Science 122 Program 17 Gravitation

# Gravitation Program 17 Lesson 3.2

 2. Universal 8. Halley's comet

## Coming Up

Before we're done with this lesson we will have seen some of the implications and uses of Newton's law of universal gravitation. We will see how weight is explained by combining the second law with gravitation.

We will see why astronauts are weightless and the relationship between inertial and gravitational mass, how the tides are a result of gravitation,how atmospheric pressure is caused by the weight of air molecules, and how Halley used Newton's equations to predict the return of the comet that bears his name.

We will learn why planets are flattened at the equator, how the motions of projectiles and satellites are related how Newton's theory of gravitation was tested and used to find new planets, and how gravitation is used to navigate between the planets.

Last, but not least, we will learn how the gravitational equation was used to find the mass of earth and other planets.

## 1. Introduction

Newton's theory of gravitation was accepted immediately by the scientific community although it could not be tested quantitatively in the laboratory.

The compelling argument by Newton left few doubts because he had presented it so thoroughly, stitching up all of the seams with the combined induction and deduction of the analytic geometry. The synthesis left few holes.

Not only that, it simply explained many phenomena such as freefall and planetary motion, which had been problems from the beginning. It also brought under one principle other phenomena, such as the tides and the flattening of planets, earth included, at the poles.

## 2. Universal

The theory was not just successful in explaining the motion of the planets, it was truly Universal. By that we mean that it not only applied to the earthly realm as well as the heavenly, but it also unified all types of motion, projectile, violent, and planetary while it unified the solar system under one law. At the same time it provided a unified problem solving method which applied to all of physics, not just astronomy and mechanics.

Now let's take a tour through some of the effects and implications of the law of universal gravitation. You need not be familiar with the details of all of these examples. Pick two or three of them and study them. Be aware of the significance of the others.

## 3. Weight

Weight is seen to be the result of the gravitational force acting on a particular mass.

### 3.1. Gravitational force necessary to accelerate a mass at the rate of g

It is the gravitational force necessary to accelerate a mass at the rate of g, on earth 10 meters per second squared.

It is one one hand a consequence of the gravitational equation applied to the earth (m1) the object (m2) and the distance to the center of the earth from the surface, or the radius of the earth (r).

### 3.2. Second law relationship

On the other hand, weight can be viewed as a second law relationship (F=ma), where F is the gravitational force and a is the acceleration of gravity, which we have called g. So the weight of an object can be seen as the force of gravity acting on the object which is just sufficient to accelerate it downward at a constant rate of 10 meters per second squared at sea level.

### 3.3. Uniform rate for all objects

It is at a uniform rate for all objects, as observed by Galileo. The reason why can be seen from these two relationships.

Doubling the mass of any object doubles the gravitational force on it (from the gravitational equation), but also doubles it's inertial resistance to being accelerated (from the second law).

### 3.4. Not the same as mass

Weight is not the same as mass. Mass is an intrinsic property of mass, invariant anywhere, anytime. It is related on one hand to the ability of an object to exert mutual gravitational attraction with another mass. It remains constant, even during "weightlessness" in freefall or in orbit. This fact has been painfully recognized by astronauts who playfully launch themselves in "weightless" space, only to recognize that upon striking the opposite wall, a force is still necessary to stop their motion.

It still hurts to run into a wall even if you are "weightless".

Weight, on the other hand, is an effect of gravitation and changes depending upon the gravitational environment. Weight is the force which you or any other object is pulled towards the earth or other object. One the moon's surface, your weight would be about 18% of what it is here on earth because of the moon's smaller mass and smaller radius.

## 4. Weightlessness

We hear of astronauts being "weightless" while orbiting the earth. This can't be true, can it? In the gravitational equation or in the second law, in order for one to be weightless some term must be equal to zero. Which one becomes zero while in orbit?

It can't be mass because we know that is an intrinsic property, constant everywhere in the universe. It can't be the acceleration of gravity because that decreases with the square of the distance from earth's center, and orbiting satellites are only a few hundred miles up. It can't be the gravitational constant because it is constant.

So how can anything be weightless?

It is another one of those misunderstood concepts. We should say "apparent weightlessness". To see what happens, consider the following:

### 4.1. elevators

In an elevator at rest, or moving at a constant speed, the floor exerts an upward force equal to your weight, and holds you up just like a solid floor. What happens if the elevator is in freefall?

The picture on the right shows what happens. If the floor is also in freefall then it cannot exert an upward force because it will be accelerating downward at the same rate as everything inside it. It is as if a trapdoor opens in the floor and the passenger is suddenly falling, accelerating at the rate of 10 m/s2. The floor exerts no upward force and so there is no pair of forces between the passenger and the floor. From the passenger's point of view she is "weightless".

How would you recognize whether you were in freefall, or in the absence of any gravity? Could you?

