Transcript Ch10 - Rotation - Chabot College

Chapter 10
Rotation
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Goals for Chapter 10
• To describe rotation in terms of:
angular coordinates (q)
angular velocity (w)
angular acceleration (a)
• To analyze rotation with constant angular
acceleration
• To relate rotation to the linear velocity and linear
acceleration of a point on a body
Goals for Chapter 10
• To understand moment of inertia (I):
how it depends upon rotation axes
how it relates to rotational kinetic energy
how it is calculated
10-1 Rotational Variables

A rigid body rotates as a unit, locked together

We look at rotation about a fixed axis

These requirements exclude from consideration:
o
The Sun, where layers of gas rotate separately
o
A rolling bowling ball, where rotation and translation occur
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-1 Rotational Variables


Fixed axis is called axis of rotation
Angular position of this line (& of object) is taken
relative to a fixed direction: zero angular position
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-1 Rotational Variables


Fixed axis is called
axis of rotation
Angular position of
this line (& of object)
is taken relative to a
fixed direction: zero
angular position
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-1 Rotational Variables

Measure using radians (rad): dimensionless

Do not reset θ to zero after a full rotation

If you know θ(t), you can get:

Angular displacement

Angular Velocity
w

Angular Acceleration
a
© 2014 John Wiley & Sons, Inc. All rights reserved.
Units of angles
• Angles in radians
q = s/r.
• One complete revolution is
360° = 2π radians.
Angular coordinate
• Consider a meter with a
needle rotating about a fixed
axis.
• Angle q (in radians) that
needle makes with +x-axis is
a coordinate for rotation.
Angular coordinates
45
• Example: A car’s speedometer
needle rotates about a fixed axis.
• Angle q the needle makes with
negative x-axis is the coordinate
for rotation.
• KEY: Define your directions!
35
25
15
q
55
10-1 Rotational Variables

“Clocks are negative”:
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10-1 Rotational Variables

“Clocks are negative”:
Answer: Choices (b) and (c)
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10-1 Rotational Variables




Average angular velocity: angular displacement
during a time interval
Instantaneous angular velocity: limit as Δt → 0
If the body is rigid, these equations hold for all points
on the body
Magnitude of angular velocity = angular speed
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-1 Rotational Variables

Calculation of average angular velocity:
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Angular velocity
• The rotation axis matters!
• Subscript z means that the
rotation is about the z-axis.
• Instantaneous angular velocity is
wz = dq/dt.
• Counterclockwise rotation is
positive;
• Clockwise rotation is negative.
Calculating angular velocity
• Flywheel diameter 0.36 m;
• Suppose q(t) = (2.0 rad/s3) t3
Calculating angular velocity
• Flywheel diameter 0.36 m; q = (2.0 rad/s3) t3
Find q at t1 = 2.0 s and t2 = 5.0 s
Find distance rim moves in that interval
Find average angular velocity in rad/sec &
rev/min
Find instantaneous angular velocities at
t1 & t2
10-1 Rotational Variables


Average angular acceleration: angular velocity
change during a time interval
Instantaneous angular acceleration: limit as Δt → 0
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10-2 Rotation with Constant Angular Acceleration

Same equations as for constant linear acceleration

Change linear quantities to angular ones
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10-2 Rotation with Constant Angular Acceleration
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10-2 Rotation with Constant Angular Acceleration
Answer: Situations (a) and (d); the others do not have constant angular
acceleration
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-1 Rotational Variables

If body is rigid, equations hold for all points on body

Use right-hand rule to determine direction

Angular velocity & Acceleration are vectors

If body rotates around axis, vector points along
axis of rotation
© 2014 John Wiley & Sons, Inc. All rights reserved.
Angular velocity is a vector
• Angular velocity is defined as a vector whose
direction is given by the right-hand rule.
Angular acceleration as a vector
• For a fixed rotation axis, angular acceleration a and
angular velocity w vectors both lie along that axis.
Angular acceleration as a vector
• For a fixed rotation axis, angular acceleration a and
angular velocity w vectors both lie along that axis.
• BUT THEY DON’T HAVE TO BE IN THE SAME
DIRECTION!
w Speeds
up!
w Slows
down!
10-3 Relating the Linear and Angular Variables

