# What is the mathematical relationship between distance and gravity

### Newton's Law of Universal Gravitation

The amount of gravity that something possesses is proportional to its mass and distance between it and another object. This relationship was first published by. Express the Law of Universal Gravitation in mathematical form about the inverse-square relationship of the force between the earth and the moon, . So when finding the force of gravity exerted on a ball of 10 kg, the distance measured from. The inverse-square law, in physics, is any physical law stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution It is also a fading of the distance and mathematical to the source.

So as two objects are separated from each other, the force of gravitational attraction between them also decreases. If the separation distance between two objects is doubled increased by a factor of 2then the force of gravitational attraction is decreased by a factor of 4 2 raised to the second power.

If the separation distance between any two objects is tripled increased by a factor of 3then the force of gravitational attraction is decreased by a factor of 9 3 raised to the second power. Thinking Proportionally About Newton's Equation The proportionalities expressed by Newton's universal law of gravitation are represented graphically by the following illustration.

Observe how the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation. Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below. The constant of proportionality G in the above equation is known as the universal gravitation constant.

The precise value of G was determined experimentally by Henry Cavendish in the century after Newton's death. This experiment will be discussed later in Lesson 3. Using Newton's Gravitation Equation to Solve Problems Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance.

As a first example, consider the following problem. The solution of the problem involves substituting known values of G 6. The solution is as follows: This would place the student a distance of 6.

Two general conceptual comments can be made about the results of the two sample calculations above. First, observe that the force of gravity acting upon the student a. This illustrates the inverse relationship between separation distance and the force of gravity or in this case, the weight of the student.

The student weighs less at the higher altitude. However, a mere change of 40 feet further from the center of the Earth is virtually negligible. A distance of 40 feet from the earth's surface to a high altitude airplane is not very far when compared to a distance of 6. This alteration of distance is like a drop in a bucket when compared to the large radius of the Earth.

### Inverse-square law - Wikipedia

As shown in the diagram below, distance of separation becomes much more influential when a significant variation is made. The second conceptual comment to be made about the above sample calculations is that the use of Newton's universal gravitation equation to calculate the force of gravity or weight yields the same result as when calculating it using the equation presented in Unit 2: In the above figure, the figure on the left hand side indicates the effect of "mass" if the diatnce between the two objects remains fixed at a given value "d".

The right hand figure shows the effect of changing the distance while keeping the mass constant, and the last part of it shows the effect of changing both the distance and the mass. Check your understanding of the inverse square law as a guide to thinking by answering the following questions below. Check Your Understanding 1. Suppose that two objects attract each other with a force of 16 units like 16 N or 16 lb. If the distance between the two objects is doubled, what is the new force of attraction between the two objects?

If the distance is increased by a factor of 2, then distance squared will increase by a factor of 4. Therefore, the force of gravity becomes 4 units. Suppose the distance in question 1 is tripled.

What happens to the forces between the two objects? Again using inverse square law, we get distance squared to go up by a factor of 9.

The force decreases by a factor of 9 and becomes 1. If you wanted to make a profit in buying gold by weight at one altitude and selling it at another altitude for the same price per weight, should you buy or sell at the higher altitude location?

What kind of scale must you use for this work? To profit, buy at a high altitude and sell at a low altitude.

Explanation is left to the student. Check Your Understanding 4. Your weight is nothing but force of gravity between the earth and you as an object with a mass m. As shown in the above graph, changing one of the masses results in change in force of gravity. The planet Jupiter is more than times as massive as Earth, so it might seem that a body on the surface of Jupiter would weigh times as much as on Earth. But it so happens a body would scarcely weigh three times as much on the surface of Jupiter as it would on the surface of the Earth.

Explain why this is so. The effect of greater mass of Jupiter is partly off set by its larger radius which is about 10 times the radius of the earth. This means the object is times farther from the center of the Jupiter compared to the earth. Inverse of the distance brings a factor of to the denominator and as a result, the force increases by a factor of due to the mass, but decreases by a factor of due to the distance squared.

The net effect is that the force increases 3 times. Planetary and Satellite Motion After reading this section, it is recommended to check the following movie of Kepler's laws. Kepler's three laws of planetary motion can be described as follows: Law of Orbits Kepler's First Law is illustrated in the image shown above. The Sun is not at the center of the ellipse, but is instead at one focus generally there is nothing at the other focus of the ellipse.

The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet earth goes around its orbit. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this.

Law of Areas Kepler's second law is illustrated in the preceding figure. The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun. Thus, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion.

**Newtonian Gravity: Crash Course Physics #8**

Hence, by Kepler's second law, the planet moves fastest when it is near perihelion and slowest when it is near aphelion. Law of Periods In this equation P represents the period of revolution for a planet in some other references the period is denoted as "T" and R represents the length of its semi-major axis.

The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The periods for the two planets are assumed to be in the same time units and the lengths of the semi-major axes for the two planets are assumed to be in the same distance units. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit.

Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet Pluto requires years to do the same. The Seasons There is a popular misconception that the seasons on the Earth are caused by varying distances of the Earth from the Sun on its elliptical orbit.

### Why do mass and distance affect gravity?

This is not correct. One way to see that this reasoning may be in error is to note that the seasons are out of phase in the Northern and Southern hemispheres: Seasons in the Northern Hemisphere The primary cause of the seasons is the This means that as the Earth goes around its orbit the Northern hemisphere is at various times oriented more toward and more away from the Sun, and likewise for the Southern hemisphere, as illustrated in the following figure. Thus, we experience Summer in the Northern Hemisphere when the Earth is on that part of its orbit where the N.

Hemisphere is oriented more toward the Sun and therefore the Sun rises higher in the sky and is above the horizon longer, and the rays of the Sun strike the ground more directly. Likewise, in the N. Hemisphere Winter the hemisphere is oriented away from the Sun, the Sun only rises low in the sky, is above the horizon for a shorter period, and the rays of the Sun strike the ground more obliquely.

In fact, as the diagram indicates, the Earth is actually closer to the Sun in the N. Hemisphere Winter than in the Summer as usual, we greatly exaggerate the eccentricity of the elliptical orbit in this diagram.

## Space Environment

The Earth is at its closest approach to the Sun perihelion on about January 4 of each year, which is the dead of the N. On June 21, the summer solstice, the top of the axis the North Pole is pointed directly toward the sun. Areas north of the equator experience longer days and shorter nights. On December 21, the winter solstice, the top of earth's axis is pointed directly away from the sun. Areas north of the equator experience shorter days and longer nights.

Halfway in between the summer and winter solstices are the equinoxes. At these times the earth's axis is pointing neither toward nor away from the sun. On both equinoxes, all locations on earth receive exactly 12 hours of daylight and 12 hours of night. Southern Hemisphere Seasons As is clear from the preceding diagram, the seasons in the Southern Hemisphere are determined from the same reasoning, except that they are out of phase with the N.

Hemisphere seasons because when the N. Hemisphere is oriented toward the Sun the S. Hemisphere is oriented away, and vice versa: It is strongly recommended that you also visit the following web address for more details.

If the links do not take you to the referenced web page, please copy the link and paste it to a new page Ocean Tides Lunar Tides The tides at a given place in the Earth's oceans occur about an hour later each day.