Title: 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)?
1If 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
0
- The same
- Twice as much (2x)
- Half as much (1/2x)
- 4 x
- 8 x
- 16 x
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2Physics 1710Chapter 12 Apps Gravity
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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
3Physics 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.
4Physics 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.
5Which best corresponds to the actual shape of a
planets orbit?
Physics 1710Chapter 13 Apps Gravity
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-
-
-
- None of the above.
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6Physics 1710Chapter 12 Apps Gravity
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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
7Physics 1710Chapter 12 Apps Gravity
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Keplers First Law The orbits of the planets are
ellipses. What is the significance?
Repudiation of Aristotle
F is inverse square law
8Physics 1710Chapter 13 Apps Gravity
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- Isaac Newtons
- Universal Law of GravitationF - G M m/ d 2
d moon
9Physics 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.
10Physics 1710Chapter 12 Apps Gravity
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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.
11Physics 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
12Physics 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
13Physics 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?
14Physics 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
15Physics 1710Chapter 12 Apps Gravity
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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.
16Physics 1710Chapter 13 Apps Gravity
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- Henry Cavendish
- And the
- Cavendish Experiment
M
m
d
17Physics 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
18Physics 1710Chapter 13 Apps Gravity
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- What is g? on Mars? M ? 0.107 M? , R ? 0.532
R ?
Peer Instruction Time
19What is g? on Mars?
Physics 1710Chapter 13 Apps Gravity
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- 25.9 m/sec2
- 9.80 m/sec2
- 3.70 m/sec2
- 1.97 m/sec2
- None of the above.
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20Physics 1710Chapter 13 Apps Gravity
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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
21Physics 1710Chapter 13 Apps Gravity
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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
22What 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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- 9.8 N.
- 980. N
- 6.7 X 10 - 7 N
- 6.7 X 10 - 9 N
- None of the above
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23Physics 1710Chapter 13 Apps Gravity
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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
24Physics 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
25Physics 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
26Physics 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)
27Physics 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
28Physics 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 .
29Physics 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
30Physics 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