Download AChapter 7 notes2017

Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
6.2 – Circular Motion
Uniform circular motion is the movement of an object at a constant speed around a circle with a
fixed radius. The position at any moment in time is represented by a vector r (radius). In a given
circle, the length of r stays the same, but the direction changes. (constant velocity??)
Rotation vs. Revolution:
Rotation: When an object turns about an internal axis
Revolution: When an object turns about an external axis.
Examples: ferris wheel? People in the ferris wheel? Skater? Earth? Record / CD?
See below. Notice the velocity vectors at any point in time is tangent to the circle. Why?
Tangential velocity is due to inertia
v
r
t
where r is the displacement between r1 and r2
Centripetal Acceleration
Acceleration changes, but is always pointing towards the center of the circle.“center seeking”
a
so this may
v
where v is the subtraction of the velocity vectors – we have only added vectors,
t
be a bit confusing.
Using similar triangles, vector addition and some simple algebra, the magnitude of the
acceleration can be calculated like this:
v2
ac 
r
Honors Physics: 2014-2014
Page 1 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
Remember Newton’s second law: F = ma
If there is acceleration, there must be a net force as well.
We call this Newton’s second law for circular motion
Fnet  mac This is called, centripetal force
net force is always in the direction of the accel.
When I spin the tennis ball, what is providing the force?
Car turning a sharp corner?
Centrifugal “Force”? doesn’t exist
This is actually due to inertia. Things stay in a circular motion because of the balance
between centripetal force and inertia.
“Gravity is an attractive field force that acts between objects with mass”
Usain Bolt vs. gravity – minute physics
https://www.youtube.com/watch?v=9YUtFpLpGfk&list=PLED25F943F8D6081C&index=76
Chapter 7 – Gravitation
***Everything pulls on everything else.***
Newton didn’t “discover” gravity, but he discovered that it was a universal force. It’s not unique
to
Earth.
Remember Newton’s third law? “For every action, there is an equal and opposite reaction”
Gravity is merely a force! If the Earth has a certain gravitational pull on the moon, how much
is the gravitational force of the moon on the Earth? Less, more or equal?
The Falling Apple
Newton told at least four people that he got the idea of universal gravitation by seeing an apple
fall out of a tree. Nobody knows if it actually hit him on the head!
Newton understood the concept of inertia, (what is inertia?) and he knew that things
remained
at constant speed in a straight line unless acted upon by a force. If anything changes
speed
or direction (acceleration), a force is responsible.
a = F/m Newton’s second law
Apple falling towards Earth: accelerates because of gravity
Moon is pulled by the same force of gravity – why doesn’t it fall as well?
Honors Physics: 2014-2014
Page 2 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
The moon is moving, but doesn’t travel in a straight line path. Newton had the
insight to see that the moon DOES fall towards the Earth just like the apple,
for the same reason. Is this true?
The Falling Moon
Newton realized that if the moon didn’t fall toward the Earth, inertia would
pull it off in a straight line away from the Earth. (graphic shown in class)
The moon falls in the sense that it falls beneath the straight line it would follow if no force acted on
it.
The moon is actually falling toward Earth but has great enough tangential velocity to
avoid hitting the Earth.
Side note:
Newton’s Test: Hypothesis to Theory – must be tested.
Remember that it doesn’t matter what mass an object has, it will still
2
fall at 9.8m / s (accel. due to gravity).
Newton’s test was to see if the moon would “fall” in the same proportion to the fall of an
apple.
Newton’s Calculation: Using geometry, his value did end up to be about 1.4-mm
He was unsure about a lot of things, so stuck all of this in a drawer for 20 years!
After he developed a new branch of math (Calculus), he was able to prove his centerof-gravity hypothesis, and then he was able to publish his Law of Universal
Gravitation.
Newton’s theory of gravity confirmed the Copernican theory of the solar system!!
Why don’t the planets fall towards the sun?
Newton’s Law of Universal Gravitation
Newton did not discover gravity, but he did discover that gravity is universal. Everything
pulls on everything else in the universe in a way that involves only mass and distance.
Newton’s Law of Universal Gravitation can be stated mathematically as follows:
The force of gravity is directly proportional to the masses of the two objects and
indirectly
proportional to the square of the distances between them.
Fg  G
m1m2
d2
Where m1 is the mass of one object, m2 is the mass of the other
object and d is the distance between their centers.
Honors Physics: 2014-2014
Page 3 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
The Universal Gravitational Constant, G:
G  6.67 1011 Nm2 / kg2 = 0.0000000000667 Nm2 / kg2
Measuring G: G wasn’t measured until about 150 years after Newton’s discovery, by an
English physicist, Henry Cavendish.
