1. Chapter 10
Rotational Motion

2. Rigid Object A rigid object is one that is nondeformable
The relative locations of all particles making up the object remain constant
All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible

3. Angular Position Axis of rotation is the center of the disc
Choose a fixed reference line
Point P is at a fixed distance r from the origin

4. Angular Position, 2
Point P will rotate about the origin in a circle of radius r
Every particle on the disc undergoes circular motion about the origin, O
Polar coordinates are convenient to use to represent the position of P (or any other point)
P is located at (r, q) where r is the distance from the origin to P and q is the measured counterclockwise from the reference line

5. Angular Position, 3
As the particle moves, the only coordinate that changes is q
As the particle moves through q, it moves though an arc length s.
The arc length and r are related:
s = q r

6. Radian This can also be expressed as
q is a pure number, but commonly is given the artificial unit, radian
One radian is the angle subtended by an arc length equal to the radius of the arc

7. Conversions Comparing degrees and radians
Converting from degrees to radians

8. Angular Position, final
We can associate the angle q with the entire rigid object as well as with an individual particle
Remember every particle on the object rotates through the same angle
The angular position of the rigid object is the angle q between the reference line on the object and the fixed reference line in space
The fixed reference line in space is often the x-axis

9. Angular Displacement
The angular displacement is defined as the angle the object rotates through during some time interval
This is the angle that the reference line of length r sweeps out

10. Average Angular Speed
The average angular speed, , of a rotating rigid object is the ratio of the angular displacement to the time interval

11. Instantaneous Angular Speed
The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

12. Angular Speed, final Units of angular speed are radians/sec
rad/s or s-1 since radians have no dimensions
Angular speed will be positive if  is increasing (counterclockwise)
Angular speed will be negative if  is decreasing (clockwise)

13. Average Angular Acceleration
The average angular acceleration, a,
of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

14. Instantaneous Angular Acceleration
The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0
Units of angular acceleration are rad/s2 or s-2 since radians have no dimensions

15. Angular Motion, General Notes
When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration
So q, w, a all characterize the motion of the entire rigid object as well as the individual particles in the object

16. Directions, details
Strictly speaking, the speed and acceleration (w, a) are the magnitudes of the velocity and acceleration vectors
The directions are actually given by the right-hand rule

17. Directions, final The direction of is along the axis of rotation
By convention, its direction is out of the plane of the diagram when the rotation is counterclockwise
its direction is into of the plane of the diagram when the rotation is clockwise
The direction of is the same as if the angular speed is increasing and antiparallel if the speed is decreasing

18. Rotational Kinematics
Under constant angular acceleration, we can describe the motion of the rigid object using a set of kinematic equations
These are similar to the kinematic equations for linear motion
The rotational equations have the same mathematical form as the linear equations

19. Rotational Kinematic Equations

20. Comparison Between Rotational and Linear Equations

21. Relationship Between Angular and Linear Quantities
Displacements
Speeds
Accelerations
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion

22. Speed Comparison
The linear velocity is always tangent to the circular path
called the tangential velocity
The magnitude is defined by the tangential speed

23. Acceleration Comparison
The tangential acceleration is the derivative of the tangential velocity

24. Speed and Acceleration Note
All points on the rigid object will have the same angular speed, but not the same tangential speed
All points on the rigid object will have the same angular acceleration, but not the same tangential acceleration
The tangential quantities depend on r, and r is not the same for all points on the object

25. Centripetal Acceleration
An object traveling in a circle, even though it moves with a constant speed, will have an acceleration
Therefore, each point on a rotating rigid object will experience a centripetal acceleration

26. Resultant Acceleration
The tangential component of the acceleration is due to changing speed
The centripetal component of the acceleration is due to changing direction
Total acceleration can be found from these components

27. Rotational Kinetic Energy
An object rotating about some axis with an angular speed, , has rotational kinetic energy even though it may not have any translational kinetic energy
Each particle has a kinetic energy of
Ki = 1/2 mivi2
Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute vi = wi r

28. Rotational Kinetic Energy, cont
The total rotational kinetic energy of the rigid object is the sum of the energies of all its particles
Where I is called the moment of inertia

