1. Lecture 10 Chapter 10 Rotational Energy, Moment of Inertia, and Torque
Demo
SHOW DIFFERENT OBJECTS ROTATING DOWN AN INCLINED PLANE
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2. Chapter 10 Rotation
Angular variables analogous to linear motion for constant
Angle
Angular velocity
Angular acceleration
Kinetic energy including rotation
Moment of Inertia
Parallel axis theorem
Torque
Rolling without Slipping

3. EXAMPLE -ROTATION WITH CONSTANT ANGULAR ACCELERATION
Restrict discussion to a fixed axis of rotation but also applies if the axis is in translation as long as the axis of rotation goes throuigh the center of mass

4. Rotation with constant angular acceleration: Consider some string wound around a cylinder with a weight attached to the string. There is no slippage between string and cylinder. θ v s r mg T ma

5. Red dot indicates a spot on the cylinder that is rotating as I apply a force to the massless string
Front view
Isometric view ω θ v s r

6. Define Angle (radians)θ v s r
Conversion from degrees to radians is radians per degree
or 57.3 degrees per radian
For s=r theta to be valid theta must be in radians.
Pull a string off a spool and show that the length of string pulled out is the arc length thru which a point on the rim moved
Differentiate s=r theta
All points in a rigid body have the same omega and alpha.
at is the tangential componet and is zero unless the point on the rim speeds up or slows down
ar is the radial component or centripetal acceleration component and is zero unless the object is turning.
s and θ are not vectors

7. Define Angular Velocity ω
Take derivative holding r constant
Tangential velocity
Angular velocity:
Pseudo vector
For s=r theta to be valid theta must be in radians.
Pull a string off a spool and show that the length of string pulled out is the arc length thru which a point on the rim moved
Differentiate s=r theta
All points in a rigid body have the same omega and alpha.
at is the tangential componet and is zero unless the point on the rim speeds up or slows down
ar is the radial component or centripetal acceleration component and is zero unless the object is turning.

8. Use Right hand rule to get direction of ωv ω r v
Counterclockwise is + for angular displacement θ
and angular velocity ω.

10. Also called the tangential acceleration
Define Angular Acceleration
Also called the tangential acceleration
is called the angular acceleration
α is in the same or opposite direction as ω
Recall there is also the radial acc.

11. Two Kinds of Accelerationθ s
Tangential acceleration at ar
Radial acceleration ω at ar a
Radial and tangential accelerations are perpendicular to each other

12. Angular Quantities
Conceptual Example : Is the lion faster than the horse?
On a rotating carousel or merry-go-round, one child sits on a horse near the outer edge and another child sits on a lion halfway out from the center. (a) Which child has the greater linear velocity? (b) Which child has the greater angular velocity?
Answer: The horse has a greater linear velocity; the angular velocities are the same.

13. Angular Quantities
Objects farther from the axis of rotation will move faster.
Figure Caption: A wheel rotating uniformly counterclockwise. Two points on the wheel, at distances RA and RB from the center, have the same angular velocity ω because they travel through the same angle θ in the same time interval. But the two points have different linear velocities because they travel different distances in the same time interval. Since RB > RA, then vB > vA (because v = Rω).

14. Angular Quantities
If the angular velocity of a rotating object changes, it has a tangential acceleration:
Even if the angular velocity is constant, each point on the object has a centripetal acceleration:
Figure Caption: On a rotating wheel whose angular speed is increasing, a point P has both tangential and radial (centripetal) components of linear acceleration. (See also Chapter 5.)

15. For constant acceleration
We have an analogous set of formulas for angular variables

16. Example Example 10-5: Given ω as function of time.
A disk of radius R = 3.0 m rotates at an angular velocity ω = ( t) rad/s, where t is in seconds. At the instant t = 2.0 s, determine:
(a) the angular acceleration. Is it constant?
(b) the speed v and the components of the acceleration a of a point on the edge of the disk
© the initial angular velocity
(d) the angular displacement after 2.00 s. Assume initial displacement=0
Solution: a. The angular acceleration is the derivative of the angular velocity: 1.2 rad/s2.
b. V = 12.0 m/s; atan = 3.6 m/s2; aR = 48 m/s2.

17. We call mr2 the moment of inertia I.
How do we define kinetic energy of a rotating body? Consider what is the kinetic energy of a small rigid object moving in a circle? v ar θ
We call mr2 the moment of inertia I. r
Kinetic energy of rotation and rotational inertia
Kinetic energy of rotation
Suppose we have 2 bodies? How do we define it?

18. Calculation of Moment of Inertia
Figure 10.19
Calculation of Moment of Inertia
Example 10-8
Two Cases

19. Suppose our rotating body is a rigid rodv s θ
I is called the moment of inertia
about an axis through the end of the rod.

