PHYS 1403 Introduction to Astronomy
Basic Physics
Chapter 5
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Things that Move in Circles: Units
Radian: An angle at the center of a circle
whose arc is equal in length to the radius
Units of Measure: Radian, Degrees
and Revolutions
1 radian = 57.3 degrees
Source: wikipedia
Things that Move in Circles
Angular velocity (ω) = change in angle / change in time
rad/s or rev/s, or deg/s
Angular Acceleration (α) = change in ω / change in time
rad/s2 or rev/s2, or deg/s2
Newton’s Second Law for
Rotating Bodies
• Any object that moves
in a circle or an arc has
centripetal force
• Centripetal force (Fc)
r
measured in Newton’s
Fc = ma = m v2 / r
• Centripetal (radial)
Circumference = 2 π r
acceleration
ac = v 2 / r
v=2πr/t
v
Source: share.ehs.uen.org
Source: faculty.wcas.northwestern.edu
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Visual Demo of Centripetal Force
Consequences of Centripetal Force
Why planets are not perfect spheres?
https://www.youtube.com/watch?v=Tctr8CIMOZA
Image source: NASA, STSI
Angular Momentum
Rotating and or orbiting object poses angular momentum
Solar System
Source: Wikipedia
Galaxy
Angular Momentum is
Conserved
I is called Moment of Inertia ~ MR2
M is mass and R is radius of disk
Source: wikipedia
https://www.youtube.com/watch?v=0RVyhd3E9hY
Source: wikipedia
Angular Momentum?
Which of the following equation gives angular
momentum>
a) L = mv
b) L = Iω
c) L = mα
d) L = mr2
http://hea-www.harvard.edu/~pgreen/
Angular Momentum?
Since angular momentum is conserved, the orbital speed
of a planet at perihelion as compared to aphelion
a) is larger
b) is smaller
c) is the same
d) approaches infinity
http://hea-www.harvard.edu/~pgreen/
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Newton’s Law of Gravitation
Any two bodies are attracting each other
through gravitation, with a force (Fg)
proportional to the product of their masses
(M,m) and inversely proportional to the square
of their distance (r). G is called gravitational
constant.
Fg = GMm
r2
Gravity and Distance: The InverseSquare Law
Inverse-square law -• relates the intensity of an effect to the inversesquare of the distance from the cause
• in equation form: intensity = 1/distance2
• for increases in distance, there is decreases in
force
• even at great distances, force approaches but
never reaches zero
YouTube
Inverse-Square Law
Gravitational Constant
How to find the value of G?
G=6.67x10-11 N.m2/kg2
Fg = Gm1m2
d2
Force of Gravity and Inverse-Square Law
Gravity?
Gravity is
a) sometimes a repulsive force and sometimes an
attractive force.
b) always a repulsive force.
c) always an attractive force.
d) none of the above.
Gravitational force is significant only for very large masses and small
separation distance
http://hea-www.harvard.edu/~pgreen/
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Gravity?
Compared to your mass here on Earth, your mass out in
the space between the stars would be __________.
a) zero
b) negligibly small
c) much much greater
d) the same
e) the question cannot be answered from the
information given
http://hea-www.harvard.edu/~pgreen/
ClassAction: Astronomy Education at the University of Nebraska-Lincoln Web Site (http://astro.unl.edu)
How to find the Weight on Earth?
ClassAction: Astronomy Education at the University of Nebraska-Lincoln Web Site (http://astro.unl.edu)
ClassAction: Astronomy Education at the University of Nebraska-Lincoln Web Site (http://astro.unl.edu)
How to find g of a Planet?
Weight is a Force = mass x acceleration due to gravity
Weight in Newtons = mass in kg x 9.8 m/s2 for Earth
Mass is a property of matter, it is not equal to weight
Weight = Gravitational Force
=
Apple (m)
Apple (m)
=
Earth (M)
Weight of apple = mg = m x 9.8 m/s2 = Fg
Earth (M)
If we know G, M and r then we can find g
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How to find the mass of Earth?
Attendance
Weight = Gravitational Force
mg = GMm
Apple (m)
r2
M=
g
r2
Earth (M)
G
Since g, r and G are known we can calculate the mass of Earth = 6 x 1024 kg
Center of Gravity and Mass
Campus.kellerisd.net
Sun – Earth System
Earth- Sun System
Balance point of any System
Atvconnection.com
Wikipedia.com
https://www.youtube.com/watch?v=1iSR3Yw6FXo
karatechoach.com
Wikipedia.com
ClassAction: Astronomy Education at the University of Nebraska-Lincoln Web Site (http://astro.unl.edu)
ClassAction: Astronomy Education at the University of Nebraska-Lincoln Web Site (http://astro.unl.edu)
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Center of Mass and Newton's Laws
• Why Earth has to Move around Sun?
– asun = F/Msun
– aearth = F / Mearth
– Msun is 300,000 times larger than Mearth
– Therefore asun is much small than aearth
• The sun also moves slowly just as you
would move if you swing a child in a circle.
• The center of mass is inside the Sun,
therefore we see only a wobble motion.
Types of Orbits
In order to stay on a
closed orbit, an object
has to be within a
certain range of
velocities:
Understanding Orbital Motion
The universal law of gravity allows us to
understand orbital motion of planets and
moons:
Example:
• Earth and moon attract each other through gravitation.
Dv
• Since Earth is much more
massive than the moon, the moon’s
effect on Earth is small.
v
v’
• Earth’s gravitational force
Moon
constantly accelerates the moon
towards Earth.
F
• This acceleration is constantly
Earth
changing the moon’s direction of
motion, holding it on its almost
circular orbit.
Circular Velocity
A speed a planet or a satellite must have to
remain in a circular orbit
2/
Too slow => Object falls
back down to Earth
Too fast => Object escapes
Earth’s gravity
=
=
/
How to calculate velocity of escape?
Moons circular velocity is 1.02 km/s
Orbital Velocities of Planets
Geosynchronous Orbits
A geosynchronous
satellite orbits
eastward with the
rotation of Earth and
remains above a fixed
spot on the
equator, which is ideal
for communications
and weather satellites
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Gravitational Potential Energy
• It is the gravitational energy the object
would have above the earth surface.
GPE = Gravitational Force x Distance
GPE=
×
=
Satellite (m)
Distance (r)
Earth (M)
ClassAction: Astronomy Education at the University of Nebraska-Lincoln Web Site (http://astro.unl.edu)
Escape Velocity
Conservation of Energy and Kepler's Second Law
of Planetary Motion
Escape velocity is the velocity required to escape an astronomical body
For the case of Rocket on Earth
If mass of the spaceship << the mass of Earth, then the escape velocity is
KE
1
mv
2
v 
 PE
2
GMm
r
2 GM
r

M = mass of the central body in kg
G = gravitational constant (6.67ˣ10-11 m3/s2/kg)
r = radius of the central body
For Earth it is 11.2 km/s or 24,600 miles/hour
Challenge Question: Gravity?
If the radius of the Earth were to double, with no change
in its mass, a person's weight would
a) be unchanged.
b) increase by factor of 4
c) decrease by a factor of 4
d) double
e) be cut in half
Acknowledgment
• The slides in this lecture is for Tarleton:
PHYS1411/PHYS1403 class use only
• Images and text material have been
borrowed from various sources with
appropriate citations in the slides,
including PowerPoint slides from
Seeds/Backman text that has been
adopted for class.
http://hea-www.harvard.edu/~pgreen/
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