Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Chapter 12

Gravitation

Goals for Chapter 12

•  To study Newton’s Law of Gravitation •  To consider gravitational force, weight, and

gravitational energy •  To compare and understand the orbits of

satellites and celestial objects

Introduction •  Looking at the picture of

Saturn, we see a very organized ring around the planet. Why do the particles arrange themselves in such orderly fashion?

•  From Copernicus and Galileo to Hubble and NASA, centuries of scientists have struggled to characterize gravitation and celestial motion.

Gravity: what is different about it? What is the difference between Dr. Sinova and gravity?

Can be attractive Can be repulsive

Gravity is ALWAYS attractive

Newton’s Law of Gravitation •  The gravitational force

is always attractive and depends on both the masses of the bodies involved and their separations.

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Fg =Gm1m2

r2

Henry Cavendish determines G

•  Gravitational forces were relative until 1798 when Henry Cavendish made the sensitive measurement to determine a numerical value for the constant G.

G remains the least precise known of all the universal constants of nature!!

Calculate gravitational forces: superposition

Calculate the force on the object at the origin by two other objects: The two forces add up (superposition)

Weight (skip Weight Watchers, just climb upward)

•  Gravity (and hence, weight) decreases as altitude rises.

Gravitational potential energy •  Objects changing their distance from earth are also

changing their potential energy with respect to earth.

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PE = −G m1m2

r12This is the true potential energy. The zero level is set when they are very far apart

Escaping from the Earth What is the velocity you need to shoot straight up from the surface of the earth an not come back (conservation of energy)?

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KE1 + PE1 = KE∞ + PE∞

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12m v2−G mME

RE

= 0 + 0

⇒ v =2GME

RE

=1.12 ×104 m/s = 25,000 mph

Satellite motion

Satellite circular motion

The force is radial so Newton’s 2nd law reads:

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r : G MEms

R2= ms

v2

R= ms

4π 2RT 2

⇒ v = G ME

R

⇒ T =4π 2R3

GME

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T ∝ R3 / 2

Kepler’s laws for planetary motion •  Each planet moves in an

elliptical orbit with the sun at one focus.

•  A line connecting the sun to a given planet sweeps out equal areas in equal times (due to conservation of angular momentum).

•  The periods of the planets are proportional to the 3/2 powers of the major axis lengths in their orbits.

How big is a black hole?

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v =2GME

RE

Remember the escape velocity?

But the maximum speed in the universe is the speed of light, c. What does this mean?

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c =2GMR

⇒ R =2GMc 2

The Earth has a mass of 5.97 x1024 Kg

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R =2(6.6742 ×10−11)(5.97 ×1024 )

(3 ×108)2 m = 8.9 mm