20. Evaluation of the rotational inertia
for a rod rotating about an axis through
the end perpendicular to the length L . θ

21. Now consider rod rotating about an axis through the center of mass of the rodθ L v .

22. Parallel Axis Theorem . Notice that the difference General relation vθ L v . r
com
General relation

23. Determining Moments of Inertia
Example : Parallel axis.
Determine the moment of inertia of a solid cylinder of radius R0 and mass M about an axis tangent to its edge and parallel to its symmetry axis.
Figure
Solution: ICM = ½ MR02, so I = 3/2 MR02.

24. Perpendicular-axis Theorem
The perpendicular-axis theorem is valid only for flat objects.
Figure Caption: The perpendicular-axis theorem.

26. Torque: What is it?
A longer lever arm is very helpful in rotating objects.
Figure Caption: (a) A tire iron too can have a long lever arm. (b) A plumber can exert greater torque using a wrench with a long lever arm.

27. Torque. It is similar to force but it also depends on the axis of rotation. Why do we have to define torque?
Use towel to open stuck cap on jar
Door knob far from hinge
Screw driver with large fat handle
Lug wrench to unscrew nuts on rim for tires

28. Consider Torque τ for Opening a Door
the lever arm for FA is the distance from the knob to the hinge;
the lever arm for FD is zero;
the lever arm for FC is as shown.
Lever Arm
Figure Caption: (a) Forces acting at different angles at the doorknob. (b) The lever arm is defined as the perpendicular distance from the axis of rotation (the hinge) to the line of action of the force.

29. Torque
Figure Caption: Torque = R┴ F = RF┴.

30. Newton’s 2nd law for Rotation τ=Iα
Suppose I have small mass at the end of a massless rod

31. Newton’s 2nd law for rotation
Suppose the rod does have mass and we have the same
axis of rotation, then you simply add the moment of inertia

32. Work Energy Theorem for Rotations

33. Figure 10.21
Do Example 10-9

34. Figure 10.22
Do Example 10-10

35. What is the acceleration of the mass m
What is the acceleration of the mass m? Now we can take into account the rotation of the pulley? v +x -y M r θ ma T mg

36. Frictionless Sideways Atwood machine with a pulley with mass+x -y
Now take into account the rotation of the pulley.
new equation T1 Mp T2 a

37. Now include friction between block M and surface+x -y
new equation
Ia = (T2-T1) R T1 Mp T2 a

38. 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 or kinetic energy they have.
Figure

39. Translational and Rotational Kinetic Energyh V
V = 0
Pure translation,
K =1/2 MV at the bottom of the plane
Translation and rolling down inclined plane.
at the bottom of the plane

40. First we have to ask what is rolling without slipping?
Linear speed of the center of mass of wheel is ds/dt
without slipping
The angular speed ω about the com is dθ/dt.
From s=θR we get ds/dt = dθ/dt R or vcom = ω R for smooth rolling motion

41. Rolling can be considered rotating about an axis through the
com while the center of mass moves.
At the bottom P is instantaneously at rest. The wheel also
moves slower at the bottom because pure rotation motion and
pure translation partially cancel out
See photo in Fig Also tire tracks are clear in the
snow and are not smudged.

42. What is the acceleration of:
a) An object in free fall?
b) A block sliding down a frictionless inclined plane?
c) A block with with friction?
d) A sphere rolling down an inclined plane?
e) The tip of a tall pole when it hits the ground

43. Figure 10.36

44. What is the acceleration of a sphere smoothly rolling down an inclined plane?
Positive x

45. What is the acceleration of a sphere smoothly rolling down an inclined plane?
Find torque about the com
τnet= Iα
Positive x
Solve for fs

46. This will predict which objects will roll down the inclined faster.
Solve for
This will predict which objects will roll down the inclined faster.

47. Let θ= 30 deg
Sin 30 = 0.5
shape
Icom
1+ Icom/MR2
acom
sphere
2/5 MR2
1.4
0.71x g/2
disk
1/2MR2
1.5
0.67 x g/2
pipe
MR2
2.0
0.50 x g/2

48. Yo-yo rolls down the string as if it were inclined plane at 90 degrees
Instead of friction, tension in the string holds it back
The moment of inertia Icom is that of the
yo-yo itself.

49. Figure 11.50

50. Rules τ=Iα is useful to solve problems. Valid when:
When the axis of rotation is fixed in an inertial reference frame
When the axis of rotation passes through the center of mass and is fixed in direction even if it is accelerating

51. Other topics When does a bowling ball stop sliding and
roll without slipping
Why does a rolling sphere slow down?