violet) with many more outside what we humans can see (radio, microwaves, infrared, ultraviolet, x-ray, and gamma). Red light has a lower frequency than yellow light which has a lower frequency than blue light.

So where do the other colors come from? If all we have are those six colors to choose from, how do we get browns and pinks and even white? What may surprise you, especially if you’re artistically inclined, is that you get brighter, lighter colors by adding multiple colors together. White light is light which contains frequencies of all the other colors at once. That is why you can see every color when in a room lit only by white light–the white light contains all the other colors. Notice how this is different than what you get if you add together different pigmentslike paints. If you add pigments together you end up with a dark, blackish color. We’ll talk about why these two are different in a second, but first we must discuss what it means for an object to “be” a certain color.

For the most part, color in the natural world is a reflective property. This is because most objects don’t emit their own light (some things, like butterfly wings, are colored by diffraction but that’s a different story entirely). They only reflect light that hits them, redirecting it into our eyes. However, most objects don’t reflect all of the light that hits them, only certain frequencies or colors.

When you look at an object and see white, it means that object is reflecting all colors of light back at you. Black objects reflect little to no light (this is why wearing a black shirt in summer makes you warm). When you see a blue object, it means only blue light is reflected off of it, with all other colors being absorbed.

When we say an object is blue, we mean that it only reflects blue light. A blue object absorbs every color except for blue. This is why mixing pigments gives you darker colors. Each pigment only reflects certain ranges of colors, and if these ranges don’t overlap, less light will be reflected, giving you a darker, gray or black color.

With all of this said, we can now return to our original question: What color is the sky? To answer this, we need to know what exactly the sky is. And for the most part, it’s air. Mostly nitrogen, about 20% oxygen, some carbon dioxide, and other trace gases. It’s the same stuff that surrounds you all the time. It’s the same air that you’re breathing right now.

So look around. What color is the air around you? What colors of light are being absorbed by the “empty” space in front of you? Unless you’re living in a heavily polluted area, the answer should be clear. Literally.

Air is transparent. And thus, so is the sky.

 

Revised Question: Why does the sky appear blue?

Answer: You might think that was all a little bit pedantic, but the distinction between the sky looking blue and the sky being blue is crucial to understanding why the sky looks the way it does. The sky isn’t colored by reflection the way most other objects are. It’s a different process entirely, one you may have experienced yourself without realizing.

What lights up the entire sky is the sun. And the sun, despite what you may think from looking up at it here on Earth, is white. Not yellow, not orange, but white.

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Courtesy NASA

What you’ll notice in the picture above is that, when viewed from space, the light from the sun doesn’t spread nearly as much as it does here on Earth. In fact, even with the glare, you can see the total blackness of space as little as ten degrees away from the sun.

What allows the sun to color the entire sky is a process known as Rayleigh Scattering, and it’s exactly what it sounds like. If you think of light as a stream of tiny particles (or photons), then you should be able to imagine those particles colliding with things like molecules in our atmosphere. As a result of those many collisions, the light gets scattered, spreading out in all directions so that, no matter where you look, you can see it.

If you have ever played with a laser pointer you’ve experienced this before. Shine a laser pointer at a wall and all you’ll see is a red dot. You won’t be able to see the beam itself. But give the light something thick like smoke or fog or chalk dust, and the beam will scatter off of it, becoming visible to all.

So the scattering of light can explain why the sky has color, why it’s not just pitch black around the sun. But why is it blue?

I will try to stay away from doing too much math (there was enough of that in my last post), but there is an equation which governs the amount of scattering that occurs via this process:

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It’s a messy formula, and if you’d like more information about what each variable represents you can check here, but the important part is the dependence on frequency. The higher frequencies of light scatter more than the lower frequencies. This is why you can see blue and violet all over the sky, but red and yellow only when you look directly toward the sun. The lower frequencies don’t scatter as much.

