1. Chapter 8 Rotational Motion a.k.a. Angular Motion
a.k.a. Circular Motion
This is a carousel.

2. This is a merry go round

3. How is Rotational motion the most common in the universe?
Electrons rotate around nucleus
Moons, planets and stars spin
Moons and planets orbit other bodies
Galaxies spin

4. THE MANY Units of Chapter 8
Angular Quantities (some definitions)
Constant Angular Acceleration (kinematics)
Rolling Motion (Without Slipping) (definitions)
Torque (defined)
Rotational Dynamics; Torque and Rotational Inertia (F = ma)
Solving Problems in Rotational Dynamics

5. Units of Chapter 8 Rotational Kinetic Energy (KE = ½ m v2)
Angular Momentum and Its Conservation
Vector Nature of Angular Quantities
No new physics concepts are presented in this chapter.
Only a new geometry!
Up to now, we have done TRANSLATION
Now we do ROTATION
These two cover all there is in the universe.

6. Summer science camps and opportunities.
There are dozens, and most require application in March. Google science summer camps for a look.
Choose a place or college if you want to explore there
These look good on a resume
They will help you see college and career options
application by march 1
at the U of M, applications due march 31.
xploring Careers in Engineering and Physical Science at U of M

7. 8-1 Angular Quantities (define stuff)
In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. The angle θ in radians is defined:
where l is the arc length.
(8-1a)

8. 8-1 Angular Quantities Angular displacement:
The average angular velocity is defined as the total angular displacement divided by time:
w = Dq/Dt
The instantaneous angular velocity:
(8-2a)
(8-2b)

9. 8-1 Angular Quantities
The angular acceleration is the rate at which the angular velocity changes with time:
The instantaneous acceleration:
(8-3a)
(8-3b)

10. 8-1 Angular Quantities
Every point on a rotating body has the same angular velocity ω but linear velocity v can vary.
They are related:
(8-4)
How fast is a point at the center going?

11. 8-1 Angular Quantities
Therefore, objects farther from the axis of rotation will move faster.

12. 8-1 Angular Quantities
If the angular velocity of a rotating object changes, it has a tangential acceleration:
And, as we learned, each point on the object has a centripetal acceleration:
(8-5)
(8-6)

13. 8-1 Angular Quantities
Here is the relationship between linear and rotational quantities:
Start a clean page and reserve it all for a table:
Memorize this please

14. Conversions and units Use the fishbone method to manage units
Rpm = revolutions per minute
One revolution
= one cycle = 2p radians = 360o
Convert using fishbone:
3300 rpm = ________ rad/s

15. Some rotational kinematics practice
Bike wheel + fan car :
what is a?
What is a?
Centrifuge, same questions
You accelerate your car from 0 to 55mph in 8.0 s. Diameter of tire is 63cm.
What is a of wheel?
What is a of car?

16. 8-1 Angular Quantities, more definitions
The frequency is the number of revolutions per second:
Frequency units is Hertz.
“ “per second”
The period is the time of one revolution:
Why this?
(8-8)

17. 8-2 Constant Angular Acceleration
The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.

18. Practice kinematics The centrifuge reaches 3300 rpm in 3.0 seconds.
What is final angular speed?
What is acceleration?
What is q for centrifuge as it is runs up to full speed?

19. How a car engine works

20. 8-3 Rolling Motion (Without Slipping)
In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at rest, and the center moves with velocity v.
In (b) the same wheel is seen from a reference frame where C is at rest. Now point P is moving with velocity –v.
The linear speed of the wheel is related to its angular speed:

21. Practice rolling
Vladimir lives 14 km from the Kremlin. His tires have a radius of 0.45m. The time required is 10.5 minutes.
What is average speed?
What is the average angular speed of his tires?
How many revolutions did his tires require?
At one point, his car stops quickly, from 25m/s to 0 in 35 turns of the wheel. What is a?

22. Force  torque Push a cart in a straight direction with FORCE
Push a wheel in a rotational direction with TORQUE

23. 8-4 Torque
To make an object start rotating, a force is needed; the position and direction of the force matter as well.
The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.

24. 8-4 Torque A longer lever arm is very helpful in rotating objects.
Why aren’t door handles in the middle of the door?

25. 8-4 Torque
Here, the lever arm for FA is the distance from the knob to the hinge; the lever arm for FD is zero; and the lever arm for FC is as shown.
Only the component that is perpendicular to the rotation matters….
Which of these 3 forces can open the door?
How much of Fc is effective at opening the door?