Linear & angular variables are related by r,
perpendicular distance from the rotational axis

Position (note θ must be in radians):

Speed (note ω must be in radian measure):

We can express the period in radian measure:
© 2014 John Wiley & Sons, Inc. All rights reserved.
Relating linear and angular kinematics
• For a point a distance r from the
axis of rotation:
its linear speed is
v = rw (meters/sec)
10-3 Relating the Linear and Angular Variables

We can write the radial
acceleration in terms of
angular velocity (radians):
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10-3 Relating the Linear and Angular Variables

Tangential acceleration
(radians):
© 2014 John Wiley & Sons, Inc. All rights reserved.
Relating linear and angular kinematics
• For a point a distance r from the
axis of rotation:
its linear tangential
acceleration is
atan = ra (m/s2)
its centripetal (radial)
acceleration is
arad = v2/r = rw
10-3 Relating the Linear and Angular Variables
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10-3 Relating the Linear and Angular Variables
Answer: (a) yes (b) no (c) yes (d) yes
© 2014 John Wiley & Sons, Inc. All rights reserved.
Rotation of a Blu-ray disc
• A Blu-ray disc is coming
to rest after being played.
• @ t = 0, w = 27.5 rad/sec;
a = -10.0 rad/s2
• What is w at t = 0.3
seconds?
• What angle does PQ make
with x axis then?
An athlete throwing a discus
• Whirl discus in circle of r = 80 cm; at some time t
athlete is rotating at 10.0 rad/sec; speed increasing
at 50.0 rad/sec/sec.
• Find tangential and centripetal accelerations and
overall magnitude of acceleration
Designing a propeller
• Say rotation of propeller is at a constant 2400 rpm, as
plane flies forward at 75.0 m/s at constant speed.
• But…tips of propellers must move slower than 270 m/s
to avoid excessive noise.
• What is maximum propeller radius?
• What is acceleration of the tip?
Designing a propeller
• Tips of propellers must move slower than than 270 m/s
to avoid excessive noise. What is maximum propeller
radius? What is acceleration of the tip?
Moments of inertia
How much force it takes to get something rotating, and how
much energy it has when rotating, depends on WHERE the
mass is in relation to the rotation axis.
Moments of inertia
• Getting MORE mass, FARTHER from the axis, to rotate will
take more force!
• Some rotating at the same rate with more mass farther away
will have more KE!
10-4 Kinetic Energy of Rotation



Apply kinetic energy formula for a point & sum over all
K = Σ ½mivi2
But… same angular velocity for all points doesn’t
mean same linear velocities! Points could be at
different radii from axis!
Express vi (linear velocity) in terms of angular velocity:
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-4 Kinetic Energy of Rotation

We call the quantity in parentheses on the right side
the rotational inertia, or moment of inertia, I

I is constant for a rigid object and given rotational axis

Caution: the axis for I must always be specified
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-4 Kinetic Energy of Rotation

We can write:

And rewrite the kinetic energy as:

Use these equations for a finite set of rotating particles

Rotational inertia corresponds to how difficult it is to
change the state of rotation (speed up, slow down or
change the axis of rotation)
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-4 Kinetic Energy of Rotation
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10-4 Kinetic Energy of Rotation
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10-4 Kinetic Energy of Rotation
Answer: They are all equal!
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Moments of inertia
Moments of inertia
Moments of inertia of some common bodies
Rotational kinetic energy example 9.6
What is I about A?
What is I about B/C?
What is KE if it rotates through A with w=4.0 rad/sec?
An unwinding cable
• Wrap a light, non-stretching cable around solid
cylinder of mass 50 kg; diameter .120 m. Pull for 2.0 m
with constant force of 9.0 N. What is final angular
speed and final linear speed of cable?
More on an unwinding cable
Consider falling mass “m” tied
to rotating wheel of mass M and
radius R
What is the resulting speed of
the small mass when it
reaches the bottom?
More on an unwinding cable
Consider falling mass “m” tied to
rotating wheel of mass M and
radius R
What is the resulting speed of
the small mass when it reaches
the bottom?
More on an unwinding cable
mgh
½ mv2 + ½ Iw2
More on an unwinding cable
The parallel-axis theorem
• What happens if you rotate about
an EXTERNAL axis, not internal?
• Spinning planet orbiting
around the Sun
• Rotating ball bearing orbiting
in the bearing
• Mass on turntable
The parallel-axis theorem
• What happens if you rotate about an EXTERNAL
axis, not internal?
• Effect is a COMBINATION of TWO rotations
• Object itself spinning;
• Point mass orbiting
• Net rotational inertia combines both:
• The parallel-axis theorem is:
IP = Icm + Md2
The parallel-axis theorem
• The parallel-axis theorem is: IP = Icm + Md2.
M
d
The parallel-axis theorem
• The parallel-axis theorem is: IP = Icm + Md2.
• Mass 3.6 kg, I = 0.132 kg-m2 through P.
What is I about parallel axis through center of mass?
10-5 Calculating the Rotational Inertia
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10-5 Calculating the Rotational Inertia
Answer: (1), (2), (4), (3)
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-5 Calculating the Rotational Inertial
Example Calculate the moment of inertia for Fig. 10-13 (b)
o
Summing by particle:
o
Use the parallel-axis theorem
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-6 Torque

