Transcript Centripetal Force and Gravity

Centripetal Force and
Gravity
Chapter 5
How do the planets move?
 Newton
developed mathematical
understanding of planets using:


Dynamics
Astronomy
 Overcame
the idea of “Centrifugal” force –
objects are throw outward

Items released from a circle move TANGENT
to the curve
Centripetal Force
 Center-seeking
force exerted that allows
an object to move in a curved path
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Can comes from
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Pull of string
Gravity
Magnetism
Friction
Normal Force
Force acts towards the center
Centripetal Acceleration
 Centripetal
force causes the object to
move in a curved line
 Acceleration caused by

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
Increasing velocity
Decreasing velocity
Changing direction
Centripetal acceleration
 Centripetal
acceleration formula
ac = v²/r
ac = centripetal acceleration (m/s²)
v = velocity (m/s)
r = radius (m)
Center-Seeking Forces
 If
a mass is accelerating it must have a
force acting on it
Centripetal Force
Fc = mac = mv²/r
 This
is the force that tugs a body off its
straight-line course
Example #1: Strings and Flat Surfaces
Suppose that a mass is tied to the end of a string and is being whirled in a circle along the top of a
frictionless table as shown in the diagram below.
A freebody diagram of the forces on the mass would show
The tension is the unbalanced central force: T = Fc = mac, it is supplying the centripetal force
necessary to keep the block moving in its circular path.
Example #2: Conical Pendulums
Our next example is also an object on the end of string but this time it is a conical pendulum.
Notice, that its path also tracks out a horizontal circle in which gravity is always perpendicular to
the object's path.
A freebody diagram of the mass on the end of the pendulum would show the following forces.
T cos θ is balanced by the object's weight, mg. It is T sin θ that is the unbalanced central force
that is supplying the centripetal force necessary to keep the block moving in its circular path: T
sin θ = Fc = mac.
Example #3: Flat Curves
Many times, friction is the source of the centripetal force. Suppose in our initial example that a car is
traveling through a curve along a flat, level road. A freebody diagram of this situation would look very
much like that of the block on the end of a string, except that friction would replace tension.
Friction is the unbalanced central force that is supplying the centripetal force necessary to keep the car
moving along its horizontal circular path: f = Fc = mac.
Since f = μN and N = mg on this horizontal surface, most problems usually ask you to solve for the
minimum coefficient of friction required to keep the car on the road.
Banked Curves
 “Bank”
a turn so that normal force exerted
by the road provides the centripetal force
 To calculate the angle to bank at a set
speed:
tan θ = v²/gr
 As long as you aren’t going over the
recommended velocity, you should never
slip off a banked road (even if the surface
is wet)
Great Notes
 http://spiff.rit.edu/classes/phys211/lectures
/bank/bank_all.html
Example #4: Banked Curves
If instead, the curve is banked then there is a critical speed at which the coefficient of friction can equal zero and
the car still travel through the curve without slipping out of its circular path.
A freebody diagram of the forces acting on the car would show weight and a normal. Since the car is not sliding
down the bank of the incline, but is instead traveling across the incline, components of the normal are examined.
N sin θ is the unbalanced central force; that is, N sin θ = Fc = mac. This component of the normal is supplying the
centripetal force necessary to keep the car moving through the banked curve.
Circular Motion
 Understand
Gravity
the math behind the
force

Newtonian
• reliable and simple
• fails on the “Grand” scale of the galaxy
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Einstein’s Theory of Relativity
• Relates gravity to “fabric” of space and
time
• Complex math – not needed for daily
experience
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Today – still exploring
• String theory
• Dark Energy
Law of Universal Gravitation
 Gravity
force is related to masses of two
bodies and the distance
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FG α mM/r²
Center-to-Center attraction between all forms
of matter
Evolution of the Law

Many scientists worked to develop
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Copernicus and Galileo– Similar matter attracted
Kepler
• Argued that two stones in space would attract to each other,
proportional to their mass
• Noticed that force decreases with distance
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Bullialdus – Attraction was in a line dropping off
inversely squared
Newton – related centripetal acceleration to
gravitational acceleration
Gravitational Constant
 By
adding a constant the proportion can
be made into a equality
 Universal Gravitational Constant
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6.672 x 10-¹¹ Nm²/kg²
 Measured
by Cavendish
But G is so small…
 Only
really noticed when one of the
masses is REALLY BIG
 Unlimited range
 Purely attractive – not weakened by
repulsion
Cool Conclusions
 Cavendish
wanted to find the density of
earth when he did his “G” experiment

g(surface) = GM/R² (solve for M  D=M/V)
 Newton
(although he didn’t have
Cavendish’s experiment) made a guess at
density to come up with “g” for earth
Imperfect Earth
 Not
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a uniform sphere
Hills and valleys
Bulge at the North (pearshaped)
The spin of earth “throws”
the center out
Moon interferes
 Gravity
is not constant
everywhere
The Cosmic Force

Johannes Kepler
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Interesting family life
“Inherited” his life’s
work from Tycho
Brahe
Took two decades to
formulate his “Three
Laws of Planetary
Motion”
Laws of Planetary Motion
 First
Law– The
planets move in
elliptical orbits
with the Sun at
one focus

The orbits are
NEARLY
circular, but an
oval makes a
difference
Laws of Planetary Motion
 Second
Law- As a
planet orbits the
Sun it moves in
such a way that a
line drawn from
the Sun to the
planet sweeps out
equal areas in
equal time
intervals
Second law
 The
speed will be
greater when near
the sun
 As it moves away,
gravity slows it
down
 Idea is used to
“sling-shot” rockets
and probes through
space
Laws of Planetary Motion
law – The ratio of the average
distance from the Sun cubed to the period
squared is the same constant value for all
planets
r³/T² = C
r – distance to Sun
T – time to travel around the Sun
C – Solar Constant*
 Third
*Different constants for sun, earth, other planets or stars
Third law
Satellite Orbits
 Projectiles
– Sail in a parabola until it hits
the earth
 Fire it faster – go farther
 Finally – the earth would “fall away”
Different Velocities
Orbital speed
 When
centripetal force equals gravitational
force – the object stays in orbit
 GmM/r² = mv²o/r
 Simplified
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
v = √GM/r
Circular orbital speed
o
Varying Orbitals
 If
the velocity is more or less than the
circular orbital
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Circle – speed v = vo
Elliptical – speed v < v
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Large elliptical – speed v > vo and < √2v
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Parabola – v = √2v
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Hyperbola - v > √2v
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o
o
o
o
Effectively Weightless
 When
in free-fall, you have no weight
 If you stand on a scale in a free falling
elevator
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The scale would drop to zero
No normal force pushing back-up
Only gravity is acting
Vomit Comit
 When
Gravitational Field
an object experiences forces over a
continuous range of locations
 Graviton – hypothetical massless carrier of
gravitational interaction
 Gravity – elusive study in physics