26. Torque is defined: = F r 8-4 Torque
If you look at the geometry, this is equivalent to :

27. Practice torque
2-3 objects with force on them, find net torque

28. Solving torque problems
Draw the body in isolation, the FBD
Draw all forces on the body
Find components that are perpendicular to radius
Identify the axis of rotation and decide what direction of rotation is positive
Find radius to the axis for each force
Calculate each torque and add them up
(note that frictional torque is always opposite to the rotation direction)
Look back at the steps to solve a force problem……

29. Demo of wheel and fan How fast is the wheel turning?
How do we describe it’s acceleration?
How to quantify the force of the fan to cause rotation?
How much energy in the turning?
When I drop a weight on the wheel, how do we account for the collision?

30. Practice with a moving cart and a rolling thing
I form teams. Take data.
Take data, 10 min max
Calculate for 10 min.
A. push steady speed at ~1 m/s for 162cm
B. Accelerate up to ~1.m/s in 162cm.
For each case, calculate a, vf, wf, a, q for the rotating thing

31. Mass  rotational inertia
There is only one way to push a cart and change its motion
There are many ways to push a ball and change its motion
Slide with no rolling
Roll
Spin on a string

32. 8-5 Rotational Dynamics; Torque and Rotational Inertia St = Ia
Knowing that , show
(8-11)
This is for a single point mass; what about an extended object?
As the angular acceleration is the same for the whole object, we can write:
(8-12)
So…… St = Ia

33. 8-5 Rotational Dynamics; Torque and Rotational Inertia
The quantity is called the rotational inertia of an object.
The distribution of mass matters here – these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.

34. Finding d
For any given body, we can find I by adding up (mr2) for every bit of mass.
For simple bodies, this is possible
Do a dumbbell
For extended or distributed mass, this would be tedious or require calculus.
So, let someone else calculate them….

35. 8-5 Rotational Dynamics; Torque and Rotational Inertia
The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.
Memorize a, c, and e

36. Vernier rotary motion device
Sketch in your notebook, side view
Where is force applied?
Measure and record everything that might be quantified by the physics you know.
Observe motion of weight drop and falling off
Describe each stage
Hold onto this, we have a minilab coming

37. Check it… I give you 1.0 kg of clay. What shape and how rotate to give
Highest I
Lowest I

38. 8-6 Solving Problems in Rotational Dynamics
Draw a diagram.
Decide what the system comprises.
Draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act.
Find the axis of rotation; calculate all the torques around it.

39. 8-6 Solving Problems in Rotational Dynamics
5. Apply Newton’s second law for rotation. If the rotational inertia is not provided, you need to find it before proceeding with this step.
6. Apply Newton’s second law for translation and other laws and principles as needed.
7. Solve.
8. Check your answer for units and correct order of magnitude.

40. Practice rotational dynamics
Problem 8-35

41. Minilab rotational dynamics
The large bike wheel with a fan and balance weight.
Measure F of fan, measure r of fan
Measure t and q
Calculate I
Repeat for case with two lead weights added. Measure the weights and their I.
Capture data in class, write up as one page handwritten summary with a sketch and plenty of detail
for next Tuesday.

42. Practice calculating I
Build two weighted meter sticks, each with same mass total taped on
Everyone turn both from center
Calculate the I of each for rotation about the center, ignore the stick
Then calculate I and include the stick
Calculate I of each for rotation about one end

43. Check it…
A merry go round is pushed from 0 to 10 rpm in 30s. It has m=400kg, including children. Model as a ring.
r1 = 1.0m, r2 = 2.5m
What is a?
What is tnet needed to do this?
Answer under here
1/30 rad/s2
48 N m

44. Practice rolling Roll ball down table R = 10.cm t = 3.58s L = 1.62m
Calculate a = .25
Vf = .91
a 2.5
wf 9.1 q
rotations

45. Minilab: rotary motion Vernier device
Using physics, relate forces to find equation for I
Assuming no friction
Including frictional torque
Calculate applied torque from falling mass
Using Loggerpro data collected.
Find aacc and afriction
Calculate I using dynamics equation
Calculate I using physical measurements of device
Ignore the connecting rod
Include connecting rod

46. Now link linear and rotational dynamics?
Ex bucket in a well

47. Linear kinetic energy  angular kinetic energy
Push a cart and it has energy
Rotate a ball and it has energy
Rotate around center axis
Rotate around other axis (on a string)

48. 2 apr: 8-7 Rotational Kinetic Energy
The kinetic energy of a rotating object is given by
By substituting the rotational quantities, we find that the rotational kinetic energy can be written:
A object that has both translational and rotational motion also has both translational and rotational kinetic energy:
(8-15)
(8-16)

49. 8-7 Rotational Kinetic Energy
When using conservation of energy, both rotational and translational kinetic energy must be taken into account.
All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational
inertia they have.

