Gravitation Applications

Lecturer: Professor Stephen T. Thornton

Reading Quiz

Astronauts Astronauts float in the float in the space space shuttle shuttle because:because:

A) They are so far from Earth that Earth’s gravity doesn’t act any more.

B) Gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward.

C) While gravity is trying to pull them inward, they are trying to continue on a straight-line path.

D) Their weight is reduced in space so the force of gravity is much weaker.

Astronauts in the space shuttle float because they are in “free fall” around Earth, just like a satellite or the Moon. Again, it is gravity that provides the centripetal force that keeps them in circular motion.

Reading Quiz

Astronauts Astronauts float in the float in the space shuttle space shuttle because:because:

A) They are so far from Earth that Earth’s gravity doesn’t act any more.

B) Gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward.

C) While gravity is trying to pull them inward, they are trying to continue on a straight-line path.

D) Their weight is reduced in space so the force of gravity is much weaker.

Follow-up:Follow-up: How weak is the value of How weak is the value of gg at an altitude of at an altitude of 300 km300 km??

Last TimeHistory of gravitation

Newton’s law of universal gravitation

Kepler’s laws

Free floating in space

TodayOrbital maneuvers

Ocean tides

Geophysical applications

Free floating in space

Satellites and weightlessnessPrinciple of EquivalenceBlack holes

Conceptual QuizA) It’s in Earth’s gravitational fieldA) It’s in Earth’s gravitational fieldB) The net force on it is zeroB) The net force on it is zeroC) It is beyond the main pull of C) It is beyond the main pull of

Earth’s gravityEarth’s gravityD) It’s being pulled by the Sun as D) It’s being pulled by the Sun as

well as by Earthwell as by EarthE) none of the above is precise E) none of the above is precise

enoughenough

The Moon does The Moon does not crash into not crash into Earth because:Earth because:

The Moon does not crash into Earth because of its high speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity providing the centripetal force.

Conceptual Quiz

The Moon does The Moon does not crash into not crash into Earth because:Earth because:

Follow-up:Follow-up: What happens to a satellite orbiting Earth as it slows? What happens to a satellite orbiting Earth as it slows?

A) It is attracted to EarthA) It is attracted to EarthB) The net force on it is zeroB) The net force on it is zeroC) It is beyond the main pull of C) It is beyond the main pull of

Earth’s gravityEarth’s gravityD) It’s being pulled by the Sun as D) It’s being pulled by the Sun as

well as by Earthwell as by EarthE) None of the above is precise E) None of the above is precise

enoughenough

The Global Positioning System

Orbital Maneuvers

Move to higher orbit

Orbital Maneuvers

Move to lower orbit

Ocean TidesArrows denote force due to the moon relative to the force at the center of Earth.

Newton finally correctly explained tides!

Gravitation Force Due To Ring. A mass M is ring shaped with radius r. A small mass m is placed at a distance x along the ring’s axis as shown in the figure. Show that the gravitational force on the mass m due to the ring is directed inward along the axis and has magnitude

[Hint: Think of the ring as made upof many small point masses dM;sum over the forces due to each dMand use symmetry.]

322 2

GMmxF .

x r

Vector Form of Newton’s Universal Gravitation

If there are many particles, the total force is the vector sum of the individual forces:

21

4

15F

3 51F

14F

13F

12F

We can relate the gravitational constant to the local acceleration of gravity. We know that on the surface of the Earth:

Solving for g gives:

g can be measured to 1 part in 109 so that mineral and oil deposits can be detected using sensitive gravitometers.

2E

E

mmmg G

r=

2E

E

mg G

r=

The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical.

Geosynchronous satellite.

A geosynchronous satellite stays above the same point on the Earth, which is possible only if it is above a point on the equator. Such satellites are used for TV and radio transmission, for weather forecasting, and as communication relays. They must have an orbit of precisely 24 hours. In order to do that, they must be about 22,000 miles above the Earth and have a precise speed.