Only a few more questions to clear up. If higher frequencies of light scatter more, why is the sky blue instead of violet? The answer is because the sun emits more blue light than it does violet. The violet does scatter, but it is overwhelmed by all the blue light around it.

Why is the sky red during sunrise/sunset? At these times, when the sun is reaching the horizon, the blue light scatters too much, having to pass through too much of the atmosphere to reach us. Once all of the blue and purple light has been scattered away, all we’re left with are those beautiful red, orange, and golden sunsets.

Well, this was a fun one. What I like about questions like this is how trying to answer one question opens up–and then answers–many more. For example, you may have thought that it would be simple to explain why the sky is blue, but in the process of doing so we got to talk about what color is, how things get their color, and the scattering of light in a medium. These are the kinds of questions I like most, where the answer takes a winding road through different areas and disciplines. It’s because of questions like this that I started teaching. Unfortunately, given our prescriptive and standardized attitude toward education, it’s rare that kids be given the chance to really explore a topic like this in school.

That about wraps it up for this week. If you have any other questions you want answered, let me know.

The Mystery of the Murderous Monies

It’s been a long time since I’ve posted anything on here. Life got in the way (as it tends to). One thing I would like to do more of here is explaining things. I am a teacher, after all. So starting today, whenever I get a chance I’m going to answer some common questions or address some misconceptions that are out there. I apologize that this one ended up being a little bit maths-intensive. I’ll try to limit that in the future as much as possible.

 

Question: Can a penny dropped off the roof of a tall building kill someone standing on the street below?

 

Answer: No. Not even if we were to remove the atmosphere. Without the atmosphere, a falling penny would be in a state of free fall in which the only force acting on it is gravity pulling it down toward the ground. A modern penny, with its zinc center and copper core, has a mass of 2.5000 g according to the US mint. Let’s say this penny is dropped from the Empire State Building, specifically from the 102nd floor observatory 1,224 feet (373.0752 meters). A penny dropped from this height in the absence of atmosphere would hit the ground with a speed of just over 120 meters per second (around 265mph).

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Notice that nowhere in this calculation does the mass or weight of the penny appear. That is because the final speed of an object in free fall is independent of how much that object weighs. In perfect free fall, every object dropped from the 102nd floor of the Empire State Building would reach this speed before it hit the ground, regardless of its weight or shape.

But this does not mean that the mass of the penny is irrelevant, even in this atmosphere-less approximation. The mass of the penny may not matter when finding its final speed, but it certainly matters to the person standing underneath it, because the mass of the penny determines how much energy the penny has when it hits the ground. A penny moving at that speed would have around 18 Joules of energy, which is about as much energy as a 60-Watt light bulb uses in twenty seconds.

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Finding an exact number for the amount of energy or pressure required to fracture or break a human skull is difficult (not many people willing to test it out, after all), but the lowest estimate I’ve found is 45 Joules, so even neglecting air resistance the penny just doesn’t have enough mass to kill someone when dropped off the roof of a building.

So let’s go higher. What if we dropped the penny out of an airplane flying at a typical cruising altitude of 30,000 feet (9,144 meters)? I could repeat the calculations, but there is no need to considering the kinetic energy of a falling object increases linearly with its drop height (see the equation below for proof and note you could prove the same thing using conservation of energy).

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Going from 1,224 feet to 30,000 feet is an increase of 24.5 times, meaning the penny would have that much more energy when dropped from the plane than it would when dropped from the building. This would give us an energy of 442.9 Joules, well above our minimum requirement to bust some poor soul’s skull open like a melon.

So does this mean that a penny dropped from a sufficient height can actually kill someone? Alas, no. Keep in mind, we ignored the atmosphere when performing our calculations, and if we were to perform this test in a place without air, suffocation would kill the person long before we could even drop the penny. In order to give an actual answer, we must take into account the effects of air resistance.

You may wonder why exactly it matters whether there is atmosphere or not, and that’s because, for the most part in your life, it doesn’t. We’re so accustomed to being surrounded by atmosphere that most of us tend to think of it as empty space. You can’t see it, and if you reach out your hand you don’t feel it, but that doesn’t mean nothing’s there.

