If two (spherical) asteroids are in contact and are attracted with a force of 1 Newton, how much more or less force would they experience if they both had been twice as large (assuming the same density)?

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If two (spherical) asteroids are in contact and
are attracted with a force of 1 Newton, how much
more or less force would they experience if they
both had been twice as large (assuming the same
density)?
Physics 1710Warm-up Quiz

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1. The same
2. Twice as much (2x)
3. Half as much (1/2x)
4. 4 x
5. 8 x
6. 16 x

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Physics 1710Chapter 12 Apps Gravity
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• Solution

F G M1 M2 /r2 M1 M2 4p/3 ?R3 R r F G
(4p/3 ?R3)2 /R2 F?R4 F2/F1 (R2/R1)4 24 16
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Physics 1710Chapter 13 Apps Gravity
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• 1' Lecture
• Fg - rG Mm/d 2
• G 6.673 x 10 11 N m2 /kg2 2/3 x 10 10 N m2
  /kg2
• The gravitational force constant g is equal to
  g G M/(Rh) 2, M and R are the mass and radius
  of the planet.
• The gravitation field is the force divided by the
  mass.
• The gravitation potential energy for a point mass
  is proportional to the product of the masses and
  inversely proportional to the distance between
  their centers.

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Physics 1710Chapter 13 Apps Gravity
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• 1' Lecture (contd.)
• Keplers Laws
• The orbits of the planets are ellipses.
• The areal velocity of a planet is constant.
• The cube of the radius r 3 of a planets orbit
• is proportional to the square of the period T 2.

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Which best corresponds to the actual shape of a
planets orbit?
Physics 1710Chapter 13 Apps Gravity

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4. None of the above.

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Physics 1710Chapter 12 Apps Gravity
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• Solution

Keplers First Law The orbits of the planets are
ellipses. r ro /1e cos ? e eccentricity
Planet e ? 0.2056? 0.0068? 0.0934? 0.0483
? 0.0560? 0.0461? 0.0097? 0.2482

ro
ro/(1e)
ro/(1-e)
ro
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Physics 1710Chapter 12 Apps Gravity
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• History

Keplers First Law The orbits of the planets are
ellipses. What is the significance?
Repudiation of Aristotle
F is inverse square law
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Physics 1710Chapter 13 Apps Gravity
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• Isaac Newtons
• Universal Law of GravitationF - G M m/ d 2

d moon
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Physics 1710Chapter 13 Apps Gravity
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• Keplers Laws
• The orbits are ellipses.
• The areal velocity is a constant.
• T 2 ? r 3 implies F ? 1/ r 2, only.

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Physics 1710Chapter 12 Apps Gravity
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• Comment

Keplers Second Law The areal velocity is
constant. ½ r2 ? constant.
Why? mr2 ? L constant. L is conserved if no
torque, i.e. F is central force.

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Physics 1710Chapter 13 Apps Gravity
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• Keplers Laws
• The orbits are ellipses. (Contrary to Aristotle
  and Ptolemy.)
• A central force F ? 1/ r 2 or r 2
• The areal velocity is a constant.
• Angular momentum is conserved
• ½ v r ?t constant implies that
• rmv L constant.
• T 2 ? r 3 implies F ? 1/ r 2, only.

?
r3
T 2
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Physics 1710Chapter 12 Apps Gravity
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• How did Newton figure out UL of G?
• Fact a moon circling a planet has an
  acceleration of a v 2 /r
• Fact a F/m.
• Fact Kepler had found that the square of the
  period T was proportional to the cube of the
  radius of the orbit r
• T 2 k r 3 .
• Thus
• v 2p r / T

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Physics 1710Chapter 12 Apps Gravity
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• And
• T 2 (2pr) 2 /(F r /m) k r 3
• Thus
• F (2p) 2 m/(k r 2 )
• An inverse square law, with k 1/ (2p) 2G
  M
• F G Mm/ r 2 ,
• But what value is G?