Force necessary to rotate
an object depends on:

Angle of the force

Where it is applied
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10-6 Torque





Torque takes these factors into account:
A line extended through the applied force is called the
line of action of the force
The perpendicular distance from the line of action to
the axis is called the moment arm
The unit of torque is the newton-meter, N m
Note that 1 J = 1 N m, but torques are never
expressed in joules, torque is not energy
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-6 Torque


Force necessary to rotate
an object depends on:

Angle of the force

Where it is applied
We can resolve the force
into components to see
how it affects rotation
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10-6 Torque
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10-6 Torque


Again, torque is positive if it would cause a
counterclockwise rotation, otherwise negative
For several torques, the net torque or resultant
torque is the sum of individual torques
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-6 Torque


Again, torque is positive if it would cause a
counterclockwise rotation, otherwise negative
For several torques, the net torque or resultant
torque is the sum of individual torques
Answer: F1 & F3, F4, F2 & F5
© 2014 John Wiley & Sons, Inc. All rights reserved.
10-7 Newton's Second Law for Rotation

Rewrite F = ma with rotational variables:
Eq. (10-42)

It is torque that
causes angular
acceleration
Figure 10-17
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10-7 Newton's Second Law for Rotation
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10-7 Newton's Second Law for Rotation
Answer: (a) F2 should point downward, and
(b) should have a smaller magnitude than F1
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10-8 Work and Rotational Kinetic Energy

The rotational work-kinetic energy theorem states:
Eq. (10-52)

The work done in a rotation about a fixed axis can be
calculated by:
Eq. (10-53)

Which, for a constant torque, reduces to:
Eq. (10-54)
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10-8 Work and Rotational Kinetic Energy

We can relate work to power with the equation:
Eq. (10-55)

Table 10-3 shows corresponding quantities for linear
and rotational motion:
Tab. 10-3
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10
Summary
Angular Position

Angular Displacement
Measured around a rotation
axis, relative to a reference
line:
• A change in angular position
Eq. (10-4)
Eq. (10-1)
Angular Velocity and
Speed
Angular Acceleration
• Average and instantaneous
values:
• Average and instantaneous
values:
Eq. (10-5)
Eq. (10-6)
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Eq. (10-7)
Eq. (10-8)
10
Summary
Kinematic Equations
• Given in Table 10-1 for constant
acceleration
• Match the linear case
Rotational Kinetic Energy
and Rotational Inertia
Linear and Angular
Variables Related
• Linear and angular
displacement, velocity, and
acceleration are related by r
The Parallel-Axis Theorem
• Relate moment of inertia around
any parallel axis to value around
com axis
Eq. (10-34)
Eq. (10-36)
Eq. (10-33)
© 2014 John Wiley & Sons, Inc. All rights reserved.
10
Summary
Torque
• Force applied at distance from
an axis:
Newton's Second Law in
Angular Form
Eq. (10-39)
• Moment arm: perpendicular
distance to the rotation axis
Work and Rotational
Kinetic Energy
Eq. (10-53)
Eq. (10-55)
© 2014 John Wiley & Sons, Inc. All rights reserved.
Eq. (10-42)