You can feel this for yourself if you swing your hand fast enough—you’ll be able to feel the air being pushed out of the way. A better (although slightly more dangerous example of this) is to stick your hand out of the window of a moving vehicle. If the vehicle is moving fast enough, you can feel the air pushing your hand back. That is the force of air resistance that slows down a falling penny.

Air is a fluid, a term often incorrectly used interchangeably with liquid in daily life (as a side note, it is this misuse of the term that leads to some people believing that glass is a liquid, when it is actually a solid fluid). In physics, a fluid is just anything that flows, and that’s exactly what air does. The space that looks empty to us is filled with air molecules—nitrogen and oxygen and carbon dioxide among many others. In order for an object to move through the air, it must first push these molecules out of its way. Because of Newton’s 3rd Law, the air molecules push back on the moving object with equal force, causing it to slow down.

If you’ve ever tried to wade through water then you’ve felt this force before, known as a drag force. Water is a much denser fluid than air and thus pushes on you much more when you try to displace it. And the faster you try to move, the more force you feel pushing you back because you’re trying to displace more of the fluid at a time.

Applying this back to our murderous penny, the air pushes up on the penny with a resistive force that gets stronger the faster the penny moves. The equation for air resistance, shown below, gives the relationship between the force of drag an object experiences and its speed.

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Plugging in values for the drag coefficient of a flat disk, the density of air, and the dimensions of a penny as given by the US mint, we can calculate that the force of air resistance acting on the penny should be somewhere between the two values shown below (the minimum value assumes the penny fell the whole way with its thin side pointing down while the maximum value assumes it was flat-side down the whole time). Both of these cases are extremely unlikely, but we’ll use the numbers, just to prove our point.

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We now have two forces acting on our penny—gravity pulling it down toward the ground, and air resistance pushing it back up. The gravitational force is constant (the weight of the penny, 0.0245 newtons) while the drag force increases as the penny gains speed. Eventually these two forces will equal each other, and the penny will enter a state of dynamic equilibrium (equilibrium because all of the forces acting on it balance each other out, dynamic because the penny happened to be moving when the forces became equal and thus will continue to move).

Once the penny enters equilibrium, the air will be pushing it up just as hard as gravity is pulling it down. As a result, the penny’s speed will become constant. We have a special name for the speed at which this happens: terminal velocity. Once a falling object reaches its terminal velocity, it stops accelerating and just falls with that speed until it hits the ground. And unlike objects in free fall which we discussed before, the terminal velocity of an object does depend on its weight. A heavier object will fall faster than a lighter one because it has a higher terminal velocity.

Using the two values for the drag force above, we can calculate the range of our penny’s terminal velocity to be between 10 and 85 meters-per-second (21.6 and 190 miles per hour). Notice that the minimum value of the drag force leads to the maximum terminal velocity.

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Given these results, it makes no difference whether the penny is dropped from the top of the Empire State building, from a plane, or even from outer space because in all of these cases the penny will reach its terminal velocity before it hits the ground.

If you would like to look at drag forces for yourself, I’d recommend a simple experiment. All you need are a few clear glasses or bottles and some liquids of different densities (regular water, salt water, mineral oil, vinegar, even clear alcohols can work well for this, and taller glasses will give you a better chance to see what is happening). Drop the same object into each fluid and see which ones slow it down the most. Drop objects of different weights into the fluids and see which ones fall fastest. Change the shape of your weights by using something like aluminum foil and see how the geometry of the object affects how quickly it falls. If you have a tall enough glass or a light enough object, see if you can spot the moment it hits terminal velocity.

How much does the Earth Weigh?

I actually heard this question in a smartphone commercial and it bothered me. They were showing off how easy it is to do a Google search from the newest tablet or whatever; just ask your question and it speaks the answer to you. In this case 5.97E24kg.