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Physics 1710Chapter 13 Apps Gravity
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• Isaac Newtons
• Universal Law of GravitationF - G M m/ d 2

d moon
F g m g G M ? / R? 2
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Physics 1710Chapter 12 Apps Gravity
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• Gravitation

g? G M?/ R?2 G M? gR? 2 (9.80 N/kg)(6.37x10
6 m) 3.99x10 14 N m 2/kg
Need to know G or M.
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Physics 1710Chapter 13 Apps Gravity
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• Henry Cavendish
• And the
• Cavendish Experiment

M
m
d
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Physics 1710Chapter 13 Apps Gravity
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• G 6.673 x 10 -11 N ?m 2 /kg 2
• G 2/3 x 10 -10 N ?m 2 /kg 2
• (to an accuracy of 0.1)

So, M? (3.99x10 14 N m 2/kg)/(6.673 x 10-11 N
?m 2/kg 2) 5.98 x10 24 kg
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Physics 1710Chapter 13 Apps Gravity
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• What is g? on Mars? M ? 0.107 M? , R ? 0.532
  R ?

Peer Instruction Time
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What is g? on Mars?
Physics 1710Chapter 13 Apps Gravity

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1. 25.9 m/sec2
2. 9.80 m/sec2
3. 3.70 m/sec2
4. 1.97 m/sec2
5. None of the above.

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Physics 1710Chapter 13 Apps Gravity
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• What is g? on Mars?

g?/g? (M?/M?)(R?/R ?)2 (0.107)(1/0.532)2
0.378 g? 3.70 m/sec2
M?
M?
Peer Instruction Time
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Physics 1710Chapter 13 Apps Gravity
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• Gravitational Force

F - G M m/ d 2 What is the order of magnitude
of the attraction between two people (m 100 kg)
separated by a distance of 1m?
M
m
d
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What is the order of magnitude of the attraction
between two people (m 100 kg) separated by a
distance of 1m?
Physics 1710Chapter 13 Apps Gravity

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1. 9.8 N.
2. 980. N
3. 6.7 X 10 - 7 N
4. 6.7 X 10 - 9 N
5. None of the above

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Physics 1710Chapter 13 Apps Gravity
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• Gravitational Force

F - G M m/ d 2 F - (6.67 x10 11 N ?m 2 /kg
2)(100 kg)(100kg )/(1 m) 2 F - 6.67 x10 7 N
Equivalent weight F/g 67 ng
M
m
d
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Physics 1710Chapter 13 Apps Gravity

• Gravitational Potential Energy
• U -?8R Fd r
• U -?8R G Mm/r 2d r
• U GMm/R
• U 0 as r ?8

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Physics 1710Chapter 13 Apps Gravity

• Total Energy
• for Gravitationally Bound Mass
• E K U
• E ½ m v 2 GMm/r
• E L2/2mr 2 GMm/r
• Bound orbit if E 0 and
• - dE/dr ro 2 (L2/2mro) - GMm 0
• E - GMm/2ro

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Physics 1710Chapter 13 Apps Gravity
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• Total Energy
• for Gravitationally Bound Mass
• E K UE - GMm/2ro

E - L2 /2mr 2 lt GMm/r
ro - 2E/(GMm)
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Physics 1710Chapter 13 Apps Gravity

• Escape Velocity
• If K is such that E gt 0, then
• K ½ m v 2 GMm/ r
• Thus at r R
• v v 2GM/R

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Physics 1710Chapter 13 Apps Gravity

• Summary
• The force of attraction between two bodies with
  mass M and m respectively is proportional to the
  product of their masses and inversely
  proportional to the distance between their
  centers squared.
• F - G M m/ r 2
• The proportionality constant in the Universal
  Law of Gravitation G is equal to 6.673 x 10 11 N
  m2 /kg2 .

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Physics 1710Chapter 13 Apps Gravity

• Summary
• The gravitational force constant g is equal to
• G M/(Rh) 2, R is the radius of the planet.
• Keplers Laws
• The orbits of the planets are ellipses.
• The areal velocity of a planet is constant.
• The cube of the radius of a planets orbit
• is proportional to the square of the period.
• The gravitation field is the force divided by
  the mass.
• g Fg / m

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Physics 1710Chapter 13 Apps Gravity

• Summary
• The gravitation potential energy for a point
  mass is proportional to the product of the masses
  and inversely proportional to the distance
  between their centers
• U GMm / r
• The escape velocity is the minimum speed a
  projectile must have at the surface of a planet
  to escape the gravitational field.
• v v 2GM/R