Download lecture 18 rotational motion

LECTURE 18
ROTATIONAL MOTION
Instructor: Kazumi Tolich
Lecture 18
2
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Reading chapter 10-1 to 10-2
¤ Angular
displacement
¤ Angular velocity and speed
¤ Angular acceleration
¤ Rotational kinematics with constant angular acceleration
Angular position
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Angular position (𝜃) is the angle between a
reference line on a rotating rigid body and a
reference line in space.
Angular position is measured counterclockwise
from the reference line in space.
By convention:
𝜃 > 0 counterclockwise rotation
𝜃 < 0 clockwise rotation
Arc length
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The arc length 𝑠 for an arbitrary
angle 𝜃 measured in radians is given
by
𝑠 = 𝑟𝜃
Quiz: 1
5
Demo 1
6
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Radian Disk
¤ Demonstration
of unit, radian
¤ 1 radian is approximately 57.3 degrees.
Quiz: 2
7
Angular displacement
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Angular displacement (∆𝜃) is the
change in the angular position with
respect to a reference line as the
object rotates.
∆𝜃 = 𝜃* − 𝜃,
¨
Angular displacement is measured in
radians.
Angular velocity and speed
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Angular velocity (𝜔) is defined as
𝜔./ =
∆0
∆1
𝜔 = lim∆1→6
¨
¨
(average)
∆0
∆1
(instantaneous)
The magnitude of angular velocity is called angular speed.
If a point on a rotating object moves in
¤
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counterclockwise: 𝜔 > 0.
clockwise: 𝜔 < 0.
Period
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The time to complete one revolution, 𝑇, is called period.
2𝜋
𝑇=
𝜔
Example: 1
11
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One of the most studied objects in the night
sky is the Crab nebula, the remains of a
supernova explosion observed by the
Chinese in 1054. In 1968 it was discovered
that a pulser – a rapidly rotating neutron
star that emits a pulse of radio waves with
each revolution – lies near the center of the
Crab nebula. The pulses are emitted every
33 ms. What is the angular speed of the
Crab nebula pulser in rad/s, in degrees/s,
and in revolutions/s?
Angular acceleration
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Angular acceleration (𝛼) is defined as
∆;
∆1
∆;
lim
∆1→6 ∆1
𝛼./ =
𝛼=
¨
(average)
(instantaneous)
Angular acceleration can be positive or negative, depending on the sign of
change in angular velocity.
Quiz: 3 & 4
13
Rotation with constant angular acceleration
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For an object rotating with an initial angular position, 𝜃0 , initial angular velocity, 𝜔0 , and a
constant angular acceleration, 𝛼,
¤
the angular velocity, 𝜔, as a function of time, 𝑡, is given by
𝜔 = 𝜔0 + 𝛼𝑡
¤
the angular position, 𝜃, as a function of time, is given by
𝜃 = 𝜃6 + 𝜔6 𝑡 + >?𝛼𝑡 ?
¤
The angular velocity as a function of angular displacement, ∆𝜃, is given by
𝜔? = 𝜔6? + 2𝛼∆𝜃
Example: 2
15
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A wheel has a constant angular
acceleration α = 0.35 rad/s2. It
starts from rest with an arbitrary
reference line horizontal, at
angular position θ0 = 0.
a)
b)
What is the angular
displacement Δθ of the reference
line at t = 18 s?
What is the wheel’s angular
velocity at t = 18 s?
Example: 3
16
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The drill used by most dentists today is powered
by a small air turbine that can operate at
angular speeds of 350,000 rpm. These drills,
along with ultrasonic dental drills, are the fastest
turbines in the world – far exceeding the angular
speeds of jet engines. Suppose a drill starts at
rest and comes up to operating speed in 2.1 s.
a)
Find the angular acceleration produced by the drill,
assuming it to be constant.
b)
How many revolutions does the drill bit make as it
comes up to speed?