Transcript Circular Motion and Gravitation

Uniform Circular Motion and
Gravitation
Yet another reason why Star Wars
is fictitious…
Features of UCM
• Constant speed
• “center-seeking” acceleration
• Tangential velocity
Rotation with Rigid Objects
• For our purposes, we’ll talk about rigid
objects
• Rigid objects have the same rotational
period at all points
• Ex: merry-go round, CD
Exceptions?
• http://upload.wikimedia.org/wikipedia/com
mons/a/a3/7901060203_Voyager_58M_to_31M_reduced.gif
Imagine sitting 0.5 m from the center of
a merry-go-round. How does your
speed change if you double your
distance from the center?
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It doubles
It decreases by half
It remains the same
It increases by a factor of 4
It decreases by 1/4
What happens to your acceleration if
you double your distance from the
center of the merry-go-round?
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1.
2.
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5.
It doubles
It decreases by half
It remains the same
It increases by a factor of 4
It decreases by 1/4
As a car rounds a corner, which
wheels are spinning faster, the
wheels on the inside of the turn (1)
or the outside (2)?
Bonus: how can they do this if
attached to the same axle?
Demos/Applications
• Rail road tracks
• CD tracking:
• http://electronics.howstuffworks.com/cd6.h
tm
The Driving Force of UCM
• Think about twirling a tennis ball above
your head with a rope
• In which direction do you have to pull to
keep the tennis ball in a uniform circular
path?
• A center-seeking force is called a
centripetal force
What is it?
• The centripetal force is similar to the net
force (in fact, it is a type of net force)
• It is the result of the external forces acting
upon an object
• Centripetal force is not a separate force
like gravity or friction
• Fc is caused by external forces like gravity
Newton’s 2nd, Remixed
• Fnet = ma
• For circular motion:
• Fc = mac = m(vT2/R)
A tennis ball has a mass of
0.060kg. What tension force is
required to swing the ball around in
a circular path of radius 0.5m at a
speed of 5m/s?
A car (m = 2000kg) is about to round a
curve of radius 20m. Assuming the
coefficient of friction between the road
and tires is 0.7, what is the maximum
speed the car can round the curve
without slipping?
Centrifugal “Forces”
• Say you’re in a car, rounding a curve
• What does it feel like?
• We sometimes call this sensation a
centrifugal force
• What is the origin of this force?
It’s All About Perspective
• From inside the car, it appears that you
are being pulled upon by some imaginary
“force”
• From above, what are you doing?
• Following Newton’s first law
• Centrifugal forces are not forces at all
• They are a by-product of our inertia when
placed in accelerated situations
• This is an example of the dangers that
occur when analyzing physics in an
accelerated situation (reference frame)
Examples
• Centrifuges
• Amusement park rides
• Washing machine
• So as much as I love Star Wars, I can’t
help but notice that people tend to walk
normally while riding in space ships
• What’s wrong with this picture?
Artificial Gravity
• How could we use rotation to create
“artificial” gravity in space?
• We could use the centrifugal effect to
produce a faux gravitational force
Vertical Circles
So How Much Do You Weigh on
the Sun?
Universal Gravitation
• Gravity doesn’t stop at Earth’s boundaries
• It is the dominant force in the universe
over large scales
• What are gravity’s origins?
• How do we calculate gravitational force?
Gravity’s Origins
• Object’s with mass, separated by some
distance R
• Fg = Gm1m2/R2
Say I have two objects, A and B,
separated by distance R, which exert a
gravitational force F on one another.
By what factor does that force change if
they are moved to a distance of 2.5R
apart?
Calculate the gravitational pull
between you and your neighbor.
Acceleration due
to gravity?
Who has a stronger
gravitational pull: a person
beside you (with the same mass
as you), or the neighbor behind
you (twice as massive, but twice
as far away). Assume 1 is for
the neighbor beside you, while 2
is for the neighbor across
• But I thought Fg = mg ?!
