The Universe Adventure - Gravitational Equation
Gravity and Mass. Mass v. Force. From the graph we can see that as the Newton combined the inverse square relation between distance and. 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. GRAVITY AND MASS Which mass exerts a greater force of gravity and why? 7. JUST THE OPPOSITE The relationship between distance and.
ALL objects attract each other with a force of gravitational attraction. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance which separates their centers. Weight and the Gravitational Force We have seen that in the Universal Law of Gravitation the crucial quantity is mass.
In popular language mass and weight are often used to mean the same thing; in reality they are related but quite different things. What we commonly call weight is really just the gravitational force exerted on an object of a certain mass. We can illustrate by choosing the Earth as one of the two masses in the previous illustration of the Law of Gravitation: Thus, the weight of an object of mass m at the surface of the Earth is obtained by multiplying the mass m by the acceleration due to gravity, g, at the surface of the Earth.
The acceleration due to gravity is approximately the product of the universal gravitational constant G and the mass of the Earth M, divided by the radius of the Earth, r, squared.
We assume the Earth to be spherical and neglect the radius of the object relative to the radius of the Earth in this discussion. Mass and Weight Mass is a measure of how much material is in an object, but weight is a measure of the gravitational force exerted on that material in a gravitational field; thus, mass and weight are proportional to each other, with the acceleration due to gravity as the proportionality constant.
It follows that mass is constant for an object actually this is not quite true as described by the Relativity Theorybut weight depends on the location of the object. For example, if we transported the preceding object of mass m to the surface of the Moon, the gravitational acceleration would change because the radius and mass of the Moon both differ from those of the Earth.
Thus, our object has mass m both on the surface of the Earth and on the surface of the Moon, but it will weigh much less on the surface of the Moon because the gravitational acceleration there is a factor of 6 less than at the surface of the Earth.
Using Equations as a Guide to Thinking The inverse square law proposed by Newton suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centers.
Altering the separation distance r results in an alteration in the force of gravity acting between the objects. Since the two quantities are inversely proportional, an increase in one quantity results in a decrease in the value of the other quantity. That is, an increase in the separation distance causes a decrease in the force of gravity and a decrease in the separation distance causes an increase in the force of gravity.
Furthermore, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. So if the separation distance is doubled increased by a factor of 2then the force of gravity is decreased by a factor of four 2 raised to the second power. And if the separation distance r is tripled increased by a factor of 3then the force of gravity is decreased by a factor of nine 3 raised to the second power.
Thinking of the force-distance relationship in this way involves using a mathematical relationship as a guide to thinking about how an alteration in one variable effects the other variable. Equations can be more than merely recipes for algebraic problem-solving; they can be "guides to thinking.
Newton's Law of Universal Gravitation
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. 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.
Relationship Between Gravity & the Mass of the Planets or Stars | Sciencing
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.
- Planetary and Satellite Motion
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. 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.
This is not correct. 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.
Newton's Law of Universal Gravitation
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. 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: