Physics 350

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Physics 350

• Chapter 5
• Circular Motion and the Law of Gravity

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Centripetal Acceleration

• Consider a car moving in a circle with constant
  velocity
• Even though the car is moving with constant
  speed, it has an acceleration
• The centripetal acceleration is due to the change
  in the direction of the velocity

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Centripetal Acceleration

• Using vectors, rearrange to determine the change
  in velocity (direction)
• The vector change is directed towards the center
  of motion

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Pictorial Derivation of Centripetal Acceleration
a Dv/Dt
v2
Top view
v1
a v2/r (r is radius of curve)
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Centripetal Acceleration

• The magnitude of the centripetal acceleration is
  given by
• This direction is toward the center of the circle

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Centripetal Acceleration

• An object can have a centripetal acceleration
  only if some external force acts on it.
• In the case in the figure, the force is the
  tension in the string

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Centripetal Acceleration

• For the car moving on a flat circular track, the
  force is the friction between the car and the
  track.

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Centripetal Acceleration

• Forces that act inward are considered to be
  centripetal forces
• Examples
• Tension in example above
• Gravity on a satellite orbiting the Earth
• Force of friction

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Centripetal Acceleration

• Applying Newtons Second Law along the radial
  direction we can determine the net centripetal
  force Fc
• Fc mac m (v2 / r)

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Centripetal Acceleration

• If the centripetal force were removed, the object
  would leave its circular path and move in a
  straight line tangent to the circle
• Merry go round

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Curves, Centrifugal, Centripetal Forces

• Going around a curve smushes you against window
• Understand this as inertia you want to go
  straight

your body wants to keep going straight
but the car is accelerating towards the center of
the curve
Car acceleration is v2/r ? you think youre
being accelerated by v2/r relative to the car
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Centripetal, Centrifugal Forces, continued

• The car is accelerated toward the center of the
  curve by a centripetal (center seeking) force
• In your reference frame of the car, you
  experience a fake, or fictitious centrifugal
  force
• Not a real force, just inertia relative to cars
  acceleration

Centripetal Force on car
velocity of car (and the way youd rather go)
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Centripetal Forces

• Fictitious Force - Centrifugal
• Driving in a car around a curve feels like you
  are applying a force to the car outward
• Not really a force, the force one feels is the
  car applying a force on you from the frictional
  force it applies to the road
• Inertia keeps our bodies wanting to move forward,
  the car applies a force to push it inwards

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Rotating Drum Ride

• Vertical drum rotates, youre pressed against
  wall
• Friction force against wall matches gravity
• Seem to stick to wall, feel very heavy

The forces real and perceived
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Centripetal Acceleration

• Gravitron
• Accelerating upwards?
• Climbing car

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Works in vertical direction too

• Roller coaster loops
• Loop accelerates you downward (at top) with
  acceleration greater than gravity
• You are pulled into the floor, train stays on
  track
• its actually the train being pulled into you!

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Vertical Circular Motion

• Consider the forces acting on a motorcycle
  performing a loop-to-loop

Can you think of other objects that undergo
similar motions?
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Old-Fashioned Swings

• The angle of the ropes tells us where the forces
  are
• Ropes and gravity pull on swingers
• If no vertical motions (level swing), vertical
  forces cancel
• Only thing left is horizontal component pointing
  toward center centripetal force
• Centripetal force is just mv2/r (F ma a v2/r)

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What about our circular motions on Earth?

• Earth revolves on its axis once per day
• Earth moves in (roughly) a circle about the sun
• What are the accelerations produced by these
  motions, and why dont we feel them?

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Earth Rotation

• Velocity at equator 2?r / (86,400 sec) 463 m/s
• v2/r 0.034 m/s2
• 300 times weaker than gravity, which is 9.8 m/s2
• Makes you feel lighter by 0.3 than if not
  rotating
• No rotation at north pole ? no reduction in g
• If you weigh 150 pounds at north pole, youll
  weigh 149.5 pounds at the equator
• actually, effect is even more pronounced than
  this (by another half-pound) owing to stronger
  gravity at pole earths oblate shape is the
  reason for this

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ConcepTest 5.1 Tetherball

• In the game of tetherball, the struck ball
  whirls around a pole. In what direction does the
  net force on the ball point?

