Transcript Chapter 10 - SFSU Physics & Astronomy

Rotational Kinematics
Angular Position, Velocity, and
Acceleration
Angular Position, Velocity, and
Acceleration
Degrees and revolutions:
Angular Position, Velocity, and
Acceleration
Arc length s,
measured in
radians:
10-1 Angular Position, Velocity, and
Acceleration
Angular Position, Velocity, and
Acceleration
Angular Position, Velocity, and
Acceleration
Angular Position, Velocity, and
Acceleration
Rotational Kinematics
If the angular
acceleration is
constant:
Rotational Kinematics
Analogies between linear and rotational
kinematics:
Connections Between Linear and
Rotational Quantities
Connections Between Linear and
Rotational Quantities
Connections Between Linear and
Rotational Quantities
Connections Between Linear and
Rotational Quantities
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:
Torque
From experience, we know that the same force
will be much more effective at rotating an
object such as a nut or a door if our hand is not
too close to the axis.
This is why we have
long-handled
wrenches, and why
doorknobs are not
next to hinges.
Torque
We define a quantity called torque:
The torque increases as the force increases,
and also as the distance increases.
Note:  has the same unit (N . M) as work but it
is a very different thing!
Torque
Only the tangential component of force causes
a torque:
Torque
This leads to a more general definition of torque:
Torque
If the torque causes a counterclockwise angular
acceleration, it is positive; if it causes a
clockwise angular acceleration, it is negative.
Rotational Kinetic Energy and the Moment
of Inertia
For this mass,
Rotational Kinetic Energy and the Moment
of Inertia
We can also write the kinetic energy as
Where I, the moment of inertia, is given by
Rotational Kinetic Energy and the Moment
of Inertia
Moments of inertia of various regular objects can
be calculated:
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.
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.
Summary
• Describing rotational motion requires analogs
to position, velocity, and acceleration
• Average and instantaneous angular velocity:
• Average and instantaneous angular
acceleration:
Summary
• Period:
• Counterclockwise rotations are positive,
clockwise negative
• Linear and angular quantities:
Summary
• 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.