Chapter 10

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ROTATIONAL KINEMATICS AND
ENERGY
Chapter 10
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Units of Chapter 10
•  Angular Position, Velocity, and Acceleration
•  Rotational Kinematics
•  Connections Between Linear and Rotational Quantities
•  Rolling Motion
•  Rotational Kinetic Energy and the Moment of Inertia
•  Conservation of Energy
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Angular Position, Velocity, and
Acceleration
10-1
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Angular Position, Velocity, and
Acceleration
10-1
Degrees and revolutions:
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Angular Position, Velocity, and
Acceleration
10-1
•  Arc length s, measured
in radians:
•  Radian: angle for which the arc length on a circle of
radius r is equal to the radius of the circle
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Angular Position, Velocity, and
Acceleration
10-1
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Angular Position, Velocity, and
Acceleration
10-1
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Angular Position, Velocity, and
Acceleration
10-1
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Angular Position, Velocity, and
Acceleration
10-1
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10-2
Rotational Kinematics
If the angular
acceleration is
constant:
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10-2
Rotational Kinematics
Analogies between linear and rotational kinematics:
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Example
A high speed dental drill is rotating at 3.14×104 rads/sec.
Through how many degrees does the drill rotate in 1.00
sec?
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Connections Between Linear and
Rotational Quantities
10-3
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Connections Between Linear and
Rotational Quantities
10-3
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Connections Between Linear and
Rotational Quantities
10-3
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Connections Between Linear and
Rotational Quantities
10-3
•  This merry-go-round has
both tangential and
centripetal acceleration.
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10-4
Rolling motion
•  If a round object rolls without slipping, there is a fixed
relationship between the translational and rotational speeds:
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10-4
Rolling motion
•  We may also consider rolling motion to be a combination of pure
rotational and pure translational motion:
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Rotational Kinetic Energy and the
Moment of Inertia
10-5
•  For this mass,
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Rotational Kinetic Energy and the
Moment of Inertia
10-5
•  We can also write the kinetic energy as
•  Where I, the moment of inertia, is given by
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Example
(a) Find the moment of inertia of the system below. The
masses are m1 and m2 and they are separated by a
distance r. Assume the rod connecting the masses is
massless.
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Example continued
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Rotational Kinetic Energy and the
Moment of Inertia
10-5
•  Moments of inertia of various regular objects can be
calculated:
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Example
•  What is the rotational inertia of a solid iron disk of
mass 49.0 kg with a thickness of 5.00 cm and a radius
of 20.0 cm, about an axis through its center and
perpendicular to it?
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10-6
Conservation of energy
•  The total kinetic energy of a rolling object is the sum
of its linear and rotational kinetic energies:
•  The second equation makes it clear that the kinetic
energy of a rolling object is a multiple of the kinetic
energy of translation.
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10-6
Conservation of energy
•  If these two objects, of the same mass and
radius, are released simultaneously, the disk
will reach the bottom first – more of its
gravitational potential energy becomes
translational kinetic energy, and less
rotational.
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Example
•  Two objects (a solid disk and a solid sphere) are
rolling down a ramp. Both objects start from rest and
from the same height. Which object reaches the
bottom of the ramp first?
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Example continued
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Example continued
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Summary of Chapter 10
•  Describing rotational motion requires analogs to position,
velocity, and acceleration
•  Average and instantaneous angular velocity:
•  Average and instantaneous angular acceleration:
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Summary of Chapter 10
•  Period:
•  Counterclockwise rotations are positive, clockwise
negative
•  Linear and angular quantities:
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Summary of Chapter 10
•  Linear and angular equations of motion:
•  Tangential speed:
•  Centripetal acceleration:
•  Tangential acceleration:
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Summary of Chapter 10
•  Rolling motion:
•  Kinetic energy of rotation:
•  Moment of inertia:
•  Kinetic energy of an object rolling without slipping:
•  When solving problems involving conservation of energy,
both the rotational and linear kinetic energy must be taken
into account.