Chapter 10

Transcription

Chapter 10

Copyright © 2010 Pearson Education, Inc.
ROTATIONAL KINEMATICS AND
ENERGY
Chapter 10
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
Angular Position, Velocity, and
Acceleration
10-1
Acceleration
10-1
Degrees and revolutions:
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
Acceleration
10-1
Acceleration
10-1
Acceleration
10-1
Acceleration
10-1
10-2
Rotational Kinematics
If the angular
acceleration is
constant:
10-2
Rotational Kinematics
Analogies between linear and rotational kinematics:
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?
Connections Between Linear and
Rotational Quantities
10-3
10-3
10-3
10-3
•  This merry-go-round has
both tangential and
centripetal acceleration.
10-4
Rolling motion
•  If a round object rolls without slipping, there is a fixed
relationship between the translational and rotational speeds:
10-4
Rolling motion
•  We may also consider rolling motion to be a combination of pure
rotational and pure translational motion:
Rotational Kinetic Energy and the
Moment of Inertia
10-5
•  For this mass,
Moment of Inertia
10-5
•  We can also write the kinetic energy as
•  Where I, the moment of inertia, is given by
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.
Example continued
Moment of Inertia
10-5
•  Moments of inertia of various regular objects can be
calculated:
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?
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.
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.
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?
Example continued
Example continued
Summary of Chapter 10
•  Describing rotational motion requires analogs to position,
velocity, and acceleration
•  Average and instantaneous angular velocity:
•  Average and instantaneous angular acceleration:
•  Period:
•  Counterclockwise rotations are positive, clockwise
negative
•  Linear and angular quantities:
•  Linear and angular equations of motion:
•  Tangential speed:
•  Centripetal acceleration:
•  Tangential acceleration:
•  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.

Chapter 10

Transcription

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