Download CM and gravitational force summary

CIRCULAR MOTION AND NEWTON’S LAW OF GRAVITATION
I. Speed and Velocity
Speed is distance divided by time…is it any different for an object moving
around a circle?
The distance around a circle is C = 2πr, where r is the radius of the circle
So average speed must be the circumference divided by the time to get around
the circle once
C 2π r
v= =
T
T
one trip around the circle is
known as the period. We’ll use
big T to represent this time
SINCE THE SPEED INCREASES WITH RADIUS,
CAN YOU VISUALIZE THAT IF YOU WERE SITTING ON A
SPINNING DISK, YOU WOULD SPEED UP IF YOU MOVED
CLOSER TO THE OUTER EDGE OF THE DISK?
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These four dots each make one revolution
around the disk in the same time, but the
one on the edge goes the longest distance.
It must be moving with a greater speed.
Remember velocity is a vector.
The direction of velocity in circular motion
is on a tangent to the circle. The direction
of the vector is ALWAYS changing in
circular motion.
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II. ACCELERATION
If the velocity vector is always changing
in circular motion,
EQUATIONS:
THEN AN OBJECT IN CIRCULAR
MOTION IS ACCELERATING.
acceleration in circular
motion can be written as,
THE ACCELERATION VECTOR POINTS
INWARD TO THE CENTER OF THE
MOTION.
vf
a
vi
Pick two v points on the path.
Subtract head to tail…the
resultant is the change in v or
the acceleration vector
a
-vi
vf
Note that the acceleration vector
points in
 v2
a=
r
 4π 2 r
a= 2
T
Assignment: check the units
and do the algebra to make
sure you believe these
equations
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III. Centripetal and Centrifugal Force
If the acceleration vector points inward, there must be an inward force as well.
vf
a
vi
The INWARD FORCE on an object in circular
motion is called CENTRIPETAL FORCE
Without centripetal force, an object could
not maintain a circular motion path. The velocity vector
is telling you the object wants to go straight.
Centripetal force acts in a direction perpendicular
to the velocity vector (the force, as it must be, is
in the same direction as the acceleration).
Centripetal force causes the acceleration NOT by
increasing (or changing) speed but by changing direction
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The outward force is fictitious. It is called CENTRIFUGAL force.
you before
curve
you
in the
turn
you
FAST
merry-go-round
You feel like you are being
pushed outward when you
are on a circular motion path.
This is only because your
moving body WANTS to
travel tangentially to the
circular path.
The only thing keeping you
in a circle is YOU holding on.
direction before
curve
car
Banking into a left curve, you feel
like you are being thrown by an
outward force to the right.
That’s not a real force. That’s you,
wanting to continue in your
straight line path (your inertia resists
the change) meeting the side of the
car that is moving in a circle.
You stay in a circular path thanks
to your seat belt and the closed door
of the car.
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III. Some Equations for Circular Motion
We know from Newton’s 2nd Law that F = ma
Here’s how the equation applies to circular motion
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v
F=m
r
Using these equations, you can solve
problems involving force on objects
moving in a circular path.
4π 2 r
F=m 2
T
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Example problem
A
945‐kg
car
makes
a
180‐degree
turn
with
a
speed
of
10.0
m/s.
The
radius
of
the
circle
through
which
the
car
is
turning
is
25.0
m.
Determine
the
force
of
fricEon
acEng
upon
the
car.
This problem is way easier than it sounds. You just need to realize what the source
and direction of the friction force is.
A car “in a 180 degree turn” just means it is in a circular motion path.
25 m

Ffrict

Fnorm

Fgrav


F norm = F grav


F frict = F net
Ffrict
v2
=m
r
Ffrcit
m 2
)
s
= 945kg
25m
= 3780N
(10
The frictional force points inward. It is the
centripetal force required to keep the car
on the road in a circle.
Ffrict
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IV. NEWTON”S LAW OF UNIVERSAL GRAVITATION
•  Newton applied his ideas about why objects fall to earth to planetary orbits
•  Basically he hypothesized that the Moon falls around the Earth for the same
reason an object (like an apple?) falls toward the Earth…an attractive force he
called GRAVITY
path of projectile without gravity
Assuming no air resistance…
Newton reasoned that if one could launch a
projectile with sufficient speed that its horizontal path
followed the Earth’s curvature, the force of
gravity would cause it to “fall around” the Earth.
The projectile would always fall toward the
Earth without ever striking it.
Launch speeds too low, as we know from
experience, crash to the Earth due to gravity
According to Newton, the projectile falling around the
Earth was an analogy for the Moon orbiting the Earth
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•  Newton knew that objects near the Earth accelerated by 9.8 m/s2
•  He also knew the Moon accelerated toward the Earth by 0.00272 m/s2
• WHY THE DIFFERENCE IF GRAVITY CAUSES AN OBJECT TO FALL AND THE
MOON TO ORBIT THE EARTH?
Look at the ratio of the accelerations:
m
9.8 2
s
Now look at the ratio of the distances of
from the surface of the Earth to its center
and the Moon to the center of the Earth:
1
=
m 3600
0.00272 2
s
6378km
1
=
382680km 60
COMPARE THE TWO RATIOS ABOVE
9.8
m
s2
0.00272
m
s2
=(
6378km 2
1
) =
382680km
3600
THIS SAYS THAT GRAVITY
DECREASES AS THE INVERSE
SQUARE OF DISTANCE
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NEWTON SHOWED:
 1
F∝ 2
r
Force of attraction between
two objects is inversely proportional
to the square of the distance
separating them
Earth’s gravity force represented with
vectors. The arrows are smaller further
away from Earth but still directed to the
center of Earth
The Moon, being very far from the Earth, is less influenced by its gravity and so
accelerates more slowly toward the Earth.
This not explain the source of the Moon’s velocity
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Newton’s Law of Universal Gravitation is:

m1m2
F grav = G 2
r
m1
F1
F2
m2
r
The force of attraction between two objects is
DIRECTLY PROPORTIONAL TO THE PRODUCT OF THEIR MASSES
INVERSELY PROPORTIONAL TO THE SQUARE OF THE DISTANCE
BETWEEN THEM
The universal gravitational constant, G, was ingeniously determined by Cavendish in
the 1700s. The value is use today is
G = 6.67428 × 10
−11
m 2
N( )
kg
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V. Computing gravitational acceleration
Using the Universal Gravitational equation, we can derive an equation that
allows you to compute the acceleration of gravity for other objects (besides Earth)

m1mEarth
F grav = G
2
r
m1mEarth
m1g = G
r2
mEarth
g=G 2
r
generally, g = G
m planet
r
2
r is the radius of the
planet
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VI. SATTELITE MOTION
Remember the projectile falling around the Earth earlier in the notes?
Remember the discussion of the Moon falling around the Earth?
This type of motion is the same as satellite motion and we can derive an
equation for the velocity of a satellite orbiting the Earth using our equations
for circular motion, centripetal force and universal gravitation.
A satellite in circular orbit, like a car rounding a curve MUST experience a
centripetal force.

v2
F net = msat
r

m m
F grav = G sat 2 Earth
r


F net = F grav
…and the acceleration of the
the satellite is like the
acceleration due to
gravity equation on the
last page…
v
m m
= G sat 2 Earth
r
r
m
v 2 = G Earth
r
m
v = G Earth
r
mEarth
r2
m
= G Earth
r2
g=G
2
msat
asat
velocity of satellite
orbiting Earth a
distance r from the center
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