50. Solve the race of the shapes
Example 8-15

51. 8-7 Rotational energy: work
The torque does work as it moves the wheel through an angle θ:
(8-17)

52. practice Fan turns the bike wheel 2 revolutions
How much work did fan do?
What was final angular speed of the wheel?
What average power was used?

53. Measure rotational work
Merry go round, mass m and radius R, rotational inertia of I, time t
Boy pushes with force F at edge
Spins from 0 to wf
What work does he do?
How many turns does he make?
How long does this take

54. Try this….
Measure Watt-hrs on centrifuge at full speed for a few minutes.
Calculate frictional torque.
W = t q note the units!

55. Update your table of analogy between linear and angular (translation vs. rotation)
Review rotational kinematics
Review rotational dynamics
Review rotational energy
Introduce angular momentum
Note “rotational” = “angular”

56. Remember: what is momentum?
A quantity (mv) that happens to be conserved in an isolated system of bodies.
the momentum is conserved for the system even if they are colliding and losing energy.
So it’s useful especially when there are collisions
Example: m1v1 + m2v2 = m1v1’ + m2v2’

57. Linear momentum vs. angular momentum
Push a cart and it has linear momentum
Does it require force to stop it?
Dp = FDt
Spin a bowling ball and it has angular momentum
DL = ?

58. An angular collision
With the Vernier device show an angular collision with data.

59. April 2: 8-8 Angular Momentum and Its Conservation
Remember linear momentum: p = mv
In analogy with linear momentum, we can define angular momentum L:
and DL = tDt
And the Law: If the net torque on an object is zero, the total angular momentum is constant. DL = 0 for an isolated system.
(8-18)

60. 8-8 Angular Momentum and Its Conservation
Therefore, systems that can change their rotational inertia through internal forces will also change their rate of rotation:

61. Solving angular momentum problems
Decide it is a momentum problem
Verify no outside torques, L is conserved
Find how I is changing
Adding a mass (collision) with it’s own I
Any time moving masses touch, it’s a collision
Changing the I of the original body by moving mass
Calculate all the known w and I values
Set Linitial = Lfinal and solve

62. Check understanding for L = Iw
1. spinning wheel I = 1.5 kg m2
Initial speed = 3.0rad/s
Drop mass 2.0 kg at r = 0.2m
What is new speed?
2. figure skater.
Model him with reasonable numbers
50kg skater….total extension I = 7.8kg m2
Total pull in I = 1.0 kg m2
Find speed change with hands coming in

63. L is conserved even if Work is done on the body
But only if the force doing the work gives no torque.
Girl climbs to center of a spinning merry go round
Radially pull the string on a spinning mass
Skater pulls in her arms ?

64. HW and micro lab
Suspend two bowling balls from a long pipe as a barbell. (what does a bigger I look like)
Use a 1 N spring scale to add a force at a given radius.
Measure time and angle
Calculate and show me in a HW
Model I of the system
Torque
Alpha
Calculate I using Newton’s second law

65. When to use which angular law (dynamics, energy, momentum)
Law use when………
St = Ia rigid body (I = constant)
DMEsystem = 0 single body, rigid, no outside force
DLsystem = 0 isolated system, no outside torque

66. 8-9 Vector Nature of Angular Quantities
The angular velocity vector points along the axis of rotation; its direction is found using a right hand rule:

67. 8-9 Vector Nature of Angular Quantities
Angular acceleration and angular momentum vectors also point along the axis of rotation.
Look at turntable and at gyroscope

68. Bowling ball in the PAC due Wed 11 April as HW
If I roll a bowling ball down the ramp in the PAC
How fast will it be at the end? v = (10/7 gh)1/2
What acceleration will it experience? A=5/7 gh/L
Predict these with energy or dynamics or momentum
Measure h and L and M and R
Show equations and predicted answers
Then, as a class, we’ll measure v to verify

69. Summary of Chapter 8
Angles are measured in radians; a whole circle is 2π radians.
Angular velocity is the rate of change of angular position.
Angular acceleration is the rate of change of angular velocity.
The angular velocity and acceleration can be related to the linear velocity and acceleration.
The frequency is the number of full revolutions per second; the period is the inverse of the frequency.

