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Resources

Newton’s Embarrassing Secret – Start watching at 9:52 to see the chapters of The Elegant Universe about squaring up Newton’s Law of Universal Gravitation and what we know about the “universal speed limit” from Einstein’s theory of relativity.

PhET Gravity and Orbits Simulation

You can see how the force of gravity between a star and its planet changes as the masses and distances between the two objects change.

Video Transcript

Hello there! Welcome to lecture 9: gravity!

Gravity is a force we experience all the time in our lives. It allows us to understand concepts such as weight, acceleration, and the motion of planetary objects. Newton’s law of universal gravitation will allow us to understand how gravity works on both the Earth and other planets, and where the value 9.8 meters per second-squared comes from. 

Each of the following concepts will be discussed in this video: apparent weight, the inverse-square law, Newton’s law of universal gravitation, gravitational acceleration, gravitational fields, and the limits of Newton’s laws.

Apparent Weight

Weight is a topic that we’ve discussed in several lectures already. We’ve learned that weight is a force that comes about due to our gravitational attraction with the Earth, and that it’s related to our mass. While none of that is wrong, it’s also not the full story of what weight is.

As you watch this video, I’d like you to consider where you feel your weight in your body. Where does the sensation of weight come from right now? For me, standing here, I can feel my weight in my feet. If I were sitting, I’d likely be feeling my weight in my bottom and the backs of my thighs. If I were laying down, I’d feel my weight along my side or my back. 

We have weight because of gravity, but our apparent weight comes from the support force that we get from whatever surface we’re in contact with. Our apparent weight is equal to the value of the support force that is acting on us.

In this video, I jump up and down on a force plate. The force registered by the plate is recorded by Logger Pro. When I stand still, my weight records as being constant. However, when I jump up and down, my weight changes. When I’m in the air, I’m no longer in contact with the scale and my weight is zero. When I’m coming back to the surface again, my weight increases. This goes to show that weight is not necessarily constant, even while being located in pretty much the same position on the Earth’s surface.

Any time our motion contains a vertical component of acceleration, whether it be from jumping up and down, riding a roller coaster, or traveling in an elevator, our support force will change, and our apparent weight will change as well. 

Let’s consider riding in an elevator. Any time the elevator moves at a constant speed, there is no acceleration in the vertical direction, and our apparent weight will not change. The net force of the system is zero. The support force is equal to our mass times gravitational acceleration.

In equation form, we can write this as: our net force (which is equal to our mass times our acceleration) is equal to the support force plus our mass times gravitational acceleration. Recall that gravitational acceleration is negative 9.8 meters per second-squared.

As the elevator we’re riding in speeds up moving upward, at that time there is a net force pointing up. Our mass hasn’t changed, so the mg term remains the same. The support force becomes greater than our weight. mg plus the support force equals the resultant force, that net upward force causing the elevator to speed up. This is why our apparent weight increases as we speed up in an elevator moving upward.

Let’s consider a specific numerical example. Say the occupant of an elevator has a mass of 75 kilograms. Their weight in an unaccelerated condition is negative 9.8 meters per second-squared times 75 kilograms: negative 735 newtons. The elevator accelerates upward at a rate of 4 meters per second-squared. The net force on the person is 75 kilograms times 4 meters per second-squared: 300 newtons. Mg plus support force equals net force. This means the support force is equal to 300 newtons plus 735 newtons: 1,035 newtons!

If an elevator speeds up moving downward, at that time there is a net force pointing down. The support force becomes less than our weight. mg plus the support force equals a downward pointing force. This is why our apparent weight decreases as we speed up in an elevator moving downward.

Let’s consider a different numerical example. Our 75 kilogram elevator occupant has an unaccelerated weight of negative 735 newtons. The elevator accelerates downward at a rate of 3 meters per second-squared. The net force on the person is 75 kilograms times negative 3 meters per second-squared: negative 225 newtons. Mg plus support force equals net force. This means the support force is equal to -225 newtons plus 735 newtons: 510 newtons!

