1. Rotational Kinematics
Chapter 8
Rotational Kinematics

2. 8.1 Rotational Motion and Angular Displacement
The angle through which the
object rotates is called the
angular displacement.

3. 8.1 Rotational Motion and Angular Displacement
DEFINITION OF ANGULAR DISPLACEMENT
When a rigid body rotates about a fixed axis, the angular
displacement is the angle swept out by a line passing through
any point on the body and intersecting the axis of
rotation perpendicularly.
By convention, the angular displacement
is positive if it is counterclockwise and
negative if it is clockwise.
SI Unit of Angular Displacement: radian (rad)

4. 8.1 Rotational Motion and Angular Displacement
For a full revolution:

5. 8.2 Angular Velocity and Angular Acceleration

6. 8.2 Angular Velocity and Angular Acceleration
Changing angular velocity means that an angular
acceleration is occurring.
DEFINITION OF ANGULAR ACCELERATION
SI Unit of Angular acceleration: radian per second squared (rad/s2)

7. 8.2 Angular Velocity and Angular Acceleration
Example 4 A Jet Revving Its Engines
As seen from the front of the
engine, the fan blades are
rotating with an angular
speed of -110 rad/s. As the
plane takes off, the angular
velocity of the blades reaches
-330 rad/s in a time of 14 s.
Find the angular acceleration, assuming it to
be constant.

8. 8.2 Angular Velocity and Angular Acceleration

9. 8.3 The Equations of Rotational Kinematics
Recall the equations of kinematics for constant acceleration.
Five kinematic variables:
1. displacement, x
2. acceleration (constant), a
3. final velocity (at time t), v
4. initial velocity, vo
5. elapsed time, t

10. 8.3 The Equations of Rotational Kinematics
The equations of rotational kinematics for constant
angular acceleration:
ANGULAR ACCELERATION
ANGULAR VELOCITY
TIME
ANGULAR DISPLACEMENT

11. 8.3 The Equations of Rotational Kinematics

12. 8.4 Angular Variables and Tangential Variables

13. 8.4 Angular Variables and Tangential Variables

14. 8.4 Angular Variables and Tangential Variables

15. 8.4 Angular Variables and Tangential Variables
Example 6 A Helicopter Blade
A helicopter blade has an angular speed of 6.50 rev/s and an
angular acceleration of 1.30 rev/s2.
For point 1 on the blade, find
the magnitude of (a) the
tangential speed and (b) the
tangential acceleration.

16. 8.4 Angular Variables and Tangential Variables1

17. 8.4 Angular Variables and Tangential Variables1

18. 8.4 Angular Variables and Tangential Variables1 2

19. 8.5 Centripetal Acceleration and Tangential Acceleration
always

20. 8.5 Centripetal Acceleration and Tangential Acceleration
Example 7 A Discus Thrower
Starting from rest, the thrower
accelerates the discus to a final
angular speed of rad/s in
a time of s before releasing it.
During the acceleration, the discus
moves in a circular arc of radius
0.810 m. The angular acceleration is constant.
Find the magnitude of the total
acceleration.

21. 8.5 Centripetal Acceleration and Tangential Acceleration

22. 8.5 Centripetal Acceleration and Tangential Acceleration
if α is constant: aT is constant, ac is not!
not constant
constant

23. 8.5 Centripetal Acceleration and Tangential Acceleration
if α is constant: aT is constant, ac is not!
not constant
constant

24. 8.5 Centripetal Acceleration and Tangential Acceleration
if α is constant: aT is constant, ac is not!
not constant
constant
Φ=tan-1(aT/ac)

25. 8.6 Rolling Motion
Translation rotation

26. 8.6 Rolling Motion
Translation rotation
v=va +ωR
v=va -ωR va
v = - ωR
v = +ωR ω R

27. condition to roll without slipping
8.6 Rolling Motion
Translation rotation
v=va +ωR
v=va -ωR va
v = - ωR
v = +ωR ω R
v=va=ωR
if va= ωR vcontact-point=0
condition to roll without slipping
contact point: v=0

28. condition to roll without slipping
8.6 Rolling Motion
Translation rotation
v=va +ωR
v=va -ωR
v = - ωR
v = +ωR vc
v=0
v=va=ωR
if va= ωR vcontact-point=0
condition to roll without slipping

29. 8.6 Rolling Motion conditions for no slipping
speed of the car
= speed of axes of wheels
acceleration of the car
= acc. of axes of wheels

30. Example 8 An Accelerating Car Starting from rest, the car accelerates
8.6 Rolling Motion
Example 8 An Accelerating Car
Starting from rest, the car accelerates
(without slipping) for 20.0 s with a
constant linear
acceleration of m/s2. The
radius of the tires is m.
What is the angle through which
each wheel has rotated?