Since G is so small, we can see that the force of gravity is a very weak force.
Think about the attraction between you and a classmate. It’s there, but do you notice it?
The force between you and the Earth, however, is noticeable. This is your
weight!
Gravity depends on the masses AND the distance from the center of the Earth.
You would weigh a little less on the top of the Sears Tower
Gravity and Distance: The Inverse-Square Law
Look at the formula: Fg  G
m1m2
d2
Q: What happens when you double one of the masses? Triple?
Q: What happens when you double one mass and triple the other mass?
Q: What happens when you double the distance? Careful!!!
Gravity decreases according to what’s called the “inverse square law” – just like light intensity
Since it’s difficult to measure and see how gravity decreases, we’ll look at an example using a
butter gun:
https://www.youtube.com/watch?v=JW3tT0L2gpc (already watched)
http://www.cyberphysics.co.uk/general_pages/inverse_square/inverse_square.htm
Look also at light intensity which also decreases according to the inverse square law - other
examples are electric field, sound, nuclear radiation, electromagnetic radiation, potential energy
in a spring, etc.
It applies whenever something dissipates equally in all directions from a small source. (point
source)
One more thing about the inverse square law:
Look at Figure 13.13. This is a graphical representation of what happens to gravitational
force with respect to distance. ( http://www.west.net/~ger/inversesquare.html )
(the behavior is asymptotic – never becomes zero, no matter how far away the objects are
from each other!)
Honors Physics: 2014-2014
Page 4 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
Question? If you weigh 300 N. at sea level, you’ll weigh 299 N at the top of Mt. Everest.
Why so little change?
Question: If you were magically transported to the surface of Pluto, would Earth’s
gravitational pull disappear?
Lab?
Look at the formula once again:
Fg  G
m1m2
d2
What happens if you have a 1 kg mass on the surface of the Earth?
24
Mass of the Earth: 5.9810 kg
6
Radius of the Earth: 6.3810 m
Question: What if you were standing on the surface of the Earth, and it suddenly expanded
to twice its size (twice the radius), but remained the same mass. What would happen to your
gravitational attraction to the Earth? What if the mass of the Earth also doubled?
(CD 13.2 – back side of the page – class exercise)
What about below Earth’s surface? Is there still gravity?
If you fell into a tunnel through the center of the Earth from the North Pole, you would pick
up speed as you fall towards the center, overshoot and slow down until you reached the South
Pole. Repeat over and over – simple harmonic motion!
Interesting note: Each trip through the Earth would take about 45 minutes, 90 minutes to
return to your original side. It takes a satellite, in close orbit around the Earth 90 minutes to
make one complete rotation and return to its original point. Same time!
https://www.youtube.com/watch?v=jN-FfJKgis8
What if the Earth was hollow – minute physics
Question: What if you were standing on the surface of the Earth, and it suddenly expanded
to twice its size (twice the radius), but remained the same mass. What would happen to your
gravitational attraction to the Earth? What if the mass of the Earth also doubled?
Weight and Weightlessness
The force of gravity is like any other force. It causes acceleration. Objects under the
influence of gravity are pulled toward each other and accelerate. When we are standing on the
Earth, we think of gravity as something that presses us against the Earth – not as something that
accelerates us!
Honors Physics: 2014-2014
Page 5 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
Force against the Earth is the sensation we interpret as weight! (Support force)
2
Spacecraft orbiting the Earth: g = 8.7 m / s
Not that much different than on the surface of the Earth.
Gravity is not zero – in fact it is keeping the shuttle in orbit
But remember the falling moon – the shuttle is free-falling towards the Earth.
If you're standing on the surface of the Earth, you feel gravity (1g, 9.8 m/s2) because
you're not in free fall. Your feet press down against the ground, and the ground presses up against
your feet.
Inside the ISS, there's a downward gravitational pull of about 0.89g, but the station itself is
simultaneously accelerating downward at 0.89g -- because of the gravitational pull. Everyone and
everything inside the station experiences the same gravity and acceleration, and the sum is close to
zero.
There's still a little bit of atmosphere even at the height at which the ISS orbits, and that causes
some drag. Every now and then they have to re-boost the station, using rockets. During a re-boost,
the station isn't in free fall. The result is, in effect, a very small "gravitational" pull inside the station
-- which you can see in this fascinating video.
You are not “weightless”, but your apparent weight is zero.
What happens when you are standing on a scale in an elevator during accelerated motion?
Going up? Going down?
What if the cable broke? You would no longer be supported by the floor so you would feel
“weightless”. Are you?