29. Rotational Kinetic Energy, final
There is an analogy between the kinetic energies associated with linear motion (K = 1/2 mv 2) and the kinetic energy associated with rotational motion (KR= 1/2 Iw2)
Rotational kinetic energy is not a new type of energy, the form is different because it is applied to a rotating object
The units of rotational kinetic energy are Joules (J)

30. Moment of Inertia The definition of moment of inertia is
The dimensions of moment of inertia are ML2 and its SI units are kg.m2
We can calculate the moment of inertia of an object more easily by assuming it is divided into many small volume elements, each of mass Dmi

31. Moment of Inertia, cont
We can rewrite the expression for I in terms of Dm
With the small volume segment assumption,
If r is constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known

32. Moment of Inertia of a Uniform Solid Cylinder
Divide the cylinder into concentric shells with radius r, thickness dr and length L
Then for I

33. Moments of Inertia of Various Rigid Objects

34. Torque
Torque, t, is the tendency of a force to rotate an object about some axis
Torque is a vector
t = r F sin f = F d
F is the force
f is the angle the force makes with the horizontal
d is the moment arm (or lever arm)

35. Torque, cont
The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force
d = r sin 

36. Torque, final
The horizontal component of the force (F cos f) has no tendency to produce a rotation
Torque will have direction
If the turning tendency of the force is counterclockwise, the torque will be positive
If the turning tendency is clockwise, the torque will be negative

37. Net Torque
The force F1 will tend to cause a counterclockwise rotation about O
The force F2 will tend to cause a clockwise rotation about O
tnet = t1 + t2 = F1d1 – F2d2

38. Torque vs. Force Forces can cause a change in linear motion
Described by Newton’s Second Law
Forces can cause a change in rotational motion
The effectiveness of this change depends on the force and the moment arm
The change in rotational motion depends on the torque

39. Torque Units The SI units of torque are N.m
Although torque is a force multiplied by a distance, it is very different from work and energy
The units for torque are reported in N.m and not changed to Joules

40. Torque as a Vector Product
Torque is the vector product or cross product of two other vectors

41. Vector Product, General
Given any two vectors, and
The vector product
is defined as a third vector, whose magnitude is
The direction of C is given by the right-hand rule

42. Properties of Vector Product
The vector product is not commutative
If is parallel (q = 0o or 180o) to then
This means that
If is perpendicular to then

43. Vector Products of Unit Vectors
The signs are interchangeable
For example,

44. Problem Solving Strategy – Rigid Object in Equilibrium
Conceptualize
Identify the forces acting on the object
Think about the effect of each force on the rotation of the object, if the force was acting by itself
Categorize
Confirm the object is a rigid object in equilibrium

45. Problem Solving Strategy – Rigid Object in Equilibrium, 2
Analyze
Draw a free body diagram
Label all external forces acting on the object
Resolve all the forces into rectangular components
Apply the first condition of equilibrium

46. Problem Solving Strategy – Rigid Object in Equilibrium, 3
Analyze, cont
Choose a convenient axis for calculating torques
Choose an axis that simplifies your calculations
Apply the second condition of equilibrium
Solve the simultaneous equations
Finalize
Be sure your results are consistent with the free body diagram
Check calculations

47. Torque and Angular Acceleration on a Particle
The magnitude of the torque produced by a force around the center of the circle is
t = Ft r = (mat) r
The tangential acceleration is related to the angular acceleration
St = S(mat) r = S(mra) r = S(mr 2) a
Since mr 2 is the moment of inertia of the particle,
St = Ia
The torque is directly proportional to the angular acceleration and the constant of proportionality is the moment of inertia

48. Work in Rotational Motion
Find the work done by a force on the object as it rotates through an infinitesimal distance ds = r dq
The radial component of the force does no work because it is perpendicular to the displacement

49. Work in Rotational Motion, cont
Work is also related to rotational kinetic energy:
This is the same mathematical form as the work-kinetic energy theorem for translation
If an object is both rotating and translating, W = DK + DKR

