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Chapter 10
Rotation
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2. To describe rotation in terms of:
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

3. To understand moment of inertia (I):
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

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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:
The Sun, where layers of gas rotate separately
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
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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
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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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8. Units of angles Angles in radians  = s/r.
One complete revolution is 360° = 2π radians.

9. Angular coordinate
Consider a meter with a needle rotating about a fixed axis.
Angle  (in radians) that needle makes with +x-axis is a coordinate for rotation.

10. Angular coordinates 55 45
Example: A car’s speedometer needle rotates about a fixed axis.
Angle  the needle makes with negative x-axis is the coordinate for rotation.
KEY: Define your directions! 35 25 15 q

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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
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10-1 Rotational Variables
Calculation of average angular velocity:
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15. Angular velocity The rotation axis matters!
Subscript z means that the rotation is about the z-axis.
Instantaneous angular velocity is z = d/dt.
Counterclockwise rotation is positive;
Clockwise rotation is negative.

16. Calculating angular velocity
Flywheel diameter 0.36 m;
Suppose q(t) = (2.0 rad/s3) t3

17. 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

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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
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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
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23. Angular velocity is a vector
Angular velocity is defined as a vector whose direction is given by the right-hand rule.

24. Angular acceleration as a vector
For a fixed rotation axis, angular acceleration a and angular velocity w vectors both lie along that axis.

25. 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!

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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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27. Relating linear and angular kinematics
For a point a distance r from the axis of rotation:
its linear speed is v = r (meters/sec)

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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):
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30. Relating linear and angular kinematics
For a point a distance r from the axis of rotation:
its linear tangential acceleration is atan = r (m/s2)
its centripetal (radial) acceleration is arad = v2/r = r

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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
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33. 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 = rad/s2
What is w at t = 0.3 seconds?
What angle does PQ make with x axis then?

34. 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

35. 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?

36. 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?

37. 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.

38. 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!

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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:
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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
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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)
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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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45. Moments of inertia

46. Moments of inertia

47. Moments of inertia of some common bodies

48. 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?

49. 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?

50. 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?

51. 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?

52. More on an unwinding cable
mgh
½ mv2 + ½ Iw2

53. More on an unwinding cable

54. 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

55. 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

56. The parallel-axis theorem
The parallel-axis theorem is: IP = Icm + Md2. M d

57. The parallel-axis theorem
The parallel-axis theorem is: IP = Icm + Md2.
Mass 3.6 kg, I = kg-m2 through P. What is I about parallel axis through center of mass?

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10-5 Calculating the Rotational Inertia
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10-5 Calculating the Rotational Inertia
Answer: (1), (2), (4), (3)
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10-5 Calculating the Rotational Inertial
Example Calculate the moment of inertia for Fig (b)
Summing by particle:
Use the parallel-axis theorem
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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
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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
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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
Answer: F1 & F3, F4, F2 & F5
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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:
The work done in a rotation about a fixed axis can be calculated by:
Which, for a constant torque, reduces to:
Eq. (10-52)
Eq. (10-53)
Eq. (10-54)
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10-8 Work and Rotational Kinetic Energy
We can relate work to power with the equation:
Table 10-3 shows corresponding quantities for linear and rotational motion:
Eq. (10-55)
Tab. 10-3
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Summary
Angular Position
Measured around a rotation axis, relative to a reference line:
Angular Displacement
A change in angular position
Eq. (10-4)
Eq. (10-1)
Angular Velocity and Speed
Average and instantaneous values:
Angular Acceleration
Average and instantaneous values:
Eq. (10-7)
Eq. (10-5)
Eq. (10-6)
Eq. (10-8)
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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
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)
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Summary
Torque
Force applied at distance from an axis:
Moment arm: perpendicular distance to the rotation axis
Newton's Second Law in Angular Form
Eq. (10-42)
Eq. (10-39)
Work and Rotational Kinetic Energy
Eq. (10-53)
Eq. (10-55)
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