Gravitation

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CHAPTER-13

• Gravitation

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Ch 13-2 Newtons Law of Gravitation

• Newton's Law of Gravitation- a key in
  understanding gravitational force holding Earth,
  moon, Sun and other galactic bodies together
• Magnitude of gravitational force F between two
  mases m1 and m2 separated by a distance r
• FG(m1m2/r2)
• Gravitational constant
• G 6.67x10-11 m3/kg.s

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Ch 13-2 Newtons Law of Gravitation

• Shell theorem A uniform spherical shell attracts
  a particle outside the shell as if all the shell
  mass was concentrated at the shell center
• For any particle located inside the shell, the
  net gravitational force between the particle and
  shell is zero

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Ch 13 Checkpoint 1

• A particle is to be placed , in turn, outside
  four objects, each of mass m (1) large uniform
  solid sphere, (2) large uniform spherical shell,
  (3) a small uniform solid sphere, (4) a small
  uniform shell. In each situation, the distance
  between the particle and the center of the object
  is d. Rank the objects according to the magnitude
  of the gravitational force they exert on the
  particles, greatest first

• all tie

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Ch 13 Checkpoint 2

• The figure shows four arrangements of three
  particles of equal masses. (a) Rank the
  arrangements according to the magnitude of the
  net gravitational force on the particle labeled
  m, greatest first. (b) In arrangement 2, is the
  direction of the net force closer to the line of
  length d or to the length D.

• (1) FnetGm2(1/d21/D2)i
• (2) FnetGm2(1/d2 )i(1/D2) j
• (3) FnetGm2-(1/d2 )(1/D2)i
• (4) FnetGm2-(1/d2 )j(1/D2)i
• (a) 1, tie of 2 and 4, then 3
• (b) line d

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Ch 13 Checkpoint 3
In the figure her, what is the direction of the
net gravitational force on the particle of mass
m1 due to other particles, each of mass m and
arranged symmetrically relative to the y axis?
Net force downward along y-axis with all
x-components cancelled out
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Ch 13-4 Gravitation near Earth Surface

• Principal of superposition of Gravitational force
• For n interacting particles, the net
  gravitational force on particle 1 due to other is
  F1,net?i2F1i
• Gravitation near Earth Surface
• Force of attraction between the Earth and a
  particle of mass m located outside Earth at a
  distance of r from Earths center
• FGmME/r2 but Fmag where ag is
  gravitational acceleration given by
• ag GME/r2

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Ch 13-4 Gravitation Near Earth

• Acceleration of gravity g differs from g because
• Earths mass is not uniformly distributed
• Earth is not a sphere
• Earth is rotating
• Analyze forces on a crate with mass m located at
  the equator
• FN-mag-FR-mv2/R-mR?2
• But FNmg then
• mg mag-mR?2
• g ag-R?2

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Ch 13-5 Gravitation Inside Earth

• For any particle located inside uniform shell of
  matter , the net gravitational force between the
  particle and shell is zero

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Ch 13-6 Gravitational Potential Energy

• Gravitational Potential Energy of two-particles
  system
• Work done on the ball when the ball move from
  point P to a point at infinity from earth center
• W ?F(r).dr ?F(r)dr cos?
• For ?180? and F(r)GMm/r2
• W ?R? -F(r).dr?R?-(GMm/r2)dr
• -GMm?R?(dr/r2) GMm1/rR?
• W0-GMm/r-GMm/r

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Ch 13-6 Escape Speed

• Potential Energy and Force
• Force F(r)-dU/dr-d/dr(GMm/r)-GMm/r2
• Negative sign indicates direction of force
  opposite to that increasing r
• Earth Escape Speed
• Minimum initial speed required at Earth surface
  to send an object to infinity with zero kinetic
  energy (velocity) and zero potential energy. Then
• KiUimvesc2/2-GMm/R0
• vesc?2GM/R

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Ch 13 Checkpoint 4
You move a ball of mass m away from a sphere of
mass M. (a) Does the gravitational potential
energy of the ball-sphere system increase or
decrease? (b) Is positive or negative work done
by the gravitational force between the ball and
the sphere?

• U -(GmM)/r
• U0 for r? and U becomes more negative as
  particles move closer.
• U becomes less negative and it increases
• (b) Wg-Wa
• negative

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Ch 13-7 Planets and Satellites Keplers Laws

• Keplers Law of Planetory Motion
• Three laws namely Law of Orbits, Law of Areas
  and Law of Periods
• Law of Orbits
• All planets move in elliptical orbits , with the
  sun at one focus.
• Semi major axis a, semi minor axis b
  eccentricity e, ea distance of one of the focal
  point from the center of the ellipse
• For a circle eccentricity e is zero

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Ch 13-7 Planets and Satellites Keplers Law-
Law of Areas

• Law of Areas
• A line that connects a planet to the sun sweeps
  out equal areas in the plane of the planets
  orbit in equal time intervals that is the rate
  dA/dt at which it sweeps out area A is constant
• Area ?A of the wedge is the area of the triangle
  i.e.
• ?A(r2??)/2
• dA/dt r2(d?/dt)/2 r2?/2
• But angular momentum Lmr2?
• Then dA/dt r2?/2L/2m

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Ch 13 Checkpoint 5
Satellite 1 is in a certain circular orbit around
a planet, while satellite 2 is in a large
circular orbit. Which satellite has (a) the
longer period and (b) the greater speed
T2(4?2/GM)R3 Since R1ltR2 then T1ltT2 Longer
period for satellite 2 (b) Kmv2/2GmM/2R
v2GM/R Greater v for smaller R i.e R1 the
greater speed for satellite 1
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Ch 13-7 Planets and Satellites Keplers Law-
Law of Periods

• Law of Periods
• The square of period of any planet is
  proportional to the cube of the semi major axis
  of the orbit
• Considering the circular orbit with radius R (the
  radius of a circle is equivalent to the semi
  major axis of an ellipse)
• Newtons law applied to an orbiting planet gives
• GmM/R2mv2/Rm?2/R
• GM/R3 ?2(2?/T)2
• R3/GMT2 /4?2
• T2 (4?2/GM) R3

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Ch 13-8 Satellites Orbits and Energy

• For an orbiting satellite, speed fixes its
  kinetic energy K and its distance from earth
  fixes its potential energy U. Then mechanical
  energy E (E KU) of the Earth-satellite system
  remains constant.
• Kmv2/2 but mv2/RGmM/R2
• Then Kmv2/2GmM/2R
• U-GmM/R -2K we have
• K- U/2 and
• EKU K(-2K)-K(circular orbit)
• For an elliptical orbit Ra
• Then E-K-GmM/2a
• For same value of a, E is constant