Transcript Simple Harmonic Motion Universal Gravitation

Simple Harmonic
Motion
Universal Gravitation
1. Simple Harmonic Motion
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Vibration about an equilibrium
position with a restoring force that is
proportional to the displacement
from equilibrium.
It is a back-and-forth motion over the
same path.
A force causes the motion to
continue to cycle.
Equilibrium position is when the
object is at rest.
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A spring vibrating is SHM. The restoring force is
the spring.
Pendulum swinging is SHM. The restoring force
is gravity.
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Velocity is maximum
 acceleration is zero
 displacement is zero (passing through
equilibrium).
 balanced forces – inertia keeps it going
Velocity is zero
 acceleration is maximum (change in direction)
 displacement is maximum
 Net force is maximum
Simple Harmonic Motion creates waves!
2. Pendulum
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A mass, called a bob, attached to a string with a
pivot point.
Swings side to side – simple harmonic motion
Restoring force is the gravity.
Angle of swing should be 15° or less for SHM.
The period is based on the length of the
pendulum, not the mass.
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l
T  2
g
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l = length of pendulum
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xT  2 l
g
2
4
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l
 T2 
g
2
4

l
 g
T2
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Note that the mass of the bob is not a
factor in calculating acceleration due to
gravity!
3. Mechanical Resonance
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Application of small forces at regular intervals to
a vibrating object causes the size (amplitude) of
the vibration to increase.
The time interval between applying the force
must be the same as the period of oscillation.
This is how you keep the swing going!
Soldiers do not march across bridges
Tacoma Narrows Bridge
4. Universal Gravitation
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Newton studied the planets and their motion.
He proposed a gravitational force.
Attractive force that exists between all objects in
the universe.
Law of Universal Gravitation
m1m2
Fg  G 2
d
Inverse square law between Fg and d
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Fg : gravitational force, N
G : Universal gravitational constant
G
= 6.67 x
Nm 2
10-11 kg2
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d : distance between the centers of the masses
of the objects.
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m1 : mass of one object
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m2 : mass of the other object
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Henry Cavendish measured the universal
gravitational constant by measuring the
attraction between spheres.
Using this constant,
they were able to
calculate the mass
of the earth.
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Calculating the mass of the Earth!
FW = myoug
Fg = G m m
y ou
earth
radius of earth 2
myoug =
 mearth 
G m y oumearth
r2
gr 2
G

m

6
 9.81 2  6.37 x10 m
s 

2
 mearth 
Nm
6.67 x10 11
kg 2

2
 5.98 x10 24 kg
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Objects orbiting the Earth:
Fc = Fg
 Fc = Fg
Gmme
Gmme
mac 
 mac 
2
2
r
r
2
2
c
v
Gme
 2
r
r
Gme
vc 
r
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4 r Gme
 2
2
T
r
4 2 r 3
T2
Gme
r3
T  2
Gme
5. Kepler’s Laws
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Tycho Brahe 1546 – 1601
 Believed Earth was the center of the Universe.
 Recorded the exact position of the planets and
the stars
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Johannes Kepler 1571 – 1630
 45 years before Newton
 Believed in a sun-centered solar system.
 Worked as Brahe’s assistant
 Used Brahe’s data to formulate the following 3
laws:
Kepler’s 1st Law
 Paths of the planets are ellipses, with the sun
at one focus.
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Kepler’s 2nd Law
 An imaginary line from the sun to a planet
sweeps out equal areas in equal time
intervals.
 Planets move faster when closer to the sun
and slower when farther away.
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Kepler’s 3rd Law
 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.
2
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 TEarth   rEarth 

  
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 TMars   rMars 
T
3
: period of the planet
 r : average distance from the sun
Antares is the 15th brightest
star in the sky. It is more
than 1000 light years away!
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Example Problems
Homework – Universal Gravitation