Uniform Circular Motion

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Uniform Circular Motion
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1.Define Uniform circular motion

• When the tangential speed is constant
• (speed a moment in time)

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2. Show the position and instantaneous
velocity vectors at several time intervals for
an object moving in a circular path.
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3. How could the magnitude of the tangential
velocity be calculated if the object is in
constant motion ?

• Velocity ?x
• ?t
• Distance traveled around a circle ?
• C 2pr
• Time ?
• Period (T) time it takes for something to
• complete a cycle so, V 2pr
• 
  T

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4. What would be the direction of the
instantaneous acceleration ?

• Acceleration ?v or vf - v0
• ?t
  ?t
• How do we subtract vectors ?

v0
?v
vf
vf
v0
The acceleration is toward the center.
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5. Give a formula that would give the
magnitude of the acceleration

• Centripetal (Center Seeking) acceleration
• ac v2 r

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Example Problem PG 236 1 A Rope attaches a
tire to an over hanging tree limb. A girl
swinging on the tire has a centripetal
acceleration of 3.0 m/s2. If the length of the
rope is 2.1 m. What is the girls tangential
speed ?

• ac v2/r

ac 3.0 m/s2
r 2.1m
V ?
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6. Combine the equations for velocity (2)
and acceleration (6)

• since v 2pr
• T
• Then ac (2pr)2 OR ac 4 p2r
• T
  T 2
• r

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Example Problem If a girl sits 1.2 m from the
center of a merry-go-round what is her
centripetal acceleration if the period of her
revolution is 3.4 s ?
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Example Problem Pg 236 4A race car moving
along a circular track has a centripetal
acceleration of 15.4 m/s2. If the car has a
tangential speed of 30.0 m/s, what is the
distance between the car and the center of the
track ?
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7. How can a motion with constant speed be
an accelerated motion?

• The object has a constant speed, but it is
  changing
• direction !!!!

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8. Show that the expression v2 / r for
centripetal acceleration has the units of
acceleration

• a m/s2
• v2 / r So, (m/s)2 / m m2
  . 1 m/s2 S2 m

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9. What is Centripetal Force and how is it
found ?

• Centripetal (Center Seeking) force is a force
  toward the center of a circle that keeps an
  object moving in a circle.
• Remember Something MAKES the centripetal force
  !!!!
• It must be a contact or a plus 1

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10. What is the formula for
centripetal force ?

• It is a force So
• F ma
• But a is a centripetal acceleration So
• Fc mac
• Remember ac v2/r which means
• Fc m v2
• r

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Example Problem Pg 238 1A 2.1m rope
attaches a tire to an overhanging tree limb. A
girl swinging on the tire has a tangential speed
of 2.50 m/s. If the magnitude of the centripetal
force is 88.o N, what is the girls mass ?
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11. Suppose the speed of an object moving in a
circular path of radius R is doubled. How
does this affect

• the magnitude of the centripetal force
• Fc m v2
• r
• Fc is directly proportional to v2
• so if v doubles Then Fc Quadruples

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12. Suppose the speed of an object moving in a
circular path stays constant and the radius r is
doubled. How does this affect the magnitude of
the centripetal force

• Fc m v2
• r
• Fc is inversely proportional to r so
• If r is doubled then Fc will be halved

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Example Problem Pg 238 3 A dog sits 1.50 m
from the center of a merry-go-round and revolves
at a tangential speed of 1.80 m/s. If the dogs
mass is 18.5 kg, what is the magnitude of the
centripetal force on the dog ?
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13. Identify the Centripetal Forces

• The Earth goes around the
  Sun
• The gravitational Force
  IS the Centripetal Force

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• The stopper in the lab activity
• How many forces are acting
• On the stopper ?
• 2
• Weight
• Tension
• Where does the tension come from ?
• The weight of the hanging Mass
• What is the centripetal force
• The x component of the tension
  IS the Centripetal Force
• What does the y component of the tension do ?
• Equals the weight

mg
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• A car going around a curve
• As the car goes around the
• Turn which way does it want to go ?
• Tangent to the circle ( 1st law)
• What keeps it from going off of the curve ?
• Friction So friction IS the centripetal force

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Example ProblemA 1000. kg car rounds a curve on
a flat road of radius 50 m at a speed of 14 m/s.
Will the car make the turn, or will it skid , if
A) the pavement is dry and the µ .60B) the
pavement is icy and µ .25
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What happens if there is not enough friction to
keep the car on the Curve ?

• NEED MORE FRICTION

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Can the banking of a road allow a car to go
around a curve if there is no friction ?

• Yes, a car can go around a curve with out
  friction as the centripetal force !!

