Download Gravitation, Potential Energy, Circular Orbits

PHYC 160
Start Gravitation — Ch 13
Lecture #32
Exam 3 results
Newton’s law of gravitation
• Law of gravitation: Every particle of
matter attracts every other particle
with a force that is directly
proportional to the product of their
masses and inversely proportional to
the square of the distance between
them.
• The gravitational force can be
expressed mathematically as
= Gm1m2/r2, where G is the
gravitational constant.
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Fg
Gravitational Force
• The gravitational force is very weak (?)…
• For two masses each of 1kg, separated by 1m:
FG
m1m2
G 2
r
G
1kg 1kg
1m
2
6.6742 10
11
N
• That is, the proportionately constant, G = 6.6742x10-11Nm2/kg2.
• So why is it that we feel such a strong force on us?
• The earth’s mass = 5.98x1024kg!!
Q13.2
The planet Saturn has 100 times the mass of the Earth and is
10 times more distant from the Sun than the Earth is.
Compared to the Earth’s acceleration as it orbits the Sun, the
acceleration of Saturn as it orbits the Sun is
A. 100 times greater.
B. 10 times greater.
C. the same.
D. 1/10 as great.
E. 1/100 as great.
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Gravitation and spherically symmetric bodies
• The gravitational interaction
of bodies having spherically
symmetric mass distributions
is the same as if all their mass
were concentrated at their
centers. (See Figure 13.2 at the
right.)
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Q13.1
The mass of the Moon is 1/81 of the mass of the Earth.
Compared to the gravitational force that the Earth exerts on the
Moon, the gravitational force that the Moon exerts on the Earth is
A. 812 = 6561 times greater.
B. 81 times greater.
C. equally strong.
D. 1/81 as great.
E. (1/81)2 = 1/6561 as great.
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Determining the value of G
• In 1798 Henry Cavendish made the first measurement of the
value of G. Figure 13.4 below illustrates his method.
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Superposition of Force
• Remember that if two (or more) forces are
acting on a body, the net force is just the
(vector) sum of all the forces:
F1
FNet
F2
=
F3
• The same is true for gravitational forces.
Example
Some gravitational calculations
• Example 13.1 shows how to calculate the gravitational force
between two masses.
• Example 13.2 shows the acceleration due to gravitational force.
• Example 13.3 illustrates the superposition of forces, meaning that
gravitational forces combine vectorially. (See Figure 13.5 below.)
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Spherically Symmetric Bodies
• We can do the same thing for continuous
distributions of mass.
FNet
dF1
dF2
Spherically Symmetric Bodies
• For a spherically symmetric distribution, the
net force is pointed to the center, and has the
magnitude as if all the mass was located at the
center:
m1m2
FG
G
FNet
m1
r
r
2
m2
Search For Oil
• If the mass is not spherically symmetric, this is
no longer the case:
FNet
m2
• This can be used to look for non-uniform
densities in the earth’s crust (oil, uranium,
etc.).
Weight
• The weight of a body is the total gravitational force exerted
on it by all other bodies in the universe.
• At the surface of the earth, we can neglect all other
gravitational forces, so a body’s weight is w = GmEm/RE2.
• The acceleration due to gravity at the earth’s surface is
g = GmE/RE2.
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Weight
• The weight of a body decreases with its distance from the earth’s
center, as shown in Figure 13.8 below.
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Interior of the earth
• The earth is approximately
spherically symmetric, but it
is not uniform throughout its
volume, as shown in Figure
13.9 at the right.
•
Follow Example 13.4,
which shows how to
calculate the weight of a
robotic lander on Mars.
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Gravitational potential energy
• Follow the derivation of
gravitational potential
energy using Figure 13.10
at the right.
• The gravitational potential
energy of a system
consisting of a particle of
mass m and the earth is U =
–GmEm/r.
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Gravitational potential energy depends on distance
• The gravitational potential
energy of the earth-astronaut
system increases (becomes less
negative) as the astronaut
moves away from the earth, as
shown in Figure 13.11 at the
right.
Copyright © 2012 Pearson Education Inc.
Q13.4
Compared to the Earth, Planet X has twice the mass and twice
the radius. This means that compared to the amount of energy
required to move an object from the Earth’s surface to infinity,
the amount of energy required to move that same object from
Planet X’s surface to infinity is
A. 4 times as much.
B. twice as much.
C. the same.
D. 1/2 as much.
E. 1/4 as much.
© 2012 Pearson Education, Inc.
From the earth to the moon
• To escape from the earth, an object must have the escape
speed.
• Follow Example 13.5 using Figure 13.12 below.
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Escape Velocity
From Conservation of energy:
The motion of satellites
•
The trajectory of a projectile fired from A toward B depends on its initial speed. If it is fired
fast enough, it goes into a closed elliptical orbit (trajectories 3, 4, and 5 in Figure 13.14
below).
Copyright © 2012 Pearson Education Inc.
Q13.5
A satellite is moving around the Earth in a circular orbit.
Over the course of an orbit, the Earth’s gravitational force
A. does positive work on the satellite.
B. does negative work on the satellite.
C. does positive work on the satellite during part of the orbit
and negative work on the satellite during the other part.
D. does zero work on the satellite at all points in the orbit.
© 2012 Pearson Education, Inc.
Q13.6
A planet (P) is moving around the Sun
(S) in an elliptical orbit. As the planet
moves from aphelion to perihelion, the
Sun’s gravitational force
A. does positive work on the planet.
B. does negative work on the planet.
C. does positive work on the planet
during part of the motion and
negative work during the other part.
D. does zero work on the planet at all
points between aphelion and
perihelion.
© 2012 Pearson Education, Inc.
Q13.7
A planet (P) is moving around the Sun
(S) in an elliptical orbit. As the planet
moves from aphelion to perihelion, the
planet’s angular momentum
A. increases during part of the
motion and decreases during the
rest of the motion.
B. increases at all times.
C. decreases at all times.
D. remains the same at all times.
© 2012 Pearson Education, Inc.
Q13.8
Star X has twice the mass of the Sun. One of Star X’s planets
has the same mass as the Earth and orbits Star X at the same
distance at which the Earth orbits the Sun.
The orbital speed of this planet of Star X is
A. faster than the Earth’s orbital speed.
B. the same as the Earth’s orbital speed.
C. slower than the Earth’s orbital speed.
D. not enough information given to decide
© 2012 Pearson Education, Inc.