The extreme condition of this would be a situation where the elevator cable snaps and the elevator moves downward at 9.8 meters per second-squared: free fall. In that case, our apparent weight would go to zero!

An object in free-fall, moving only under the influence of gravity with no other forces present, will have an apparent weight of zero. If you were to go skydiving, before the parachute deploys, and before air resistance slows you down, your apparent weight would be zero. Think about using a scale to measure your weight as you fall, there is nothing to give the scale a force reading. In those conditions you are truly weightless.

For the same reasons, astronauts in Earth orbit: whether they be in a space capsule, the Space Shuttle, or the space station, are also weightless. They are still under the influence of Earth’s gravitational field, and their acceleration due to gravity is still rather high. The reason that astronauts experience weightlessness is that they are in free-fall around the Earth during their orbit. To emphasize: astronauts in Earth’s orbit do not experience a zero-gravity condition, only a zero-support force condition.

The Inverse-Square Law

The inverse-square law describes physical properties that decrease when the distance between two objects, or the distance away from a single object, increases. The property and the distance are inversely related to each other. As distance increases, the physical property decreases. Not only that, but the decrease doesn’t just decrease with distance, but distance-squared. Any change in distance is going to have a more powerful effect as a result. Mathematically, we can represent the inverse square law as stating that a property is proportional to one divided by the distance squared. 

We can see how this looks on a graph by plotting this property. Note that when distance doubles, the intensity of the physical property decreases by one quarter. When the distance is cut in half, the intensity of the physical property increases by a factor of four.

Many physical properties obey the inverse-square law: light intensity, sound intensity, electromagnetism, and gravitational forces. From a conceptual level, this is because the effect of any single point of sound, light, electric charge, or mass, dilutes by the distance squared as it moves outward in three-dimensional space.

We’ll see in just a few moments that the inverse-square law plays a role in the effect of gravitational forces. And we’ll also see this again when we learn about electrostatics in lecture 22.

Newton’s Law of Universal Gravitation

Newton’s law of universal gravitation describes how we can quantify the force of gravity acting between any two objects in the universe. The equation states that F equals G m-one m-two over d-squared. In other words: force equals big G, the universal gravitational constant, times the mass of the first object, times the mass of the second object, divided by the distance between the two objects, squared.

Note from this equation that the divided by d-squared aspect means this equation obeys the inverse square law. If the distance between two objects were to double, the force would DECREASE by one fourth. If the distance between two objects were to be cut in half, the force would INCREASE four times.

The denominator of this equation also tells us another interesting thing about gravity; the force of gravity between two objects cannot become exactly equal to zero unless the distance between the two objects is infinite. Therefore, it can be concluded that all objects of mass that exit in the universe exert gravitational forces on each other. A planet millions of light-years away from you is exerting a gravitational force on you. However, the value of that force is infinitesimally small. We can effectively treat it as being zero, but it is not exactly zero.

The distance between two objects is more accurately stated as the distance between the center of mass of two objects. Although the density of the Earth is not constant, if we could approximate the Earth as having a uniform density throughout, we would expect the gravitational force to change as we move along the Earth’s surface. For one thing, the Earth is not a perfect sphere. The distance between the center of the Earth and the north and south poles is smaller than the distance between the center of the Earth and the equator. Therefore, the force of gravity would be larger at the poles. In addition, as we move farther away from the center of the Earth, say, by climbing a mountain, the gravitational force would decrease as well. It is important to point out that these differences are very small, and would not be perceptible unless you had a very accurate scale to stand on to weigh yourself. These effects are furthermore complicated by the fact that the Earth does not have a uniform density throughout its volume. For the most part, we can treat the gravitational force on the surface of the Earth to be approximately equal, but it is important to note that it is not.

In fact, astronauts visiting the space station are still under the influence of gravity – that is why the space station is able to orbit the Earth. Without gravity, no force would be able to create a centripetal force to cause objects to orbit, and they would not be able to move in a circular path. Because the astronauts lack a support force, they experience weightlessness. We will revisit this subject in the next lecture.