50. Power in Rotational Motion
The rate at which work is being done in a time interval dt is the power
This is analogous to P = Fv in a linear system

51. Angular Momentum
The instantaneous angular momentum of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector and its instantaneous linear momentum

52. Torque and Angular Momentum
The torque is related to the angular momentum
Similar to the way force is related to linear momentum
This is the rotational analog of Newton’s Second Law
The torque and angular momentum must be measured about the same origin
This is valid for any origin fixed in an inertial frame

53. More About Angular Momentum
The SI units of angular momentum are (kg.m2)/ s
The axes used to define the torque and the angular momentum must be the same
When several forces are acting on the object, the net torque must be used

54. Angular Momentum of a System of Particles
The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particles
Differentiating with respect to time

55. Angular Momentum of a Rotating Rigid Object, cont
To find the angular momentum of the entire object, add the angular momenta of all the individual particles
This is analogous to the translational momentum of p = m v

56. Summary of Useful Equations

57. Conservation of Angular Momentum
The total angular momentum of a system is conserved if the resultant external torque acting on the system is zero
Net torque = 0 -> means that the system is isolated
For a system of particles, Ltot = SLn = constant

58. Conservation of Angular Momentum, cont
If the mass of an isolated system undergoes redistribution, the moment of inertia changes
The conservation of angular momentum requires a compensating change in the angular velocity
Ii wi = If wf
This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system
The net torque must be zero in any case

59. Conservation Law Summary
For an isolated system -
(1) Conservation of Energy:
Ei = Ef
(2) Conservation of Linear Momentum:
(3) Conservation of Angular Momentum:

60. Gyroscope
Angular momentum is the basis of the operation of a gyroscope
A gyroscope is a spinning object used to control or maintain the orientation in space of the object or a system containing the object
Gyroscopes undergo precessional motion
The symmetry axis rotates, sweeping out a cone

61. Precessional Motion of a Gyroscope
The external forces acting on the top are the normal and the gravitational forces
The torque due to the gravitational force is in the xy plane
Only the direction of the angular momentum changes, causing the precession

62. Rolling Object
The red curve shows the path moved by a point on the rim of the object
This path is called a cycloid
The green line shows the path of the center of mass of the object

63. Pure Rolling Motion
The surfaces must exert friction forces on each other
Otherwise the object would slide rather than roll
In pure rolling motion, an object rolls without slipping
In such a case, there is a simple relationship between its rotational and translational motions

64. Rolling Object, Center of Mass
The velocity of the center of mass is
The acceleration of the center of mass is

65. Rolling Object, Other Points
A point on the rim, P, rotates to various positions such as Q and P ’
At any instant, the point on the rim located at point P is at rest relative to the surface since no slipping occurs

66. Total Kinetic Energy of a Rolling Object
The total kinetic energy of a rolling object is the sum of the translational energy of its center of mass and the rotational kinetic energy about its center of mass
K = 1/2 ICM w2 + 1/2 MvCM2

67. Parallel-Axis Theorem
For an arbitrary axis, the parallel-axis theorem often simplifies calculations
The theorem states Ip = ICM + MD 2
Ip is about any axis parallel to the axis through the center of mass of the object
ICM is about the axis through the center of mass
D is the distance from the center of mass axis to the arbitrary axis

68. Total Kinetic Energy, Example
Accelerated rolling motion is possible only if friction is present between the sphere and the incline
The friction produces the net torque required for rotation

69. Total Kinetic Energy, Example cont
Despite the friction, no loss of mechanical energy occurs because the contact point is at rest relative to the surface at any instant
Let U = 0 at the bottom of the plane
Kf + U f = Ki + Ui
Kf = 1/2 (ICM / R 2) vCM2 + 1/2 MvCM2
Ui = Mgh
Uf = Ki = 0

70. Turning a Spacecraft Here the gyroscope is not rotating
The angular momentum of the spacecraft about its center of mass is zero

71. Turning a Spacecraft, cont
Now assume the gyroscope is set into motion
The angular momentum must remain zero
The spacecraft will turn in the direction opposite to that of the gyroscope