Fc
mg
?
The centripetal force is the horizontal
component of the normal force
?
Fc mv2 r
mg
?
Tan? v2 rg
Tan? mv2 r mg
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Example ProblemWhat is the angle for an
expressway off-ramp curve of radius 50. m at a
design speed of 14 m/s
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Example Problem A 2.0 kg ball is swung in a
circle at the end of a 1.5 m string. How long
would it take the ball to complete 1.0
revolutions if the tension in the string is 3.2 N
?
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14. Describe what happens to the centripetal
force in a vertical circle
At the top T mg Fc
mg
T
At the bottom T- mg Fc
T
T
mg
mg
T
On each side T Fc
mg
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15. What is the gravitational Force ?

• The gravitational force is the pull of one
• object on another.Multimedia National
  Highway Traffic Safety Administration(NHTSA)
  U.S. Department of Transportation

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16. Give Newtons law of universal
gravitation and identify the variables.

• FG a m1m2
• ?x2
• Note this is a proportionality
• The two sides are not
• equal

• FG - Gravitational Force
• m1m2 - product of the
• masses
• ?x - distance between the centers of
• mass

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17. So what are the two things that effect the
gravitational force ? Give their relation to
centripetal force ?

• Product of the masses
• The product of the masses is directly
  proportional to the gravitational force
• FG a m1m2
• ?x2

The bigger the masses the bigger the force
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• Distance between the centers of mass
• The gravitational force is inversely proportional
  to the square of the distance between the centers
  of mass

As the distance increases the force decreases by
the square !!
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18. What is an inverse square law ?

• When a value changes another changes by the
  square.
• Universal gravitation is an example of an inverse
  square law.
• What happens to the gravitational force when
  distance is

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19. What is the universal gravitation
constant and who came up with it ?

• FG a m1m2
• ?x2
• Newton could not make the above equation equal
  only proportional
• FG Gm1m2 G makes the both sides
  equal
• ?x2
• G is the Universal Gravitation Constant 6.67
  X10-11 Nm2
• kg2
• The Universal Gravitation Constant was calculated
  by Henry Cavendish (1798)
• almost 100 years after Newton came up with
  Universal Gravitation

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Example ProblemMars has a mass of about 6.4 x
1023kg, and its moon Phobos has a mass of about
9.6 x 1015kg. If the magnitude of the
gravitational force between the two bodies is 4.6
x 1015, how far apart are Mars and Phobos ?
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Example Problem What is the force of gravity
acting on a 2000. kg spacecraft when it orbits
two Earth radii from the Earths center(rE
6380 km and m E 5.98 x 1024 kg)
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Example ProblemAt what height above the Earths
surface would a satellite be placed to be
geosynchronous ?U\Astronomy\04 Modern History
of Astronomy (ch3)\Geosynchronous Orbit.bmp
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Example ProblemFind the net force on the Moon
(mM 7.35x 1022 kg) due to the gravitational
attraction of both the Earth (mE 5.98 x 1024 kg
) and the Sun (m S 1.99 x 1030 kg), Assuming
they are at right angles ( ?x E - M 3.84 x 108
m, ?x s - M 1.50 x 1011 m)
Sun
moon
Earth
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20. So, what are the implications of Universal
Gravitation ?

• With Universal Gravitation
• The force between all objects can be calculated.
• The position of all present and future objects
  can be identified
• Explained the glue of the universe

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21. What is the relationship between the
acceleration due to gravity and
universal gravitation ?

• FG GMEmo
• ?x2
• mog GMEmo
• ?x2
• g depends on the mass of the central object and
  the
• distance to its center.
• g is also the same for ALL objects the same
  distance from the central object

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22. So, what is g really ?

• g is the acceleration of an object near a
• mass.
• g is the Gravitation Field Strength
• - the force per mass on an object in a
• gravitational field measure in N
• kg
• So near the surface of the Earth there is
• 9.8 N
• kg

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Near the surface of the Earth the Gravitational
Field Strength is 9.8 N/m For all objects
Earth
Twice the distance from the center of the Earth
the Gravitational field strength is 2.45 N/m
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23. Which experiences a greater gravitational
field strength the Moon or a satellite the same
distance as the Moon away from the Earth ?

• The field strength is the same for all objects
  the same distance from the central object.
• Just like the field strength is 9.8 N/m for each
  person near the surface.
• But F mg multiply the mass time the
• field strength and
  the
• Force is greater on more
• massive objects.

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24. So, do you weigh the same everywhere
on the Earth ?

• NO !!!!
• The Earth is not round.
• - it is an oblate spheroid

Earth
The Earth bulges on the equator and is flatter on
the top and bottom.
W mg
Remember that
g is smaller at the equator so you would weigh
less!!!!!!
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25. Why is the section on gravitation in the
same chapter as circular motion ?

• Planets move in nearly circular orbits

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26. What keeps the planet in its orbit ? What
implications does this have ?

• The gravitational force is the force that keeps
  planets in their orbit
• This means the gravitational force
• is the centripetal force !!!!!
• FG Fc
• Gm1m2 mv2
• ?x2 r

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27. Give another implication

• Your weight is a centripetal force

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