70. Summary of Chapter 8, cont.
The equations for rotational motion with constant angular acceleration have the same form as those for linear motion with constant acceleration.
Torque is the product of force and lever arm.
The rotational inertia depends not only on the mass of an object but also on the way its mass is distributed around the axis of rotation.
The angular acceleration is proportional to the torque and inversely proportional to the rotational inertia.

71. Summary of Chapter 8, cont.
An object that is rotating has rotational kinetic energy. If it is translating as well, the translational kinetic energy must be added to the rotational to find the total kinetic energy.
Angular momentum is
If the net torque on an object is zero, its angular momentum does not change.

72. A conversion puzzle
There is a conversion that applies only to planet earth, and is unique to earth or other planets exactly the size of earth.
feet/second
What does this constant refer to?
Prize for the correct answer.

73. You must know this without looking it up
Distance around a circular path l = qr
Circumference of a circular path = 2pr
= Dq/Dt = v/r (always do in radians)
Works for v of point on circle or v of rolling ball
rpm = rev/min is one unit of frequency
f = 1/T where T is the period
There are always 2p rad/rev
Acceleration:
centripital (also called radial) ac=v2/r
linear (tangential) at = Dw/Dt = a r

74. Analogies table Quantity Translational motion Rotational motion Time t
Intertia
Mass m
Rotational inertia I
(I depends on body shape)
Displacement
x or y q
Velocity
v = Dx/Dt
w = Dq/Dt
Acceleration
alinear = Dv/Dt
a is rotational = Dw/Dt
ac is centripital = v2/r
aL is linear= a r
Force
Second Law F
F = ma
= F r
= I a
Kinetic Energy
KE = ½ m v2
KE = ½ I w2
Work
W = FDx
W = tDq
Power
P = F v
P = t w
Momentum
p = mv
L = I w

75. Classics for chapter 8 Centrifuge kinematics Torque on merry-go-round
Shape race down a slope
Loop the loop with sphere vs. block
Bucket down well accelerates
Spin rock over head force on string
Jump on merry-go-round
Figure skater pull in arms

76. Classics for chapter 8, term 2 final
Accelerate a centrifuge from stop to some rpm, know t and r…..find q, find ac, find w, find a
Roll the wheel at constant v of radius r, a distance d in time t; find q, find v, find w

77. Skills for chapter 8 Read a linear graph of w vs t
Find q, w, a for various times
Solve the dynamics problems of minilab type, with both linear and rotational (bucket down well classic)

78. Strategy when starting a problem
If you know the “before” and “after”, use
Energy if I doesn’t change and no collision
Momentum if I changes or a collision
If don’t know “before” and “after”, use
Kinematics
t = Iw

79. Apply dynamics (Newton’s second Law) to rolling ball
Ball: mass M, radius R
Ramp: angle q, length L
Determine acceleration, a, of ball
Determine speed of ball at bottom

80. Practice graph analysis
Assemble simple object and model I value
Rotate simple object on Vernier rotation device.
Rotate slowly so that air resistance is low
Measure q and w vs t on Loggerpro
Determine a
Calculate tfriction

81. Problem for HW
1. merry go round of mass M, radius R, a cylinder is spinning at w.
A kid mass m jumps on in a radial direction. What is new speed?
Same kid now drags her feet with frictional force F on the ground. How many turns until it stops?

82. Problems for HW
2. Loop the loop problem. Derive h as function of R for sphere.
3. skater modeling. Trunk is cylinder of 50kg radius 0.25m. Arms outstretched are each length 1.0m from center mass 5 kg. Initial speed is 4 rad/s.
Arms pulled in, what is new speed?
Calculate energy initial and final.

83. Exercise with PHET physlet
Use to do the following experiments:
Show that Et = Ia for two applied forces and two braking forces
Show that you can model I for a rotating hollow cylinder and this matches the I from the physlet
Show that angular momentum is conserved for a given rotation as you vary the I by adjusting the inner radius. You should calculate L for each case.
Turn in a printout of the physlet for each case and a page of your calculations.

