1. Rotational Dynamics The Action of Forces and Torques on Rigid Objects
Chapter 9 Lesson 3
(b) Combined translation and rotation
(a) Translation

2. Newton's Second Law for Rotational Motion About a Fixed Axis
FT = maT
t = FTr
t = maTr
aT = ra
The constant of proportionality is I = mr2, which is called the moment of inertia of the particle. The SI unit for moment of inertia is kg · m2.

4. Requirement: a must be expressed in rad/s2
ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS
                                                                                                                                
   
                     
Requirement: a must be expressed in rad/s2
Although a rigid object possesses a unique total mass, it does not have a unique moment of inertia, for the moment of inertia depends on the location and orientation of the axis relative to the particles that make up the object.

5. Example 8. The Moment of Inertia Depends on Where the Axis Is
Two particles each have a mass m and are fixed to the ends of a thin rigid rod, whose mass can be ignored. 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

6. (a)
(r1=0, r2=L)
                                                                               
(b)

10. Example 9. The Torque of an Electric Saw Motor
The motor in an electric saw brings the circular blade from rest up to the rated angular velocity of 80.0 rev/s in rev. One type of blade has a moment of inertia of 1.41 × 10–2 kg · m2. What net torque (assumed constant) must the motor apply to the blade? q  a w
w0  t 
1508 rad (240.0 rev)  ?
503 rad/s (80.0 rev/s) 
0 rad/s 

12. Analogies Between Rotational and Translational Concepts
 Physical Concept 
 Rotational 
 Translational 
 Displacement 
 q  
 s 
 Velocity 
 w 
 v 
 Acceleration 
 a 
 The cause of acceleration 
 Torque t 
 Force F 
 Inertia 
 Moment of inertia I 
 Mass m 
 Newton’s second law 
 St  =  Ia 
 SF  =  ma 
 Work 
 tq  
 Fs 
 Kinetic energy 
 ½Iw2 
 ½mv2 
 Momentum 
 L  =  Iw 
 p  =  mv 

13. Rotational Work and Energy
DEFINITION OF ROTATIONAL WORK
The rotational work WR done by a constant torque t in turning an object through an angle q  is
Requirement: q  must be expressed in radians.
W = Fs = Frq 
SI Unit of Rotational Work: joule (J)

14. kinetic energy is

15. DEFINITION OF ROTATIONAL KINETIC ENERGY
The rotational kinetic energy KER of a rigid object rotating with an angular speed w about a fixed axis and having a moment of inertia I is
Requirement: w must be expressed in rad/s.
SI Unit of Rotational Kinetic Energy: joule (J)

17. Example 12. 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 . Both cylinders start at the same vertical height h0. All heights are measured relative to an arbitrarily chosen zero level that passes through the center of mass of a cylinder when it is at the bottom of the incline. Ignoring energy losses due to retarding forces, determine which cylinder has the greatest translational speed upon reaching the bottom.

18. h = h0, v0 = 0 m/s, w0 = 0 rad/s
                                                           

19.                                                                                                             
                                  
The solid cylinder, having the greater translational speed, arrives at the bottom first.

20. Angular Momentum
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 I and its angular velocity w with respect to that axis:
Requirement: w must be expressed in rad/s.
SI Unit of Angular Momentum: kg·m2/s

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

22. Conceptual Example 13. A Spinning Skater
An ice skater is spinning with both arms and a leg outstretched. She pulls her arms and leg inward. As a result of this maneuver, her spinning motion changes dramatically. Using the principle of conservation of angular momentum, explain how and why it changes.
Angular momentum is conserved. Thus L does not change. Mr^2 changes in that r decreases, thus I decreases. If I decreases, w must increase to keep L constant.

23. Problem 42
REASONING AND SOLUTION Newton's law applied to the 11.0-kg object gives a a T2 T1
T2 - (11.0 kg)(9.80 m/s2) = (11.0 kg)(4.90 m/s2) or T2 = 162 N
m2=
A similar treatment for the 44.0-kg object yields
T1 - (44.0 kg)(9.80 m/s2) = (44.0 kg)(-4.90 m/s2) or T1 = 216 N
m1=

24. T1r - T2r = I(a) For an axis about the center of the pulley r r
Clockwise rotation T2 T1
T1r - T2r = I(a)

25. M = (-2/a)(T2 - T1) = [-2/(4.90 m/s2)](162 N - 216 N)
Solving for the mass M we obtain
M = (-2/a)(T2 - T1) = [-2/(4.90 m/s2)](162 N N)
=22.0 kg