1) toward the top of the pole 2) toward the
ground 3) along the horizontal component of the
tension force 4) along the vertical component of
the tension force 5) tangential to the circle
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ConcepTest 5.1 Tetherball
1) toward the top of the pole 2) toward the
ground 3) along the horizontal component of the
tension force 4) along the vertical component of
the tension force 5) tangential to the circle

• In the game of tetherball, the struck ball
  whirls around a pole. In what direction does the
  net force on the ball point?

The vertical component of the tension balances
the weight. The horizontal component of tension
provides the centripetal force that points toward
the center of the circle.
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Centripetal Acceleration

• The tangential component of the acceleration is
  due to changing speed
• The centripetal component of the acceleration is
  due to changing direction
• Total acceleration can be found from these
  components

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Centripetal Acceleration

• Example
• A 1,000kg car rounds a curve on a flat road of
  radius 50.0m at a speed of 50.0km/hr (14.0m/s).
  Will the car make the turn if
• a) the pavement is dry and the coefficient of
  static friction is 0.800?
• b) the pavement is icy and the coefficient of
  static friction is 0.200?
• (Note use max static friction here for the
  extreme case of the tires almost slipping.)

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Centripetal Acceleration

• Example
• An engineer wishes to design a curved exit ramp
  for a toll road in such a way that a car will not
  have to rely on friction to round the curve
  without skidding. He does so by banking the road
  in such a way that the force causing the
  centripetal acceleration will be supplied by the
  component of the normal force toward the center
  of the circular path.
• a) Show that curve must be banked at tan ?
  v2/rg.
• b) Find the angle at which the curve needs to be
  banked for a 50m radius and a speed of 13.4 m/s.

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Planetary Motion and Newtonian Gravitation
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Keplers Laws

1. All planets move in elliptical orbits with the
   Sun at one of the focal points.
2. A line drawn from the Sun to any planet sweeps
   out equal areas in equal time intervals.
3. The square of the orbital period of any planet is
   proportional to the cube of the average distance
   from the planet to the Sun.

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Keplers Laws

• Keplers First Law
• All planets move in elliptical orbits with the
  Sun at one of the focal points.
• Any object bound to another by an inverse square
  law will move in an elliptical path
• Second focus is empty

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Keplers Laws

• Keplers Second Law
• A line drawn from the Sun to any planet sweeps
  out equal areas in equal time intervals.
• Objects near the Sun will need to cover more
  distance per time
• Faster velocity

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Keplers Laws

• Keplers Third Law
• The square of the orbital period of any planet
  is proportional to cube of the average distance
  from the Sun to the planet
• Orbital Period time it takes a planet to make
  one full orbit around the sun
• For orbit around the Sun, T2/r3 K KS
  2.97x10-19 s2/m3
• K is independent of the mass of the planet
• Therefore, all planets should have the same K

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Keplers Laws

• Keplers Third Law
• They do have the same K!

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Keplers Laws

• Planetary Data relative to Earth

Kepler's 3rd Law Kepler's 3rd Law Kepler's 3rd Law Kepler's 3rd Law Kepler's 3rd Law
T in years, a in astronomical units then T2 a3 T in years, a in astronomical units then T2 a3 T in years, a in astronomical units then T2 a3 T in years, a in astronomical units then T2 a3 T in years, a in astronomical units then T2 a3

Planet Period T Dist. a fr. Sun T2 a3
Mercury 0.241 0.387 0.05808 0.05796
Venus 0.616 0.723 0.37946 0.37793
Earth 1 1 1 1
Mars 1.88 1.524 3.5344 3.5396
Jupiter 11.9 5.203 141.61 140.85
Saturn 29.5 9.539 870.25 867.98
Uranus 84 19.191 7056 7068
Neptune 165 30.071 27225 27192
Pluto 248 39.457 61504 61429
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Newtonian Mechanics

• Keplers Laws described the kinematics of the
  motion of the planets but didnt answer why the
  planets move the way they do

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The Universal Law of Gravity

• Any two bodies are attracting each other through
  gravitation, with a force proportional to the
  product of their masses and inversely
  proportional to the square of their distance

m1m2
F G
r2
(G is the Universal constant of gravity.)
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Newtonian Gravitation

• Newtons Universal Law of Gravitation
• Fg G
• where G is the gravitational constant,
• G 6.673 x 10-11 m3 kg-1 s-2
• Example of the inverse-square law

m1m2
r2
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Newtonian Gravitation

• Applying Newtons third law to two masses
• Action-Reaction Pair
• F21 -F12
• Every pair of particles exerts on one another a
  mutual gravitational force of attraction.

