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Newton's Law of Universal Gravitation
Introduction
 Every particle
in the universe attracts every other particle with a force that is proportional
to the product of their masses and inversely proportional to the square of the
distance between them. This force acts along the line joining the two particles.
G = universal gravitational constant
= 6.67 x 10 -11 N m2
kg -2
m1
m2
F = G 覧------
r
r must be measured from the centers of the objects.
Henry Cavendish measured the Force between two balls using the apparatus
to the left. He did this in 1798 by measuring the extremely small force
necessary to twist the thin fiber created by the attraction of the two masses.
Table of Contents
Problems
| 1. |
Noel and Walton are in a spacecraft 800. km above the surface of the
Earth (mass = 5. 97 x 10 24
kg). The combined mass of the spacecraft and astronauts is 4500. kg. If the
distance from the Earth's center to surface is 6380 km, what is the force of
gravity on the spacecraft? What is the acceleration of gravity?
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| 2. |
Find the net force on the Moon (mass = 7.35 x 10 22 kg) due
to the gravitational attraction of both the Earth (mass = 5. 97 x 10 24
kg) and the Sun( mass = 1. 99 x 10 30 kg) when line between the
Earth and Moon is at right angle to the Sun. The Earth is 3. 84 x 10 5
km from the Moon and the Sun is 1.50 x 10 8 km from
the Earth.
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| 3. |
Doug, 85.0 kg, and Betty, 55.0 kg,
fall in love at first sight from a distance of 5.00 m. What
is the force of gravity pulling them closer together?
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| 4. |
After taking 98 years to reach Alpha
Centuri, the astronauts measure an acceleration of 12.4 m s -2
when in orbit 1500. km above a newly discovered planet. The planet
has a diameter of 1.80 x 10 4 km and the spacecraft has a
mass of 1.00 x 10 4 kg. What is the mass of the new
planet? |
Table of Contents
Answers
| 1. |
Noel and Walton are in a spacecraft 800. km above the surface of the
Earth (mass = 5. 97 x 10 24
kg). The combined mass of the spacecraft and astronauts is 4500. kg. If the
distance from the Earth's center to surface is 6380 km, what is the force of
gravity on the spacecraft? What is the acceleration of gravity?
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|
m1
m2
4500. kg x 5. 97 x 10 24
kg
F = G 覧------ = 6.67 x 10 -11 N m2
kg -2 -------------------------------------------
= 3.48 x 10 4 N
r2
( 8.00 x 10 5 m + 6.38 x 10 6 m)2
F 3.48 x 10 4
N
F = ma => a = ------ = --------------------- = 7.73 m
s -2
m
4500. kg
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|
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| 2. |
Find the net force on the Moon (mass = 7.35 x 10 22 kg) due
to the gravitational attraction of both the Earth (mass = 5. 97 x 10 24
kg) and the Sun( mass = 1. 99 x 10 30 kg) when line between the
Earth and Moon is at right angle to the Sun. The Earth is 3. 84 x 10 5
km from the Moon and the Sun is 1.50 x 10 8 km from
the Earth.
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|
The distance of the Moon to the Sun is same as the distance from the Earth
to the Sun to the given number of significant figures.
m1
m2
1. 99 x 10 30 kg x 7.35 x 10 22 kg
FS = G 覧---- = 6.67 x 10 -11 N m2
kg -2 ----------------------------------------------
=
r2
(1.50 x 10 11
m)2
F of Sun on Moon is 4.34 x 10 20 N
m1
m2
5. 97 x 10 24
kg x 7.35 x 10 22 kg
FE = G 覧---- = 6.67 x 10 -11 N m2
kg -2 ---------------------------------------------- =
r2
(3. 84 x 10 8 m)2
F of Earth on Moon is 1.98 x 10 20 N
Since the forces are at right angles to each
other at this point in the Moon's orbit, we can use the hypotenuse of the
right triangle as the total force on the Moon.
Force on Moon = ( (4.34
x 10 20 N)2 + (1.98
x 10 20 N)2 ) 1/2 = 4.77 x 10 20
N
The angle measured counter-clockwise from the
Moon to the Sun line is
4.34 x 10 20 N
Angle = Tan -1 ( ---------------------- ) = 65.5 o
1.98 x 10 20 N
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| 3. |
Doug, 85.0 kg, and Betty, 55.0 kg,
fall in love at first sight from a distance of 5.00 m. What is
the force of gravity pulling them closer together?
|
|
m1
m2
85.0 kg x 55.0 kg
F = G 覧------ = 6.67 x 10 -11 N m2
kg -2 ---------------------------
= 1.25 x 10 -8 N
r2
( 5.00 m )2
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| 4. |
After taking 98 years to
reach Alpha Centuri, the astronauts measure an acceleration of 12.4 m s
-2 when in orbit 1500. km above a newly discovered planet. The
planet has a diameter of 1.80 x 10 4 km and the spacecraft has
a mass of 1.00 x 10 4 kg. What is the mass of the new
planet?
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|
F = ma = 12.4 m s -2
x 1.00 x 10 4 kg = 1.24 x 10 5 N
mS
mp
F r2
1.24 x 10 5 N ( 1.50 x 10 6 m + 9.00 x 10
6 m) 2
F = G 覧------ => mp =
---------- =
---------------------------------------------------------------
r2
G mS 6.67 x 10 -11 N m2
kg -2 x 1.00 x 10 4 kg
mp = 2.05 x 10 25 kg
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Table of Contents
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