Transcript Chapter 13 Vibrations and Waves

Chapter 7 Rotational Motion and
the Law of Gravity
Ying Yi PhD
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Circular Motion
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Outline
 Angular Speed and Angular Acceleration
 Constant Angular Acceleration Motion
 Relation between Angular and Linear Quantities
 Centripetal Acceleration
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The Radian
 The radian is a unit of
angular measure
 The radian can be
defined as the arc
length along a circle
divided by the radius r

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s
 
r
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More About Radians
 Comparing degrees and radians
360
1 rad 
 57.3
2
 Converting from degrees to radians

 [rad] 
 [deg rees]
180
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Rigid Body
 Every point on the object undergoes circular
motion about the point O
 All parts of the object of the body rotate through
the same angle during the same time
 The object is considered to be a rigid body
 This means that each part of the body is fixed in
position relative to all other parts of the body
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Is this a rigid body?
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Angular Displacement
 Axis of rotation is the
center of the disk
 Need a fixed reference
line
 During time t, the
reference line moves
through angle θ
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Angular Displacement, cont.
 The angular displacement is defined as the angle
the object rotates through during some time
interval
   f   i

 The unit of angular displacement is the radian
 Each point on the object undergoes the same
angular displacement
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Average Angular Speed
 The average angular
speed, ω, of a rotating
rigid object is the ratio
of the angular
displacement to the
time interval
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Angular Speed, cont.
 The instantaneous angular speed is defined as the
limit of the average speed as the time interval
approaches zero
 Units of angular speed are radians/sec
 rad/s
 Speed will be positive if θ is increasing
(counterclockwise)
 Speed will be negative if θ is decreasing
(clockwise)
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Example 7.1 Whirlybirds
The rotor on a helicopter turns at an angular speed of
3.20×102 revolutions per minute (rpm). (a) Express this
angular speed in radians per second. (b) If the rotor has
a radius of 2.00 m, what arclength does the tip of the
blade trace out in 3.00×102 s?
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Average Angular Acceleration
 The average angular acceleration
of an object is
defined as the ratio of the change in the angular
speed to the time it takes for the object to undergo the
change:
 av 
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f  i
tf  ti


t
Angular Acceleration, cont
 Units of angular acceleration are rad/s²
 Positive angular accelerations are in the
counterclockwise direction and negative
accelerations are in the clockwise direction
 When a rigid object rotates about a fixed axis,
every portion of the object has the same angular
speed and the same angular acceleration
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Angular Acceleration, final
 The sign of the acceleration does not have to be the
same as the sign of the angular speed
 The instantaneous angular acceleration is defined as
the limit of the average acceleration as the time
interval approaches zero
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Linear and Rotational Motion
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Example 7.2 A rotating Wheel
A wheel rotates with a constant angular acceleration of
3.50 rad/s2. If the angular speed of the wheel is 2.00
rad/s at t=0, (a) through what angle does the wheel
rotate between t=0 and t=2.00s? Give your answer in
radians and in revolutions. (b) What is the angular
speed of the wheel at t=2.00 s? (c) What angular
displacement (in revolutions) results while the angular
speed found in part (b) doubles?
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Relationship Between Angular and Linear
Quantities
 Displacements
s  r
 Speeds
vt   r
 Accelerations
at   r
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 Every point on the
rotating object has the
same angular motion
 Every point on the
rotating object does not
have the same linear
motion
Example 7.4 Track Length of a compact disc
In a compact disc player, as the read head moves out from the
center of the disc, the angular speed of the disc changes so
that the linear speed at the position of the head remains at a
constant value of about 1.3 m/s (a) find the angular speed of
the compact disc when the read head is at r=2.0 cm and again
at r=5.6 cm. (b) An old-fashioned record player rotates at a
constant angular speed, so the linear speed of the record
groove moving under the detector changes. Find the linear
speed of a 45.0 rpm record at points 2.0 and 5.6 cm from the
center. (c) In both the CD and phonograph record,
information is recorded in a continuous spiral track, 1.3 m/s.
Calculate the total length of the track for a CD designed to
play for 1.0 h.
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Group Problem: Compact Discs
A compact disc rotates from rest up to an angular
speed of 31.4 rad/s in a time of 0.892 s. (a) What is
the angular acceleration of the disc, assuming the
angular acceleration is uniform? (b) Through what
angle does the disc turn while coming up to this
speed? (c) If the radius of the disc is 4.45 cm, find the
tangential speed of a microbe riding on the rim of
the disc when t=0.892 s. (d) What is the magnitude
of the tangential acceleration of the microbe at the
given time?
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Uniform Circular Motion
Speed doesn’t change!
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Centripetal Acceleration
 An object traveling in a circle, even though it moves
with a constant speed, will have an acceleration
 The centripetal acceleration is due to the change in
the direction of the velocity
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Centripetal Acceleration, cont.
 Centripetal refers to
“center-seeking”
 The direction of the
velocity changes
 The acceleration is
directed toward the
center of the circle of
motion
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Centripetal Acceleration, final
 The magnitude of the centripetal acceleration is
given by
2
v
ac 
r
 This direction is toward the center of the circle
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Centripetal Acceleration and Angular
Velocity
 The angular velocity and the linear velocity are
related (v = ωr)
 The centripetal acceleration can also be related to the
angular velocity
v
r ω
2
aC  
 rω
r
r
2
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2
2
Group Problem: Bobsled Track
The bobsled track at the 1994 Olympics
in Lillehammer, Norway, contained
turns with radii of 33 m and 24 m, as
Figure 5.5 illustrates. Find the
centripetal acceleration at each turn for
a speed of 34 m/s, a speed that was
achieved in the two-man event. Express
the answers as multiples of g=9.8 m/s2.
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Total 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
a  a a
2
t
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2
C
Vector Nature of Angular Quantities
 Angular displacement,
velocity and acceleration are
all vector quantities
 Direction can be more
completely defined by using
the right hand rule
 Grasp the axis of rotation with
your right hand
 Wrap your fingers in the
direction of rotation
 Your thumb points in the
direction of ω
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Velocity Directions, Example
 In a, the disk rotates
clockwise, the velocity
is into the page
 In b, the disk rotates
counterclockwise, the
velocity is out of the
page
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Acceleration Directions
 If the angular acceleration and the angular velocity
are in the same direction, the angular speed will
increase with time
 If the angular acceleration and the angular velocity
are in opposite directions, the angular speed will
decrease with time
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Example 7.5 At the Racetrack
A race car accelerates uniformly from a speed of 40.0
m/s to a speed of 60.0 m/s in 5.00 s while traveling
counterclockwise around a circular track of radius
4.00×102 m. When the car reaches a speed of 50.0 m/s,
find (a) the magnitude of the car’s centripetal
acceleration, (b) the angular speed, (c) the magnitude of
the tangential acceleration, and (d) the magnitude of
the total acceleration.
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Forces Causing Centripetal Acceleration



