Centripetal Force and Gravitation
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Transcript of Centripetal Force and Gravitation
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PHYSICS 3 (GENERAL PHYSICS I)
Centripetal Force and Newtons Law of Gravitation
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Centripetal Force
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CENTRIPETAL FORCE
Centripetal Force is a force that tends to keep object
in moving around a circular arc or path
The magnitude of the centripetal force is the product
of an objects mass and its centripetal acceleration as
it moves around the circular path
The direction of the centripetal force is always
directed towards the center of the circle.
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RECALL: Centripetal acceleration
An object traveling in a
circle, even though it moves
with a constant speed, will
have an acceleration (since
velocity changes direction)
This acceleration is called
centripetal (center-
seeking).
The acceleration is directed
toward the center of the
circle of motion
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Centripetal acceleration and the angular
velocity
The angular velocity and the linear
velocity are related (v = r)
The centripetal acceleration can
also be related to the angular
velocity
t
r
r
va
t
v
rr
vv
r
r
v
v
a
since,
Thus: rar
va CC
22
or
Similar triangles
Lengths of the sidesr
r
v
v
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Forces Causing Centripetal Acceleration
Newtons Second Law says that the centripetal acceleration is accompanied by a force
F stands for any force that keeps an object following a circular path Force of friction (level and banked curves)Tension in a stringGravity
r
vmmaF C
2
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FF
F
F
Net inward force to provide centripetal
acceleration
Due to contact and/or gravitational forces
Direction:
towards the center
CENTRIPETAL FORCE
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SOME EXAMPLES
For a rock whirled on the end of a string, the
centripetal force is the force of tension in the string.
For an object sitting on a rotating turntable, the
centripetal force is friction.
For the motion of the Earth around the Sun, the
centripetal force is gravity.
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Example
A small object of mass m is suspended from a
string of length L. The object revolves with constant
speed v in a horizontal circle of radius r. Find an
expression for v.
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Example a:
Given:
String length: L
radius: r
Find:
1. v=?
1. Draw a free body diagram,
introduce coordinate frame and
consider vertical and horizontal
projections
r
vmT
2
sin r
v
g
TT
2cos
sin
2
cos
sinv
T
rgT
2(tan vrg
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Newtons Law of Gravitation
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1. Why are planets, moons and the
sun all nearly spherical?
2. Why do some of the earth
satellites circle the earth in 90
minutes while the moon takes 27
days for the trip?
3. Why dont satellites fall back the
earth?
For the motion of the Earth around the
Sun, the centripetal force is gravity.
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Phenomenon of attraction between objects.
Modern physics describes gravitation using
Einsteins general theory of relativity, but the
much simpler Newtons Laws of Universal
gravitation provides an excellent
approximation in many cases.
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RECALL : Newtons Third Law
If two objects interact, the force F12 exerted by
object 1 on object 2 is equal in magnitude but
opposite in direction to the force F21 exerted
by object 2 on object 1.
Equivalent to saying a single isolated force cannot
exist
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Example: Newtons Third Law
Consider collision of
two spheres
F12 may be called the
action force and F21 the
reaction force
Actually, either force can
be the action or the
reaction force
The action and
reaction forces act on
different objects
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Newtons Law of Universal Gravitation
Every particle in the Universe attracts every other
particle with a force that is directly proportional
to the product of the masses and inversely
proportional to the square of the distance
between them.
2
21
r
mmGF
G is the universal gravitational constant
G = 6.673 x 10-11 N m /kg
This is an example of an inverse square law
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Gravitation Constant
Determined experimentally
Henry Cavendish
1798
The light beam and mirror serve to amplify
the motion
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GRAVITATION
The reason for the very existence of most macroscopic objects in the universe
1. Heavenly Bodies Kept in Orbit
2. Heating the interiors of forming stars and planes to very high temperatures
3. Tides
4. Rising of Hot Air/Sinking of Cold Air (Convection) and other similar phenomenon
5. Keeping us in our toes!!!
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JUMBO JUMBO
Find the gravitational force exerted by the Sun on the
Earth.
Me=6.0x1024kg;
Ms=2x1030 kg;
Res=1.5x108m;
G = 6.67 x 10-11 Nm2/kg2
Ans.3.56 x1028 N
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Applications of Universal Gravitation :
Mass of the Earth
Use an example of an object close to the surface of the earth
r ~ RE
11 2
E
E
GM mm g
R
2
EE
gRM
G
Assumption:
RE + h RE
gmr
mGmF
e
eg 12
1
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Gravitational Acceleration
mgr
mGmF
e
e
g 2
mgr
mGmF
e
e
g 2
re
h Assumption:
Re + h Re
2
e
e
r
Gmg
me
m
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Applications of Universal Gravitation:
Acceleration Due to Gravity
g will vary with altitude
2 2
E EmM MF G m G mgr r
2
EMg Gr
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Thus:
Weight is not an inherent property of an
object
mass is an inherent property
Weight depends upon location
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Weight
The magnitude of the gravitational force acting
on an object of mass m near the Earths
surface is called the weight w of the object
w = m g is a special case of Newtons Second Law
g can also be found from the Law of Universal
Gravitation
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The path of a satellite is an ellipse.
Circle is an ellipse with equal length of semi-major
and semi- minor axis.
r
vmma
r
mGmF
e
e
g
2
2
r
r
Gmv e
Applications of Universal Gravitation:
Motion of Satellite
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Escape Speed
The escape speed is the speed needed for an
object to soar off into space and not return
For the earth, vesc is about 11.2 km/s
Note, v is independent of the mass of the object
2 Eesc
E
GMv
R
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Every particle in the universe attracts every other particle with a
force that is directly proportional
to the product of the masses of the
particles and inversely proportional
to the square of the distance
between them.
Newtons Law of Universal Gravitation
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Example
Question: Calculate gravitational attraction between two
students 1 meter apart. Assume the student 1 has a mass of
70 kg while the other one has a mass of 90 kg.
2111 2
22 2
70 906.67 10
1
mm N m kg kgF G
r kg m
74.2 10F N
Extremely small
compared to the
weight (F = mg).