Ch 13 Gravitation
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State the Universal law of Gravitation.
"every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."
If the particles have masses m1
and are separated by a distance r
, the magnitude of this gravitational force is:
Where G is a conatant, called the universal gravitational constant. Its value in SI units is:
Describe with your own words the Inverse Square Law according to the idea of gravity.
- The Inverse Square Law tells us that the closer particles are to each other the stronger the gravity force between them will be. Ckeck out this drowing:
It shows that distance is inversely related to the intensity of the light (flash light is the source). Notice the the father the light rays travelled the light had to be more spread out. If you pay close attention you'll see that if we square the distance we get the number of panels needed to catch the light emmited by the source. The way that Newton relates this is by saying that the force of gravity will be 1/r2
= distance. In the case of gravity of course we apply the formula which tells us the same thing.
- Check out how we can find the relationship of 1/r2 here.
What does the following formula means?
Check out the drowing:
- The formula :
- is telling us that if there is a force coming from particle 1 onto particle 2, and if we consider the force from particle 1 to be going into the positive direction. Then, the force of particle 2 must be going on the negative direction since the particles are atracted to each other. Hence, the negative sign before G in the formula.
What is the difference between the gravity (g) on planet Earth and the Universal Gravitational constant G.
"The symbol g
represents the magnitude of the free-fall acceleration near the planet. At the surface of the Earth, g has and average value of 9.80 m/s2
. On the other hand, G
is the universal constant that has the same value anywhere in the Universe."
What is the magnitude of the force exerted by the Earth on a particle of mass m near the Earth's surface? Hint: think of the mathematical formula needed to calculat gravitational force on any object with in this Universe.
Well, let's see:
- We need to include the mass of the Earth = ME
- Also, we must include the mass of the particle = mpPlus, the inverse square distance of both masses times the gravitational constant G, which it just happens to be the center of the Earth itself. So, R2 earth sounds good. Ah!! Then this may work:
What do we call the magnitude of the gravitational force of an object near the Earth?
(mass x g)
We can also understand this relationship through the formulas:
- This equations relate the free-fall acceleration g to physical parameters of the Earth-its mass and radius-and explains the origin of the value 9.80 m/s2
Concider an object about of mass m located a distance h above the Earth's surface of a distance r from the Earth's center, where r = Rearth + h. State the formula required to calculate the magnitude of the gravitational force acting on this object.
If the magnitude of g decreases as altitude increases. How can we calculate the decrease of g as an object gets farther away for the Earth's surfice?
Since we know that the weight of an object is calculated by gm = weigth
. Then, we can simply separate g
and find the value of g
alone with the following formula:
This can be beter explained by looking into the primary formula to calculate the magnitude of a gravitational force felt by an object:
State the formula to find the mass of the Earth using the idea of gravity and gravitational constant.
State the formula to find the density of the earth using the idea of gravitation.
kg / m3
State Kepler's Laws
- 1. All planets move in elliptical orbits with the Sun at one focus.
- 2. The radious vector dreawn from the Sun to the planet sweeps out equal areas in equal time intervals.
- 3. The square of the orbital period of any planet is proportional top the cube of the semimajor axis of the elloptical orbit.
Describe the geometry for an Ellipse.
- An ellipse is mathematically defined by chosing two points F1 and F2, each of which is a called a focus, and then drawing a curve through points for which the sum of the distances r1 and r2 form F1 and F2, respectively, is constant.
- In other words, this means that if we draw a line from focus 1 directly to a point on the ellipse (length r1) and then form taht point back to focus 2 (length r2), the addition of the length of r1 and r2 is constant to the sum of r1 and r2 even if we choose a different point on the ellipse.
Where is the Sun located when we talk about its location within an Ellipse?
The Sun is located at one of the focuses
. Where the distance from the Sun to a random point wihin the Ellipse is call r1
and the distance from from the same point to second focus is called r2
. The sum of r1
is constant for any point within the Ellipse drawn as previously described.
Essentially the RED, YELLOW and GREEN lines have the same lenght.
What is the area sweept by from the sun to the planets orbit as time changes dt?
Describe Kepler's First Law.
Keppler' First law states that the orbit of every planet is an ellipse with the sun at the two foci.
Describe Kepler's Second Law.
A line joining the planet and the Sun sweeps out equal areas during equal intervals of time.
State Keppler's Thrid Law.
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
- The 3rd Law can also be described as (The Law of periods) that relates the rime required for a planet to make one complete trip around the Sun to its mean distance from the Sun
Find the mass of the Sun using Erth's orbit (Tgiven = period = time Earth takes to go around the Sun) and the distancegiven from Earth to the Sun.
SInce we know that Keppler's 3rd law relates the time required for a planet to make one complete trip around the Sun to its distance from the Sun. Then, the following mathematical formula can help:
- r = the a distance on an ellipse in this case the orgbit of Earth around the Sun
- T= the period of Earth going around the Sun (time it takes to make a complete circle ONCE around the Sun)
Explain the concept of a gravitational field and how do we measure it.
When trying to explain why panets exert forces into one another even thought they are so far apart we refer to a gravitational field. A gravitational field is that between the two palents and the force of gravity can at a certain point can be found by placing a test particle in order to measure the force of the gravitational field at that exact point. The following formula allows to calculate:
the formula originates from:
Imagine an object mass m, near the Earth's surfice. What is the magnitude of gravitational force acting on the object and therefore the gravitational field felt by the object.
- This formula gives us the magnitude of the gravitational force felt by the object.
If we want to know the the gravitational field (
) at a distance r
from the center of the earth we can use this formula:
State the mathematical formula to calculate the change in Potential gravitational energy where ri = infinity
= 0 at r1
Calculate the gavitational potential energy for two any particles (m1 and m2)
Explain the concept of Energy considerations in a planetary and satellite motion.
If we assume the object of mass M is at reat the total Mechanical Energy if the system when the objects are separated by r
is the sume of the Kinetic Energy of the object of mass m
and the Potential Energy of the system.
State mathematical origin and formula to calculate the Energy on ciruclar orbits.
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