### 4.2. projectiles

Now suppose the elevator was a projectile, with no air friction. Could you tell that you were moving forward while you were falling? Our concept of inertia suggests that you could not. In fact, you would be in freefall while moving forward and you would still feel as if you were weightless.

### 4.3. satellites

The same is true for the satellite. Newton reasoned that the satellite is in freefall. It is just a projectile that doesn't hit the ground because the ground falls away from it as it falls. The picture is a copy of Newton's illustration of how to launch an artificial satellite from a high mountain top.

Given a forward velocity that is too small a projectile will fall back to earth in a curved path. On a flat earth this path would be a parabola. At greater launch speeds the projectile will travel increasingly greater distances until the curvature of the earth becomes significant. At some point the launch speed will be just great enough that the earth's surface curves away from the projectile at exactly the rate that it accelerates. At that point gravitational acceleration is centripetal and the orbit is perfectly circular.

This is exactly the way we launch artificial satellites today, except that we use a big, powerful rocket to lift it to a desire height. From there it is aimed in the desired direction and given an initial velocity. It falls back towards earth. At a speed which is just right for a given height it will follow a circular orbit. A little slower or faster and the orbit will be elliptical.

Meanwhile, the satellite and everything in it are in freefall and therefore "apparently weightless".

There really is no difference between freefalling in the elevator and orbiting in the satellite, except of course that the elevator will hit the ground and the satellite will miss it.

## 5. Equivalence

It became apparent that mass is a little more complex that we first supposed.

On one hand, mass is the measure of inertia. It measures the resistance to changes in motion. The more mass the more resistance.

On the other hand it is the ability to attract other masses. Are these really the same thing?

We assume that they are. We use the symbol "m" for mass, and cancel it out of equations without regard for whether it is inertial mass or gravitational mass.

Einstein assumed that they were the same and stated it as "The Principle of Equivalence". From this he derived his general theory of relativity.

This also explains why objects freefall at the same rate regardless of mass.

Consider two objects. One has a mass of 1 kg, the other has a mass of 2 kg. The gravitational force on the 2 kg mass is twice as great as the gravitational force on the 1 kg mass. The 2 kg mass weights twice as much. It also has twice the inertial mass, which means that it takes twice as much force to accelerate it at the same rate as the 1 kg mass.

So, twice the force and twice the resistance to change in motion. The two effects cancel out, like this:

in general, from the second law.

For the two masses:

or, since W = mg,

numerically:

Twice the force divided by twice the mass will always give the same number equal to the acceleration of gravity.

## 6. The Tides

Newton also used the inverse square law to explain the tides. People had known for centuries that the moon affects the tides. No one until Newton knew how it did this.

There are several equivalent ways to describe the tidal forces, but the easiest to explain in terms of the inverse square has to do with the reality that the earth and moon are not points of mass. They are extended objects so the gravitational attraction on various parts is stronger for those parts close to the moon and weaker for those points further from the moon. The far side of earth is 8000 miles further away from the center of the moon than the near side. That means that the gravitational attraction is less. When their centers are 250,000 miles away, the near side of earth is only 246,000 miles from the moon's center while the far side of earth is 254,000 miles away. The ratio of their distances is . Since the inverse square law applies the ratio of gravitational force is or 94% as great on the far side as on the near side.

The result is that the water is pulled away from the earth on the near side and the earth is pulled away from the water on the far side. This causes a "bulge" to form under the moon on both sides of the earth. As the earth turns under this bulge, it causes high and low tides as the earth drags the water causing it to slosh around in the ocean basins.

## 7. Atmospheric Pressure

We will study atmospheric pressure in greater detail in lesson 26.4.4. At this point we can oversimplify by saying that atmospheric pressure is the result of the weight of overlying air from the surface to the top of the atmosphere. We will extend this description in program 26.

## 8. Halley's comet

In the Principia, Newton gave detailed instructions on how to plot the orbit of a comet from any three observations. Halley used this method to calculate the supposed trajectory of the comet observed in 1607 (the one that Brahe's instruments had shown to be celestial rather than sublunar). It was the data from observations made by Brahe's instruments that Halley used.

Halley predicted that the comet would appear again in 1683. It did, after Halley's death, and a little later that Halley predicted. By that time the effect of the larger planets, Jupiter and Saturn, could be taken into account. Newton's gravitational equation was used to make that correction to an accuracy never before possible.

The comet has appeared reliably every 76 years ever since. The next time will be in 2063. Look for it.

## 9. Planetary Bulges

Newton also used the balance of gravitational attraction and centripetal force to explain the planetary bulges. Galileo had observed the "flattened" shape of Saturn. By Newton's time it was known that the earth is slightly flattened as well.

Newton showed that more force was required at the equator than at the poles due to the faster velocity. Because gravity is the same everywhere on a spherical planet, the slight discrepancy cause the equatorial regions to bulge more than the poles.

## 10. Projectiles and Satellites

As we saw in the discussion of weightlessness above, projectile motion and satellite motion are really not very different from one another.

We use the gravitational equation to calculate the trajectory of ballistic missiles (isn't that great to know!) and the orbital parameters of a satellite launch.

## 11. New Planet

In 1781 it was discovered that the orbits of the planets did not quite match the predictions obtained from Newton's gravitational relationship. Here was the first paradigm crisis? Were the laws incomplete, or incorrect?

Two contrary schools of thought arose. One school declared that Universal Gravitation was a bust, should be declared incorrect and a new theory sought. The other school suggested that perhaps there was something else, as yet unobserved, exerting a gravitational tug on the planets. Calculations were done to find the mass and location of such an object if it should exist.

Within a week after that information was made available, two different astronomers had pointed their telescopes in that direction and discovered a new planet, exactly where the Newtonian calculations said it should be. This was Uranus, the first new planet to be discovered since Babylonian times, and the first ever with the telescope. For what it's worth, William Herschel, who traditionally is given credit for the planet's discovery, observed it with a reflecting telescope of Newton's design.

### 11.6. Neptune and Pluto also discovered in this way

When sending a spacecraft between planets, it is not easy to know where it is. There are no roadsigns, and the path is a complex curve representing the "shortest" path through the gravitational fields of the sun and planets for a given speed.

To navigate, spacecraft rely upon a reiterative method of using Newton's gravitational law and the second law. It works like this:

Once a position is known, near earth for example, the direction and distance to each of the planets is calculated. From that, the gravitational equation is used to calculate the direction and magnitude of the force on the spacecraft due to each planet. Those forces are summed as vectors, generating a single resultant force. Using the second law, the net effect of that force on the spacecraft is calculated and its change in speed and direction noted. That produces a new position and the process is repeated again and again. This sounds like a lot of calculations, but your desktop computer could easily handle it. In fact, your desktop has much more computing power than the primitive computers used for the Apollo moon landings in the seventies.Click here to learn how rockets steer in outer space.

## 13. Cavendish Weighs the Earth

One of the most significant of the results of the gravitational relationship came in 1798 when Henry Cavendish (who also did research on the composition of water and air, and studied the properties of a gas that he isolated and described as "inflammable air", now called hydrogen), determined the numerical value of the constant "big G" in the gravitational equation.

### 13.1. measured Gravitational Constant

Other experiments had failed to detect any measurable gravitational effects between two masses in the laboratory. That the Law of Universal Gravitation was accepted by the scientific community without experimental verification is a testament to the thoroughness with which Newton presented it.

To measure "big G" Cavendish designed a system which was isolated from air currents and kept at a constant temperature. Since the deflection was expected to be small, Cavendish used a device called an "optical lever".

A mirror was suspended from the cable which supported the small masses as shown in the figure. A beam of light aimed at the mirror was reflected and read on a scale which was as far away as feasible. This allowed the small twist of the supporting cable to be magnified, simply from the geometry of the triangle.

To translate the angle of twist into a force, Cavendish relied upon Hooke's Law. (You may recall that Hooke was an early adversary of Newton, and the inspiration for his gravitational ponderings).

Hooke's law relates the amount of stretch or twist of a metal, or any elastic substance) to the applied force. It is a linear relationship. Simply stated it is F = kx, where F is the applied force, k is a constant that measures the stiffness of the spring, and x is a measure of the stretch or twist.

Once he had experimentally determined the constant k for his thin supporting cable, it was a simple matter to convert angle of twist to amount of force.

### 13.2. can calculate mass of any object which has a satellite

Once the value of the "big G" is known many calculations are possible. One thing is that the gravitational relationships can be used to calculate the mass of any star or planet that has a satellite. This is done by equating the expression for gravitational force (the Gravitational equation) with the expression for centripetal force. Like this:

We don't need to do this, but if you examine the algebra you will see that it is sound. Furthermore, looking at the result, we see two things.

1. All of the quantities except the mass of the planet are known

2. The relationship includes the term which is the constant from Kepler's third law.

So here, Newton's gravitation explains Kepler's third law too. More correctly, Kepler's third "shakes out" of the mathematical logic.

### 13.3. can calculate densities of planets

Once the mass or a star or a planet is known, along with its diameter, we can calculate its density. From this we learn that not all planets are the same. Then we begin to learn of the differences and similarities of the earth and other astronomical objects without ever leaving our home planet.

## 14. Summary & Conclusions

We have seen how Newton's law of universal gravitation was tested, applied to understand various aspects of physical science, and used to discover new facts about the universe. There is much more, and not just related to gravity. Newton's physics showed us the way to understand and describe forces and motions, and we will see it unfold in future programs as we trace the nature of physical science.