Only that’s not the answer. That’s what bothered me. And the truth is there’s a lot of interesting physics in that question and a very common misconception. Being a physics teacher, I feel like I should address it. So let’s discuss.

First of all, the question that Google answered isn’t how much the Earth weighs, it’s how massive the Earth is. This is a common misconception, that mass and weight are the same thing. They’re related, but separate, just how you and your parents share some of the same genes yet aren’t the same person. An object’s mass is just a measure of how much stuff it contains, its density multiplied by its volume. An object’s weight is the gravitational force pulling it toward the nearest massive object, usually the Earth. The mass of the Earth isn’t all that interesting to consider, even though it is interesting to think about how exactly we figured it out.

So now that we know 5.97E24kg isn’t how much the Earth weighs, what is? Here’s where things start to get fun. An object’s weight is just the force with which the Earth pulls on that object though gravity. But forces can only exist between two separate objects. By that logic, if we consider the Earth to be a single, solid object, it doesn’t weigh anything.

But that doesn’t really make sense, does it? It can’t weigh nothing. So what if we could take the Earth and put it on top of a big bathroom scale. If we then took the reading on that scale, wouldn’t that tell us how much the Earth weighs?

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Pictured: Science

Believe it or not, you can do this yourself if you have a flat scale. All you need to do is flip it over and put it on the ground. You now have a scale with the world resting on top of it. So what does that scale read? Whatever the scale weighs. So if you’re using a 10 pound scale, the scale would read 10 pounds, a 15 pound scale would read 15 pounds and so on. Scales work by measuring the force used to compress their pressure-sensitive plates. If you flip the scale over, its own weight is the force pressing that plate down, so that’s the force it reads.

So if we take this definition to be our measurement of weight, then the Earth only weighs as much as whatever you’re using to measure it.

But that still can’t be right. Let’s assume we take all of the matter that makes up the Earth, duplicate it, and then put that on a scale. What would be the weight of all that material? Surely we can call this the weight of the world.

Sadly, this also gets us into some trouble, in that we need to consider exactly how big we make our pile of matter. The bigger the pile is, the farther away from the Earth the top of it would be. The farther away the top of the pile is from the Earth, the less it is pulled by the Earth’s gravity. The less gravity it feels, the less it weighs. If we assume our material is molded into a sphere the size of the Earth, then everything halfway up the pile would weight 4 times less than it would if it were at the bottom of the pile (gravity is an Inverse Square Law, so if we double the distance between two objects we reduce the force between them by a factor of 4). The material all the way at the top of the pile weighs 9 times less than it would on the bottom of the pile, since it’s three Earth-radii away from the Earth’s center as opposed to only one. So whatever our scale reads in this case will be much less than the Earth’s true weight.

We can fix this by taking that big ball of matter and compressing it to a size where we can measure it without noticing the changes in the Earth’s own gravitational field–say shrink it down to the size of a baseball (fortunately that’s still big enough that it doesn’t turn into a black hole and doom us all). If we plop this baseball with all the mass of the Earth crammed into it onto our scale, what would it read? What would be the weight of the world?

If you do the math: 5.85E25 Newtons or about 13,000,000,000,000,000,000,000,000 pounds.

With that out of the way, there’s really only one question left: Why am I even bothering to write about this?

The answer is two-fold. First, because this is my blog and I can write about whatever I want. But more importantly because these are the kinds of exercises that make teaching and learning and knowing physics worthwhile. In my opinion this is true physics. Notice I didn’t need to do a single calculation until the very end. Sure there are still equations, but they’re not nearly as important as the relationships they imply between the different variables. And this is what so often gets lost in the increasingly “plug and chug” nature of physics education. With both our kids’ worth as students and ours as teachers determined by the outcome of a single test at the end of the year, fewer and fewer teachers are willing to venture off and teach kids what they should be learning rather than simply what they “need” to know. To me, this is one of the most depressing things to happen to education in recent history.

So I’ll post here whatever I want to. Because no matter how they try, no one can ever stop you from learning.