31. 8.6 Rolling Motion θ α ω ωo t ?
-2.42 rad/s2
0 rad/s
20.0 s

32. 9.1 The Action of Forces and Torques on Rigid Objects
According to Newton’s second law, a net force causes an
object to have an acceleration.
What causes an object to have an angular acceleration?
TORQUE

33. 9.1 The Action of Forces and Torques on Rigid Objects
The amount of torque depends on where and in what direction the
force is applied, as well as the location of the axis of rotation.

34. 9.1 The Action of Forces and Torques on Rigid Objectsθ
DEFINITION OF TORQUE
Magnitude of Torque = (Magnitude of the force) x (Lever arm)
angle between F and d
Direction: The torque is positive when the force tends to produce a
counterclockwise rotation about the axis.
SI Unit of Torque: newton x meter (N·m)

35. 9.1 The Action of Forces and Torques on Rigid Objectsθ
angle between F and d

36. 9.2 Rigid Objects in Equilibrium
If a rigid body is in equilibrium, neither its linear motion nor its
rotational motion changes.

37. 9.2 Rigid Objects in Equilibrium
EQUILIBRIUM OF A RIGID BODY
A rigid body is in equilibrium if it has zero translational
acceleration and zero angular acceleration. In equilibrium,
the sum of the externally applied forces is zero, and the
sum of the externally applied torques is zero.

38. 9.2 Rigid Objects in Equilibrium
Reasoning Strategy
Select the object to which the equations for equilibrium are to be applied.
2. Draw a free-body diagram that shows all of the external forces acting on the
object.
Choose a convenient set of x, y axes and resolve all forces into components
that lie along these axes.
Apply the equations that specify the balance of forces at equilibrium. (Set the
net force in the x and y directions equal to zero.)
Select a convenient axis of rotation. Set the sum of the torques about this
axis equal to zero.
6. Solve the equations for the desired unknown quantities.

39. 9.2 Rigid Objects in Equilibrium
Torque can be + or -
Negative torque:
would make a clockwise rotation
Positive torque:
would make a counterclockwise rotation x x x x x x

40. 9.2 Rigid Objects in Equilibrium
Example 3 A Diving Board
A woman whose weight is 530 N is
poised at the right end of a diving board
with length 3.90 m. The board has
negligible weight and is supported by
a fulcrum 1.40 m away from the left
end.
Find the forces that the bolt and the
fulcrum exert on the board.

41. 9.2 Rigid Objects in Equilibrium

42. 9.2 Rigid Objects in Equilibrium

43. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
Moment of Inertia, I

44. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR
A RIGID BODY ROTATING ABOUT A FIXED AXIS
Requirement: Angular acceleration
must be expressed in radians/s2.

45. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
Example 9 The Moment of Inertial Depends on Where
the Axis Is.
Two particles each have mass and are fixed at the
ends of a thin rigid rod. The length of the rod is L.
Find the moment of inertia when this object
rotates relative to an axis that is
perpendicular to the rod at
(a) one end and (b) the center.

46. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

47. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

48. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

49. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

50. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

51. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

52. 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

53. 9.5 Rotational Work and Energy
DEFINITION OF ROTATIONAL KINETIC ENERGY
The rotational kinetic energy of a rigid rotating object is
Requirement: The angular speed must
be expressed in rad/s.
SI Unit of Rotational Kinetic Energy: joule (J)

54. 9.5 Rotational Work and Energy
Example 13 Rolling Cylinders
A thin-walled hollow cylinder (mass = mh, radius = rh) and
a solid cylinder (mass = ms, radius = rs) start from rest at
the top of an incline.
Determine which cylinder
has the greatest translational
speed upon reaching the
bottom.

55. 9.5 Rotational Work and Energy
ENERGY CONSERVATION

56. 9.5 Rotational Work and Energy
The cylinder with the smaller I/(mr2) ratio
will have a greater final translational speed.

57. DEFINITION OF ANGULAR MOMENTUM
The angular momentum L of a body rotating about a
fixed axis is the product of the body’s moment of
inertia and its angular velocity with respect to that
axis:
Requirement: The angular speed must
be expressed in rad/s.
SI Unit of Angular Momentum: kg·m2/s

58. PRINCIPLE OF CONSERVATION OFANGULAR MOMENTUM
The angular momentum of a system remains constant (is
conserved) if the net external torque acting on the system
is zero.

59. Conceptual Example 14 A Spinning Skater
9.6 Angular Momentum
Conceptual Example 14 A Spinning Skater
An ice skater is spinning with both
arms and a leg outstretched. She
pulls her arms and leg inward and
her spinning motion changes
dramatically.
Use the principle of conservation
of angular momentum to explain
how and why her spinning motion
changes.