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Newtonian Gravitation

• The gravitational force exerted by a uniform
  sphere on a particle outside the sphere is the
  same as the force exerted if the entire mass of
  the sphere were concentrated at its center
• This is called Gausss Law.
• Applies to electric fields also

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Why was the Law of Gravitation not obvious
(except to Newton).How big are gravitational
forces between ordinary objects?
1 Newton is about the force needed to support 100
grams of mass on the Earth About the weight of a
small apple

• Conclusion
• G is very small, soneed huge masses to get
  perceptible forces

Does gravitation play a role in atomic physics
chemistry?
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(a pretty good approximation for all the planets
because the eccentricities are much less than 1.)
Circular Orbits
(velocity)
(acceleration)
There is a subtle approximation here we are
approximating the center of mass position by the
position of the sun. This is a good approximation.
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Circular Orbits
The planetary mass m cancels out. The speed is
then
Period of revolution
Time distance / speed i.e., Period
circumference / speed
? Keplers third law T 2 ? r 3
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Generalization to elliptical orbits
(and the true center of mass!)
where a is the semi-major axis of the ellipse
The calculation of elliptical orbits is difficult
mathematics.
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Finding the Value of G

• Henry Cavendish first measured G directly (1798)
• Two masses m are fixed at the ends of a light
  horizontal rod (torsion pendulum)
• Two large masses M were placed near the small
  ones
• The angle of rotation was measured
• Results were fitted into Newtons Law

G6.67x10-11 N.m2/kg2

• G versus g
• G is the universal gravitational constant, the
  same everywhere
• g ag is the acceleration due to gravity. It
  varies by location.
• g 9.80 m/s2 at the surface of the Earth

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Superposition The net force on a point mass
when there are many others nearby is the vector
sum of the forces taken one pair at a time
All gravitational effects are between pairs of
masses. No known effects depend directly on 3 or
more masses.
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Newtonian Gravitation

• Acceleration due to gravity
• Determined experimentally
• g value varies with altitude
• ag GME / r2

Altitude (km) ag (m/s2) Altitude Example
0 9.83 Mean Earth Surface
8.8 9.80 Mt. Everest
36.6 9.71 Highest manned balloon
400 8.70 Space shuttle orbit
35,700 0.225 Comm. Satellite
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Gravitational field transmits the force

• A piece of mass m1 placed somewhere creates a
  gravitational field that has values described
  by some function g1(r) everywhere in space.
• Another piece of mass m2 feels a force
  proportional to g1(r) and in the same direction,
  also proportional to m2.

• Concepts for g-fields
• No contact needed action at a distance.
• Acceleration Field created by gravitational mass
  transmits the force as a distortion of space that
  another (inertial) mass responds to.

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Mass of the Earth

• We can use the equation
• g GMearth/Rearth2
• to solve for Mearth since we know
• g 9.8 m/s2 (from our lab experiment),
• G 6.67 x 10-11 Nt-m2/kg2 (from precise gravity
  force experiments), and
• Rearth 6,400 km (since we know the
  circumference of the earth 25,000 miles).

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Mass of the Earth

• g GMearth/Rearth2 or
• Mearth gRearth2/G
• 9.8 m/s2 (6.4 x 106 m)2 / 6.67 x 10-11
  Nt-m2/kg2
• 6.0 x 1024 kg .
• Actual value 5.9742 x 1024kg
• This value is certainly large as we expect the
  mass of the earth to be large.

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ConcepTest Fly Me Away
You weigh yourself on a scale inside an airplane
that is flying with constant speed at an altitude
of 20,000 feet. How does your measured weight in
the airplane compare with your weight as measured
on the surface of the Earth?
1) greater than 2) less than 3) same
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ConcepTest Fly Me Away
You weigh yourself on a scale inside an airplane
that is flying with constant speed at an altitude
of 20,000 feet. How does your measured weight in
the airplane compare with your weight as measured
on the surface of the Earth?
1) greater than 2) less than 3) same
At a high altitude, you are farther away from
the center of Earth. Therefore, the
gravitational force in the airplane will be less
than the force that you would experience on the
surface of the Earth.
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Keplers Laws

• Example