The universal gravitational constant, big G, is 6.67 times 10 to the negative 11 Newton meters-squared per kilogram-squared. The term universal indicates that this number is the same anywhere in the universe. It’s a very small number, indicating that the force of gravity is relatively weak. In fact, of the four fundamental forces, gravity is the weakest! In order for the force of gravity to be strong, one or both of the objects must have a lot of mass. 

One last interesting thing about gravity: because mass can only come in positive numbers, gravity is only ever an attractive force. When we learn about electricity in lecture 22, we’ll see that electric charges are capable of both attracting and repelling. However, the force of gravity is only an attractive force, causing two objects to accelerate toward each other.

Let’s do some examples using Newton’s law of universal gravitation.

First, let’s do an example where we’re given numerical values for each quantity and are asked to calculate the gravitational force. Let’s calculate the gravitational force between the Earth and the sun. The Earth has a mass of 6 times 10 to the 24 kilograms; the sun has a mass of 2 times 10 to the 30 kilograms. The Earth and sun are separated by a distance of 148 times 10 to the 9 meters. We can plug all of these numbers into the equation for force. F equals 6.67 times 10 to the negative 11 times 6 times 10 to the 24 times 2 times 10 to the 30, divided by 148 times 10 to the 9 meters, squared, which equals 3.65 times 10 to the 22 Newtons of force. This force keeps the Earth in orbit around the sun.

Now let’s do some examples without numerical quantities where instead we look at what happens to gravitational forces as a result of a change in some quantity. Let’s say we have two planets with masses of m1 and m2 separated by a distance of d. We can determine what the effect of a change in mass or separation distance will have on the force.

If the mass of one of the planets were to be cut in half, what would the effect be on the force? Force and mass are directly proportional. Therefore if the mass decreases by a factor of two, the force will decrease by a factor of two as well. The new force would be one half of the original force.

If instead of mass, the distance between the two planets were to change, what would that effect be? If the separation between the two planets is cut in half, what would happen to the force? Force and distance obey the inverse-square law. If the distance between the two planets decreases by a factor of two, the force will increase by a factor of two-squared: four. The new force would be equal to four times the value of the original force.

Gravitational Acceleration

So far in this class, we have frequently discussed little g, the acceleration due to gravity on the Earth’s surface. We’ve defined little g to be negative 9.8 meters per second-squared. Sometimes, when we do not need great accuracy in our calculations, we can round little g to be negative 10 meters per second-squared. Where does this number come from?

From Newton’s second law, we know that the net force on an object is equal to its mass times its acceleration. If the net force on an object comes from gravity, then we can use Newton’s second law and Newton’s law of universal gravitation together. We set G m-one m-two divided by d-squared equal to mass times acceleration. The mass that exists on both sides of the equation is the mass of any object on Earth: you, me, a butterfly, a tree, or a house. It doesn’t matter. The acceleration due to gravity on any object on the Earth’s surface is equal for all objects on the Earth because the mass of the object cancels out of both sides of the equation.

The second mass is equal to the mass of the Earth, and the distance is equal to the radius of the Earth. (Assuming the object is at sea level. As we discussed earlier, this value will change if we climb a mountain or go to the bottom of an ocean, but those effects are pretty small and we’ll ignore them.)

Plugging in the mass of the Earth, which is 5.97 times 10 to the 24 kilograms, and the radius of the Earth, which is 6.37 times 10 to the six meters, we can calculate that little g, the acceleration due to gravity on the Earth’s surface, is equal to 9.8 meters per second-squared.

What if we wanted to calculate the value of the gravitational acceleration on a different celestial body: the moon, the sun, another planet? In general, g equals G times m divided by d-squared. That is, the acceleration due to gravity on any celestial body is equal to the universal gravitational constant, times the mass of the celestial body, divided by the radius of the celestial body squared.

Let’s do a couple of examples.