222

2

23

2

2

4

E E

E

mM GMmv rG v

r r r T

GM Tr

24 hrT

Lagrange point

The mathematician Joseph-Louis Lagrange discovered five special points in the vicinity of the Earth’s orbit about the Sun where a small satellite (mass m) can orbit the Sun with the same period T as Earth’s (= 1 year).

One of these “Lagrange Points,” called L1, lies between the Earth and Sun on the line connecting them.

Several satellites are being placed in Lagrange points. We probably will not be able to service them like we have done with the Hubble.

Sun

Satellites and “Weightlessness”Objects in orbit are said to experience “weightlessness”. They do have a gravitational force acting on them, though! The satellite and all its contents are in free fall, so there is no normal force. This is what leads to the experience of weightlessness.

More properly, this effect is called apparent weightlessness, because the gravitational force still exists. It can be experienced on Earth as well, but only briefly:

Gravitational Field

The gravitational field is the gravitational force per unit mass:

The gravitational field due to a single mass M is given by:

Fg

m=

2ˆ

GMg r

r=-

Principle of Equivalence

Inertial mass: the mass that appears in Newton’s second law.

Gravitational mass: the mass that appears in the universal law of gravitation.

Principle of equivalence: inertial mass and gravitational mass are the same.

We can do no experiment to tell the difference between gravitational and inertial mass. Fundamental tenet of the General Theory of Relativity.

One way to visualize the curvature of space (a two-dimensional analogy):

If the gravitational field is strong enough, even light cannot escape, and we have a black hole. Einstein predicted in 1915 that light should be attracted by gravity to mass.

Light should be deflected by a massive object. On the right side, the person can not tell whether acceleration caused the light to bend or whether gravity did it.

At rest

This bending has been measured during total solar eclipses.

Gravitational Lensing

Massive stars can collapse under the gravitational force. They can become black holes, and nothing can escape even light.

Einstein showed gravity even bends light!

Gravity Assisthttp://www.youtube.com/watch?v=I3F88w3LkiI

Milky Way Galaxy. The Sun rotates about the center of the Milky Way Galaxy (see figure) at a distance of about 30,000 light-years from the center (1 ly = 9.5 x 1015 m). If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun (2 x 1030 kg), how many stars would there be in our Galaxy?

Conceptual QuizA) A) B) B) C) C) D) it’s the sameD) it’s the sameE) 2E) 2

Two satellites A and B of the same Two satellites A and B of the same mass are going around Earth in mass are going around Earth in concentric orbits. The distance of concentric orbits. The distance of satellite B from Earth’s center is satellite B from Earth’s center is twice that of satellite A. What is thetwice that of satellite A. What is the ratio ratio of the centripetal force acting of the centripetal force acting on B compared to that acting on A?on B compared to that acting on A?

18

12

14

Using the Law of Gravitation:

we find that the ratio is .we find that the ratio is .

Conceptual Quiz

2RMm

GF

A) A) B) B) C) C) D) it’s the sameD) it’s the sameE) 2E) 2

Two satellites A and B of the same Two satellites A and B of the same mass are going around Earth in mass are going around Earth in concentric orbits. The distance of concentric orbits. The distance of satellite B from Earth’s center is satellite B from Earth’s center is twice that of satellite A. What is thetwice that of satellite A. What is the ratio ratio of the centripetal force acting of the centripetal force acting on B compared to that acting on A?on B compared to that acting on A?

Note the 1/r2 factor

18

12

14

14

Conceptual QuizA planet of mass m is a distance d from Earth. Another planet of mass 2m is a distance 2d from Earth. Which force vector best represents the direction of the total gravitation force on Earth?

A BC

D

E

2d

d

2m

m

Earth

A BC

D

E

2d

d

2m

mThe force of gravity on the

Earth due to mm is greatergreater than

the force due to 22mm, which

means that the force

component pointing down in

the figure is greater than the

component pointing to the

right.

F2m = GME(22mm) / (22dd)2 = GM GMmm / / dd 22

Fm = GME mm / dd 2 = GMGMmm / / dd 22

A planet of mass m is a distance d from Earth. Another planet of mass 2m is a distance 2d from Earth. Which force vector best represents the direction of the total gravitation force on Earth?

Conceptual Quiz

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