Transcript Circular Orbits - Cloudfront.net

Sponge - I love the boat ride at
Six Flags. If the radius of the ride
is 6 meters and the boat
completes the circle in
10 seconds, what is the
centripetal acceleration? What
centripetal force is required on a
child whose mass is 30 kg?
Circular Orbits - There is
only one speed that a
satellite can have if the
satellite is to remain in an
orbit with a fixed radius.
FC = G mME
2
/r
=
2
mv /r
v = √GME/r
v = √GME/r
Mass (m) cancels out of the
equation; therefore, a satellite with
a large mass has exactly the same
orbital speed as a satellite with a
small mass.
Ex. 9 - Determine the
speed of the Hubble
space telescope
orbiting at a height of
596 km above the
earth’s surface.
Ex. 10 - The characteristics
of light detected in the galaxy M87
by the Hubble Space Telescope
indicate that matter is orbiting at a
5
speed of 7.5 x 10 m/s at a distance
17
from the center of 5.7 x 10 m.
Find the mass M of the object at
the center of this galaxy.
The period of a satellite is
the amount of time required
for one orbit.
Remember: v = 2πr/ T, and
v = √ GME/r,
So....
So . . .
2πr/ T = √ GME/r
Solving for T:
T=
3/2
2πr /√GME
The fact that the period is
proportional to the threehalves power of the
orbital radius is Kepler’s
Third Law of Planetary
Motion.
Synchronous satellites are
put into a circular orbit that
is in the plane of the
equator. All synchronous
orbits must orbit at the
same height above the
surface of the Earth, as the
equation suggests.
Ex. 11 - What is the
height above the Earth’s
surface at which all
synchronous satellites
(regardless of mass) must
be placed in orbit?
We say that people or
objects on a satellite are
“weightless”. They are
actually apparently
weightless because they
and the satellite are in
free-fall.
Rotational motion of a
space station would
produce apparent
gravity due to the
centripetal force
produced.
Ex. 13 - At what speed must
the surface of a cylindrical
space station (r = 1700 m)
move so that an astronaut
inside will experience a push
on his feet that equals his
own weight?
Ex. 14 - A space laboratory is rotating
to create artificial gravity. Its period of
rotation is chosen so the outer ring
(rO = 2150 m) simulates the
acceleration of gravity on Earth
(9.80 m/s2). What should be the radius
rI of the inner ring, so it simulates the
acceleration of gravity on the surface
of Mars (3.72 m/s2)?
In vertical circular motion the
centripetal force is the vector sum of
the normal force and the component
of the weight that pushes directly
toward the center of the circle. In the
lower half of the circular motion the
centripetal force will be less (normal
force minus weight) than in the upper
half of the circle (normal force plus
weight).
A the top of the circle, the
centripetal force is normal force plus
weight:
FC = mv2/r = FN + mg
At the correct speed, normal force
can become zero and:
FC = mv2/r = mg
Solving the last two terms for v
gives:
v= √rg
At this speed, the track
does not exert a normal
force; mg provides all the
centripetal force. The
rider at this point
experiences apparent
weightlessness.
Ex. 15 - A roller coaster
loop has a radius of 10
meters. What is the
minimum velocity
required to keep the cars
in the loop during the
ride?
Ex. 16 - An evil father is pushing
his daughter in a swing. If he
gleefully pushes her as hard as
he can, what is the minimum
velocity at which she will make
a complete vertical circle on the
swingset if the swing’s chain is
6 meters long?
Ex. 17 - After a lengthy trial, the court
decides that the punishment should fit
the crime. The father is sentenced to
be pushed in a circle on the same
swingset. If he weighs six times as
much as his daughter, what is his
minimum speed to complete the circle?
What other differences are there in the
execution of his punishment?