First, let’s calculate the gravitational acceleration on the surface of Mars. Mars has a mass of 6.4 times 10 to the 23 kilograms, and a radius of 3.4 times ten to the 6 meters. We can plug these values into the equation G times m divided by d-squared. 6.67 times 10 to the negative 11 Newton meters-squared per kilogram-squared times 6.4 times 10 to the 23 kilograms divided by 3.4 times 10 to the 6 meters, squared, equals 3.7 meters per second-squared, the gravitational acceleration on Mars.

Let’s say Mars were to magically double in mass. What would the effect be on the gravitational acceleration? Note that if we look at the equation for little-g, we see that g is directly proportional to mass. Therefore, if mass doubles, little-g doubles as well. If Mars were to magically double in mass, its gravitational acceleration would become 7.4 meters per second-squared.

Let’s assume that instead of the mass of Mars changing, the planet expands in size so that the radius doubles. What would that effect have on the gravitational acceleration? In this case, we see that the relationship between little g and the radius obeys the inverse-square law. Therefore if the radius increases by a factor of two, the gravitational acceleration would decrease by a factor of two-squared: four! In that case, little g would decrease from 3.7 meters per second-squared to 0.9 meters per second-squared.

One more thing about little g. We can use the equation for little g to determine what the gravitational acceleration would be for a person located at different distances away from the Earth’s center of mass. 9.81 meters per second-squared is valid at sea level, but what about at the top of mount Everest?

When using the equation in this manner, the mass of the Earth is treated as a constant, but the distance will be equal to the radius of the Earth PLUS whatever distance away the object is from sea level. 

The summit of Mount Everest is located approximately 8,850 meters above sea level. Plugging in to our equation, we get 6.67 times 10 to the negative 11 times 5.97 times 10 to the 24, divided by the quantity of 6.37 times 10 to the 6 + 8,850 – squared. The result is 9.79 meters per second-squared. Not a big difference from sea level!

Astronauts visiting the space station are still under the influence of gravity. To calculate their gravitational acceleration, simply add the orbital height of the space station to the radius of the Earth and plug that value in for d.

Gravitational Fields

A gravitational field is a model that we can use to describe how objects interact as a result of gravitational forces. We can draw a gravitational field by showing where any object placed around a massive object would move due to gravity. Let’s draw the gravitational field around the Earth. The field is three-dimensional in reality, but in this video we’ll just show it as two-dimensional.

Any object of mass is going to want to move toward the center of the Earth. We depict the direction of the field using arrows. Therefore, the Earth’s gravitational field is represented by arrows pointing toward the center of the Earth.

If we look at any given distance away from the center of the Earth, the distance between each of the field lines tells us the relative strength of the force. Close to the center of the Earth, the arrows are much closer together than they are far away from the center of the Earth. This makes sense based on our understanding of the inverse-square law.

The Limits of Newton’s Laws

Newton’s laws do a really great job of explaining the motion of so many objects. But there starts to be a discrepancy that becomes apparent when objects move fast – close to the speed of light. In these cases, Newton’s laws break down and we require another physical theory to take over. Einstein’s theory of general relativity can be used to accurately describe the forces and motions of objects in these cases.

The theory of general relativity explains the motion of objects due to curves in spacetime, the fabric of the universe. A massive object causes a large curving of spacetime, and anything moving near that massive object is going to travel along spacetime in a curved direction.

That might be hard to conceptualize, so let’s take a look using a model of the universe. Here, the fabric of spacetime is literally a piece of fabric stretched along a circular frame. When no massive objects exist to cause any curves in spacetime, marbles that move through space travel in straight lines.

A large piece of mass is placed in the center of the fabric. Viewed from above, it doesn’t seem like much has changed, but viewed from the side, we can see that the fabric is warped. This is similar to how largely massive objects: stars, planets, and so on, are able to curve spacetime.

Smaller objects can then be put in motion, and we can observe the effect of this spacetime warping. Instead of traveling in straight lines like they did before, the marbles now travel in circular paths. This describes the orbits of planets around suns; and the orbits of moons and satellites around planets.

Thanks for taking the time to learn about gravity! Until next time, stay well.