Download January 23

Notes
•  Office Hours: M, T 4-6+pm, NSH 414
•  Help Sessions: M,T, 6:30-8:30pm
–  O’Shaughnessy 106
•  Reminder: HW due Wednesday in lecture
•  Tonight: Major’s Night, South Dining Hall, East
–  6:00-8:00pm
•  Quick Review:
A rock, initially at rest with respect to Earth and located an
infinite distance away, is released and accelerates toward
Earth. An observation tower is built 3 Earth-radii high to
observe the rock as it plummets to Earth. Neglecting
friction, the rock s speed when it hits the ground is
1. twice
2. three times
3. four times
4. six times
5. eight times
6. nine times
7. sixteen times
its speed at the top of the tower.
Assuming that the earth is in circular orbit around the
sun, how much more velocity is necessary so that the
earth could escape the sun's gravitational pull?
Rorbit = 1.50 x 1011 m
Msun = 1.99 x 1030 kg
Mearth = 5.98 x 1024 kg
Shell Theorems
1.  A spherical shell of uniform density attracts an
external particle as if all of the mass of the shell
were concentrated at its center.
2.  A spherical shell of uniform density exerts no
gravitational force on a particle anywhere inside it.
Force and Potential for Shell:
F
r=R
r
U
r=R
r
Two concentric uniform spherical shells have masses
M1 and M2 and radii a and 2a as shown in the figure.
What is the magnitude of the gravitational force on a
point mass m located (a) a distance 3a from the center
of the shells? (b) a distance 1.9a from the center of the
shells? (c) a distance 0.9a from the center?
Find the force of gravity everywhere in space (i.e., both
inside and outside the sphere) on a point mass m due to
a sphere of uniform density. The sphere has mass M
and radius R.
Kepler s Laws
1.  All planets move in elliptical orbits having the sun at one
focus
2.  A line joining any planet to the sun sweeps out equal areas in
equal times
3.  The square of the period of a planet s orbit is proportional to
the cube of its mean distance from the sun.
A spherical hollow is made in what was formerly a
sphere of uniform density, radius R, and mass M. The
hole has radius R/2. (a) Find the force of gravity on a
point particle of mass m a distance d away from the
center of the sphere along the x axis. (b) Find the force
of gravity on a point mass m anywhere in the hole.
r = R/2
m
d
A projectile is fired vertically from the earth s surface
with an initial speed of 9.42 km/s. Neglecting
atmospheric friction, how far above the earth s surface
will it eventually rise?
A small asteroid of mass m starts from rest very far away from a
space station in the form of a ring. The station has radius R and
mass M. The asteroid moves in a line along the axis of the
station. What is the asteroid s speed when it reaches the center
of the station?
R
m
A globular cluster is a roughly spherical collection of up to
millions of stars bound together by the force of gravity.
Astronomers can measure the velocities of the stars (rotation
curves) in the cluster to get an idea of the mass distribution (as
shown Monday). Assuming the stars have roughly the same mass
and are uniformly distributed in the cluster, find the dependence
of the velocity of a star on the distance r from the center of the
cluster, (both inside and outside).
Galactic Rotation Curves
•  measure doppler shift of atomic
spectra for galaxies to see how
fast the edges are moving
•  compare v vs. R with what
expects for the mass distribution
of stars (Fgrav = Fcentrip)
Galactic Rotation Curves
•  measure doppler shift of atomic
spectra for galaxies to see how
fast the edges are moving
•  compare v vs. R with what
expects for the mass distribution
of stars (Fgrav = Fcentrip)
http://burro.astr.cwru.edu/JavaLab/RotcurveWeb/back_RC.html
NGC 2403