Chapter 10
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.

Goals for Chapter 10
•
To describe rotation in terms of:
q
w
a
•
To analyze rotation with constant angular acceleration
•
To relate
to the linear velocity and linear acceleration of a point on a body

Goals for Chapter 10
•
To understand
:
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
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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
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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 + xaxis is a coordinate for rotation.

Angular coordinates
•
Example
: A car’s speedometer needle rotates about a fixed axis .
•
Angle q the needle makes with negative xaxis is the coordinate for rotation.
25
15
35
•
KEY : Define your directions!
q
45
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 zaxis.
•
Instantaneous angular velocity is w z
= d q
/dt.
•
Counterclockwise rotation is positive ;
•
Clockwise rotation is negative .

Calculating angular velocity
•
Flywheel diameter 0.36 m;
•
Suppose q( t
)
= (2.0 rad/s 3 ) t 3

Calculating angular velocity
•
Flywheel diameter 0.36 m; q
= (2.0 rad/s 3 ) t 3
Find q at t
1
= 2.0 s and t
2
= 5.0 s
Find distance rim moves in that interval
Find average angular velocity in rad/sec & rev/min
Find instantaneous angular velocities at t
1
& t
2

10-1 Rotational Variables

Average angular acceleration : angular velocity change during a time interval

Instantaneous angular acceleration : limit as Δt → 0
© 2014 John Wiley & Sons, Inc. All rights reserved.

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 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:
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Relating linear and angular kinematics
•
For a point a distance r from the axis of rotation: its linear speed is v = r w
(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 a tan
= r a
(m/s 2 ) its centripetal (radial) acceleration is a rad
= v 2 / r = r w

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/s 2
• 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 ½ m i v i
2

But… same angular velocity for all points doesn’t mean same linear velocities! Points could be at different radii from axis!

Express v i
(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
© 2014 John Wiley & Sons, Inc. All rights reserved.

10-4 Kinetic Energy of Rotation
© 2014 John Wiley & Sons, Inc. All rights reserved.

10-4 Kinetic Energy of Rotation
Answer: They are all equal!
© 2014 John Wiley & Sons, Inc. All rights reserved.

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 mv 2 + ½ I w
2

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:
I
P
= I cm
+ Md 2

The parallel-axis theorem
•
The parallel-axis theorem is: I
P
= I cm
+ Md 2 .
M d

The parallel-axis theorem
•
The parallel-axis theorem is: I
P
= I cm
+ Md 2 .
•
Mass 3.6 kg, I = 0.132 kg-m 2 through P.
What is I about parallel axis through center of mass?

10-5 Calculating the Rotational Inertia
© 2014 John Wiley & Sons, Inc. All rights reserved.

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
© 2014 John Wiley & Sons, Inc. All rights reserved.

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
© 2014 John Wiley & Sons, Inc. All rights reserved.

10-6 Torque
© 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
© 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: F
1
& F
3
, F
4
, F
2
& F
5
© 2014 John Wiley & Sons, Inc. All rights reserved.

10-7 Newton's Second Law for Rotation

Rewrite F = ma with rotational variables:

It is torque that causes angular acceleration
Eq. (10-42)
© 2014 John Wiley & Sons, Inc. All rights reserved.
Figure 10-17

10-7 Newton's Second Law for Rotation
© 2014 John Wiley & Sons, Inc. All rights reserved.

10-7 Newton's Second Law for Rotation
Answer: (a) F
2 should point downward, and
(b) should have a smaller magnitude than F
1
© 2014 John Wiley & Sons, Inc. All rights reserved.

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)
© 2014 John Wiley & Sons, Inc. All rights reserved.

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:
© 2014 John Wiley & Sons, Inc. All rights reserved.
Tab. 10-3

10 Summary
Angular Position

Measured around a rotation axis , relative to a reference line :
Eq. (10-1)
Angular Displacement
• A change in angular position
Eq. (10-4)
Angular Velocity and
Speed
• Average and instantaneous values:
Eq. (10-5)
Angular Acceleration
• Average and instantaneous values:
Eq. (10-7)
Eq. (10-8) Eq. (10-6)
© 2014 John Wiley & Sons, Inc. All rights reserved.

10 Summary
Kinematic Equations
• Given in Table 10-1 for constant acceleration
• Match the linear case
Linear and Angular
Variables Related
• Linear and angular displacement, velocity, and acceleration are related by r
Rotational Kinetic Energy and Rotational Inertia
Eq. (10-34)
The Parallel-Axis Theorem
• Relate moment of inertia around any parallel axis to value around com axis
Eq. (10-36)
Eq. (10-33)
© 2014 John Wiley & Sons, Inc. All rights reserved.

10 Summary
Torque
• Force applied at distance from an axis:
Eq. (10-39)
Newton's Second Law in
Angular Form
Eq. (10-42)
• 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.