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

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.

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.

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

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

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 240.0 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 

                                                                                                                                                                                           

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 

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)

kinetic energy is

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)

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.

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

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

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

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.

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.

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=

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

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 - 216 N) =22.0 kg