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Tension in a string
Force of friction
Gravitational force
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Centripetal Force
 General equation
m v2
FC  m aC 
r
 If the force vanishes, the object will move in a
straight line tangent to the circle of motion
 Centripetal force is a classification that includes
forces acting toward a central point
 It is not a force in itself
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Centripetal Force Example 1: Tension
 A ball of mass m is
attached to a string
 Its weight is supported
by a frictionless table
 The tension in the
string causes the ball
to move in a circle
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Example: Tension
The model airplane in figure 5.6 has a mass of 0.90 kg
and moves at a constant speed on a circle that is parallel
to the ground. The path of the airplane and its
guideline lie in the same horizontal plane, because the
weight of the plane is balanced by the lift generated by
its wings. Find the tension in the guideline (length=17
m) for speeds of 19 and 38 m/s.
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Centripetal Force Example 1:Friction
 Friction is the force
that produces the
centripetal
acceleration
 Can find the
frictional force, µ, or
v
v  rg
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Example 7.6 Buckle Up for Safety
A car travels at a constant speed of 30.0 mi/h (13.4
m/s) on a level circular turn of radius 50.0 m. What
minimum coefficient of static friction, µs, between the
tires and roadway will allow the car to make the circular
turn without sliding?
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Centripetal Force Example 3: Normal force
A car travels on a circle of
radius r on a frictionless
banked road. The banking
angle is Ɵ, and the center of
the circle is at C.
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The force acting on the
car are its weight and
normal
force.
A
component FNsinƟ of
the normal force provide
the centripetal force.
Group Problem: Banked curves
The Daytona international Speedway in Daytona Beach,
Florida, is famous for its races especially the Daytona
500, held every February. Both of its courses feature
four-story, 31.0º banked curves, with maximum radius
of 316 m. If a car negotiates the curve too slowly, it
tends to slip down the incline of the turn, whereas if
it’s going too fast, it may begin to slide up the incline.
(a) Find the necessary centripetal acceleration on this
banked curve so the car won’t slip down or slide up the
incline. (Neglect friction) (b) Calculate the speed of the
car.
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Problem Solving Strategy
 Draw a free body diagram, showing and labeling all
the forces acting on the object(s)
 Choose a coordinate system that has one axis
perpendicular to the circular path and the other axis
tangent to the circular path
 The normal to the plane of motion is also often needed
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Problem Solving Strategy, cont.
 Find the net force toward the center of the
circular path (this is the force that causes the
centripetal acceleration, FC)
 Use Newton’s second law
 The directions will be radial, normal, and tangential
 The acceleration in the radial direction will be the
centripetal acceleration
 Solve for the unknown(s)
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Thank you
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