84. Minilab 2 See “physics I bike wheel minilab” Handout for lab report
Use I from Minilab 1 and modify
Demo and take data in class as a group
Complete handout for turning in.

85. Diagnostic quiz
Write the distance around an arc as fct of angle and radius:
Write the circumference of a circle as a function of radius:
Is this true: w = r v? If not, what is right?
Convert 1 rpm to w:
What is the period of rotation if f = 1 rpm?

86. Bowling balls and fan torque
Predict I for the setup ignoring the bar
Calculate I from dynamics using force of fan.
Find mass of the bar to correct the prediction 1.69kg
Data april 2015
M N L m
M N L m
Fan M = 142g F= 3.6g and x = 0.87m
T = 26s o
(these data didn’t work: ended up with a negative mass in the bar)

87. Some review for exam 1 Define rotational motion
Rigid Body: define and give example
Give example of rotating object that is not rigid.
Define conditions necessary to apply angular kinematic equations
Define rotational inertia using words

88. Problem fans for review
4 identical fans each with force F are spaced evenly and all pointing in same direction along massless rod of length 3L
The rod rotates around one end. What is the total torque on the rod?
The rod rotates around the middle. What is the total torque on the rod?
Each fan has mass M.
c. What is angular acceleration in each case a and b.

89. Kite string
I have a flying kite of M = 0.25kg tied to a free-spinning spool (model it as a cylinder) of string that has radius r=5.0cm, m=1.5kg. The kite, initially at rest, is pulling with constant force of 30.N on the string.
What will be the acceleration of the kite?
If the spool has a friction applying 0.25 Nm of torque, then what will be the new acceleration?
Do symbolic and then with numbers.

90. Bowling ball rolls down (do this first with symbols only)
You push a 4.0kg (10.cm) bowling ball with v = 3.0m/s and it starts down a ramp of 5o. There is no friction.
How fast will it be going after 10.0 m of the ramp?
How much time will it require to roll that 10.0m?
How many revolutions will it do during the ramp?

91. With a little challenge
Merry go round of mass M and radius R is spinning at wo. Then four friends of mass m each jump on radially.
What is new w1?
Then they crawl to the middle and hug.
What is the new w2. will find is wo
How much energy did they spend crawling to the middle? Will find is KEo

92. Practice #1 for chapter 8
A bowling ball, 10kg swings on a 2.0 m rope from the horizontal position and elastically hits another ball, 15kg, at the bottom.
How fast was the 1st ball going just before it hit the 2nd ball?
How fast was the 1st ball going just after it hit?
How fast was the 2nd ball going after the collision?
How much energy was lost in the collision?

93. Practice problem #2
I spin a ball, mass 30.g, on a 1.0m string over my head at 2.0 rev per second. I let go and it elastically strikes the Sun, mass 10.g.
What is w?
What is vball before it hits the Sun?
What is KE before?
What is v’ball after collision?
What is KEball after?
What fraction of ME is lost to heat in this collision?

94. Practice #3
Your car goes from zero to 55mph in 7.0s. Your tire radius is 35cm. Find each….. vf a wf
Linear velocity of top of wheel

95. Atwood Machine demo -- 1
An in-class demo and calculation for physics I, although Atwood is not a classic for them.
50g and 20g weight, 0.34g string, and 7.75g pully with R=24mm.
Question: using measurement of a, determine a measured value for I of the pully
So we know R = 24mm, m = 20.0g, M=50.0g, and a = 149 rad/s2

96. Solving Atwood Machine -- 2
Now solve for I = t/a.
St = T2R-T1R = Ia where from FBD, T2 = M(g-a) and T1 = m(g+a)
Also know that a = aR
So, have 4 equations and 4 unknows (T1, T2, a, and I)
Can solve for I to get I = R/a ((M(g-aR) – m(g+aR))
Which yields Ipully = 6.1x10-6 kg m2 which is very low indeed.

97. Possible errors? -- 3 Is it reasonable?
If we model the pully as a cylinder, we get Imodel = 2.2x10-6 kg m2 so correct for order of magnitude.
Could the shaft and attaching screw of the meter raise the I in a model?
Measure the mass of screw at 2.3g, and r = 4mm, so assume the shaft is same radius and 4 times the mass, gives Ishaft+screw of 0.11x10-6 kg m2, so this is not important.
Could the mass of string affect the I calculation?
Measure string mass at 0.34g, so compared to the accelerating mass of 70g, this is also not important.

98. -- 4
Atwood Machine data