• Right you are…
• So if Fg = mg = Gm1m2/R2, we can write a
new expression for g
g revisited
• g = Gm2/R2 , which is sometimes called
gravitational field strength
Say I travel to three celestial bodies:
Earth, the moon, and Saturn. Rank
my weight in each place in increasing
order
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2.
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5.
Earth, moon, Saturn
Earth, Saturn, moon
Moon, Saturn, Earth
Moon, Earth, Saturn
Saturn, Earth, Moon
Suppose I weigh 750N on Earth.
What would I weigh on the Sun’s
surface? (g = 275m/s/s)
g values around the solar system
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Earth’s Surface
9.8 m/s/s
Moon’s Surface
1.6 m/s/s
Mars’s Surface
3.8 m/s/s
Jupiter’s Surface
26.0 m/s/s
Sun’s Surface
275 m/s/s
Black Hole’s Event Horizon
4,000,000,000,000,000,000 m/s/s
• How much does a 80kg person weigh on
Mt. Everest (h = 5.51 miles above sea
level = 8.85km)?
• g = 9.79 m/s/s
• W = 783N (compared to 784N)
Black Hole Mass
• The event horizon of a “typical” black hole
is about 10 km in radius
• Given the value for g (= 4E12 m/s/s) find
the approximate mass of a “typical” black
hole
• Give in solar masses (# of times the mass
of our sun)
• 3 solar masses
Cosmic Vacuum Cleaners?
• Blackholes are sometimes thought of as
large vacuums that suck everything into
them
• From this point of view, the universe will
ultimately be consumed by the current
collection of black holes…
The Real Danger
• Fortunately, this interpretation is wrong
• The real danger of a black hole lies in its
composition: you have a TON of mass
stuffed into a very small space
• Consider this: a black hole is as heavy as
our Sun, but is about 10 miles wide
• Our Sun is about 1 million miles wide
• Outside a black hole, the “tug” of gravity
would be about 10 billion times
stronger than at the surface of our sun, or
about 1 Trillion times Earth’s gravity
Earth/Sun Gravity
• What is the force of gravity between the
Earth and the sun
• The average distance between the two is
1.5 x 108 km
• How does this compare to the number we
got last week with our circular motion
calculation (3.55E22N)?
The Galactic Center
• Astronomers have fairly good evidence of
a super massive black hole in the galactic
center
• How do they know?
• Obviously, we can’t “weigh” the galactic
center on a scale
• However, we can use the concepts of
circular motion and gravitation to indirectly
measure the galactic center’s mass
Satellite Motion
• An understanding of satellite motion will
help…
• Let’s imagine objects orbiting Earth
• What does it mean to orbit the Earth?
• You’re traveling in either a circular or
elliptical path around Earth
• Gravity is keeping you in that path
• For circular orbits, we’ve got UCM
• At what speed does the moon orbit the
earth?
• Useful information: 384,000 km between
the Earth’s center and the moon’s center
• 1020 m/s
• At what speed does a space shuttle orbit
the Earth? (altitude = 300km)
• At what speed would it need to travel to
orbit at the height of Mount Everest?
(altitude = 8.85 km)
• 7700 m/s; 7900 m/s
Geosynchronous Orbit
• Certain satellites are in what is called
geosynchronous orbit
• They literally stay above one point on
Earth’s surface
• What is their orbital radius?
• Orbital speed?
• A relationship between orbital period and
orbital radius (for one or more objects)?
• This relationship is called Kepler’s Third
Law
• Venus orbits the sun at a radius of 1.08 E
8 km
• How many days is one year on Venus?
Mass of Star X
• In an unknown solar system, astronomers
observe a planet orbiting its parent star
with a period of 200 days, at a distance of
0.5 AU
• What is the mass of the parent star?
The Galactic Center
The Catch…
• Our calculations involve uniform circular
motion
• Turns out that these stars follow elliptical
paths, meaning they travel at different
speeds during different portions of their
orbit
• The process is similar however…