LECTURE 26
GRAVITY
Instructor: Kazumi Tolich
Lecture 26
2
¨
Reading chapter 12-1 to 12-2
¤ Newton’s
law of universal gravitation
¤ Gravitational attraction of spherical bodies
Newton’s universal law of gravitation
3
¨
The force of gravity between any two objects of mass 𝑚" and 𝑚#
is attractive and of magnitude given by
𝐹=𝐺
¨
¨
𝑚"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.
Quiz: 1, 2, and 3
4
Example: 1
5
¨
A spaceship is on a straightline 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?
Example: 2
6
¨
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.
Acceleration of gravity, 𝑔
7
¨
Acceleration of gravity due to an object with a mass 𝑚, a
distance 𝑟 from its center of mass is given by
𝑔=𝐺
¨
𝑚
𝑟#
The value of 𝑔 varies slightly on the surface of Earth.
𝑔 is largest in red areas and
smallest in the dark blue areas.
𝑔 on the moon
8
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: 𝑔) = 𝐺
1.5
Vertical verlocity (m/s)
¨
y = -1.5998x + 3.8668
1
0.5
0
1.8
2.3
2.8
-0.5
-1
-1.5
time (s)
3.3
*+
,+
= 1.62 m⁄s#
Example: 3
9
¨
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.
Demo: 1/ Measurement of 𝐺
10
¨
𝐺 was first measured in 1798 by Cavandish using a torsion balance.
¨
𝐹=𝐺
*6*,-
¨
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.
Density of earth
11
¨
¨
¨
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.
Density [g/cm3]
¨
15
Upper
Mantle
Lower
Mantle
Outer
Core
Inner
Core
10
5
0
0
2000
4000
6000
Depth [km]