LECTURE 26GRAVITY

Instructor: Kazumi Tolich

Lecture 26

¨ Reading chapter 12-1 to 12-2¤ Newton’s law of universal gravitation¤ Gravitational attraction of spherical bodies

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Newton’s universal law of gravitation

¨ The force of gravity between any two objects of mass 𝑚" and 𝑚#is attractive and of magnitude given by

𝐹 = 𝐺𝑚"𝑚#𝑟#

¨ 𝐺 = 6.673× 10-11 N·∙m2/kg2 is the universal gravitational constant, and 𝑟 is the distance between the centers of mass of the objects.

¨ If a given mass is acted on by gravitational interactions with a number of other masses, the net force acting on it is the vector sum of each of the forces.

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Quiz: 1

¨ Do you exert a gravitational pull on Andromeda galaxy?A. YesB. No

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Quiz: 26-1 answer

¨ Yes¨ Since you and Andromeda galaxy both have

mass, they exert gravitational force on each other.¨ Although Andromeda galaxy is very far away,

the force of gravity is not zero.¨ Newton’s law of gravity tells you that everything

pulls on everything else in the universe.¨ You are a bunch of attractive people!

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𝐹 = 𝐺𝑚"𝑚#𝑟#

Quiz: 2

¨ Compare the gravitational force on Andromeda galaxy by you, 𝐹(), to the gravitational force on you by Andromeda galaxy, 𝐹)(.

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Quiz: 26-2 answer

¨ 𝐹() = 𝐹)(¨ Each object experiences a force of the same magnitude, acting in

opposite direction as the force of gravity between two objects forms an action-reaction pair (Newton’s 3rd law).

¨ 𝐹 = 𝐺 *+*,-,

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Quiz: 3

¨ Think about yourself standing on Earth. What is the distance between you and Earth, 𝑟, in 𝐹 = 𝐺 *+*,

-,?

A. 0B. The radius of the earth, 𝑅/C. 2𝑅/D.

"#𝑅/

E.1#𝑅/

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Quiz: 26-3 answer

¨ The radius of the earth, 𝑅/¨ For a finite-sized spherically symmetric object exerting a

force on a particle outside the sphere, the object can be treated as a particle with its entire mass located at the center.

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Example: 1

¨ A spaceship is on a straight-line path between Earth and the moon. At what distance from Earth is the net gravitational force on the spaceship by Earth and the moon zero?

¨ 𝑟2/ = 3.84×10: m¨ 𝑚2 = 7.35×10## kg¨ 𝑚/ = 5.97×10#> kg

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Example: 2

¨ When the earth, moon, and sun form a right triangle, with the moon located at the right angle as shown, the moon is in its third-quarter phase. (The earth is viewed here from above its North Pole.) Find the magnitude and direction of the net force acting on the sun. Give the direction relative to the line connecting the sun and the moon.

¨ 𝑟2/ = 3.84×10: m

¨ 𝑟?/ = 1.50×10"" m

¨ 𝑚2 = 7.35×10## kg

¨ 𝑚/ = 5.97×10#> kg

¨ 𝑚? = 2.00×101@ kg

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Acceleration of gravity, 𝑔

¨ Acceleration of gravity due to an object with a mass 𝑚, a distance 𝑟 from its center of mass is given by

𝑔 = 𝐺𝑚𝑟#

¨ The value of 𝑔 varies slightly on the surface of Earth.

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𝑔 is largest in red areas and smallest in the dark blue areas.

𝑔 on the moon

¨ Why can astronauts on the moon jump around as if they are floating?¤ https://www.youtube.com/watch?v=efzYblYVUFk

¤ The acceleration of gravity near the surface of the moon is about 1/6th of that of Earth: 𝑔2 = 𝐺 *B-B, = 1.62  m s#⁄

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y = -1.5998x + 3.8668

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Example: 3

¨ The acceleration due to gravity on the moon’s surface is known to be about one-sixth the acceleration due to gravity on the earth. Given that the radius of the moon is roughly one-quarter that of the earth, find the mass of the moon in terms of the mass of the earth.

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Demo: 1/ Measurement of 𝐺

¨ 𝐺 was first measured in 1798 by Cavandish using a torsion balance.

¨ 𝐹 = 𝐺 *+*,-,

¨ The initial measurement yielded 𝐺  = 6.754× 10-11 N·∙m2/kg2

¨ The Cavandish experiment is often referred as “weighing the earth” since by measuring 𝐺, one can calculate the mass of the earth.

¨ Modern measurement of 𝐺¤ The most accurate measurement of 𝐺 to date was made at University of

Washington, which measured 𝐺 to an accuracy of 0.0014%.¤ Using similar apparatus, the gravity group has also shown that Newton’s universal

law of gravity is correct down to separations of less than 44 μm.

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0 2000 4000 60000

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Density of earth

¨ As soon as Canvandish determined the mass of the earth, geologists calculated the average density of the earth: average mass per unit volume.

¨ The average density they found, ~5.5 g/cm3, is significantly higher than typical rocks found near the surface of the earth, ~3 g/cm3.

¨ They concluded that the interior of the earth must have a greater density than its surface.

¨ Analyzing the seismic data, now we know that the earth has a complex interior structure, including a solid inner core with a density of ~15 g/cm3.

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