Define weight as the force you and the supporting floor exert on each other. (Newton’s third
law). According to this definition, you are as heavy as you feel.
Weightlessness is not the absence of gravity, it is the absence of a support force.
https://www.youtube.com/watch?v=iQOHRKKNNLQ
Honors Physics: 2014-2014
Page 6 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
Ellipses:
Conic sections (along with parabolas, circles, hyperbolas)
http://www.mathsisfun.com/geometry/ellipse.html
Eccentricity:
http://www.mathopenref.com/ellipseeccentricity.html
Eccentric means “off center”.
Between 0 and 1 where 0 is a circle and 1 is a straight line
Two more words that you might need to know:
Perihelion – the point in an orbit of a planet, asteroid, comet or other Sun-orbiting
body where it is nearest to the Sun. (2)
Aphelion – the point where it is the farthest from the sun (1)
Another thing to know:
AU – astronomical units
One AU represents the mean distance between the Earth and our sun. The AU for
Earth then, is 1.
One AU is approximately 150 million kilometers or 93 million miles or 8 lightminutes.
More exactly, 1AU = 149,597,871 km = 92,955,807 miles
http://earthsky.org/space/what-is-the-astronomical-unit#def
Also, since Earth’s orbit is slightly elliptical, when Earth is at perihelion (in January), it is about
0.983 AU from the sun. When Earth is at aphelion (in July), it is about 1.017 AU away from the sun.
Kepler’s Laws (7.1)
Kepler’s first law states that the paths of the planets are ellipses with the Sun at one
focus.
Earth’s orbit has a very low eccentricity so is almost a circle. Ellipses shown in the book
are exaggerated.
http://astro.unl.edu/naap/pos/animations/kepler.html
Kepler’s second law states that an imaginary line from the Sun to a planet sweeps out
equal areas in time intervals.
Honors Physics: 2014-2014
Page 7 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
What does this mean with regards to the speed of the planet?
(see sim again)
Kepler’s third law states that the square of the ratio of the periods of any two planets
revolving about the Sun is equal to the cube of the ratio of their average distances from the Sun.
(
TA 2 rA 3
) ( )
TB
rB
(see sim again)
units?
Kepler’s third can be used to compare the planets’ distances from the sun or it can be
used to compare distances and periods of the Moon and artificial satellites orbiting
the Earth.
There has to be a baseline to compare it with. If you are talking about planets in the solar
system, you can compare them with Earth: TE = 1 year and rE = 1 AU
If you are talking about a moon going around Jupiter, you need information about one
moon in order to calculate the other
You will need to solve for T or r on this formula. Put the one you want to solve for on the top
(where A is above) and it will be easier.
Comet periods: 2 classifications
1. Long-period comets
example: Comet Hale-Bopp has a period of approximately 2400 years
2. Short-period comets
example: Halley’s Comet has a period of 76 years
Comets also obey Kepler’s laws, but have highly elliptical orbits
Universal Gravitation and Kepler’s Third Law
Newton was able to put together his law of universal gravitation and Kepler’s laws and was
able to definitively prove Copernicus’ observations and map out the solar system.
Honors Physics: 2014-2014
Page 8 of 9
Honors Physics Teacher Notes - Grothaus
Chapter 6.2 and 7: Circular Motion and Gravitation
Using both, you can easily calculate the period of any planet orbiting the Sun:
r3
T  2
GmS
r = the average distance from the Sun
mS = the mass of the Sun
This can be generalized to describe orbits around any star or planet if you just replace r and m with
the appropriate distances and masses.
7.2 Orbits of Planets and Satellites
The book talks about Newton’s Cannon – we looked at this in the Vsauce video we already looked at.
Do you understand the concept?
The book talks about air resistance and says that any “cannonball” would have to be at least 150 km
above Earth’s surface in order to encounter little enough air resistance to remain in orbit
Remember
ac 
v2
r and
Fnet  mac
and so
Fnet 
mv 2
r
Combine this with Newton’s Law of Universal Gravitation and you get this:
GmE m
r2

mv 2
r
The speed of a satellite orbiting Earth can be calculated as follows:
v
GmE
r
You can generalize by changing the mass of the Earth and distance to the Earth to the mass
of the planet or star you are orbiting.
You can also use
r3
T  2
GmE
from the last section, but with Earth instead of the sun.
You should now have all the tools you need to calculate anything!!
Sidereal years vs. tropical years http://www.public.asu.edu/~mjwhite/Sidereal%20and%20tropical%20years.pdf
Orbital mechanics handout – 1-7 (Universal gravitation)
Mechanics: 8,9,13,14,15,17,18 20 (8 problems)
Honors Physics: 2014-2014
Page 9 of 9