Chapter 8 Lecture 2 Rotational Inertia I.Rotational Inertia A.Newton’s 2 nd law: F = ma 1)Rewrite for Rotational Motion  = m  2)But, should we really use m (mass)? 3)In Linear motion, a)Mass = measure of inertia b)Mass = resistance to change in motion 4)In Rotational motion, mass and how far from center (r) determine inertia a)Rotational Inertia = Moment of Inertia = resistance to change in rotational motion = I b)I = mr 2 (Units = kg x m 2 ) for a point mass rotating around center

5)I depends on shape for solid objects 6)I replaces m for rotational motion  = I  is Newton’s 2 nd Law for Rotational Motion 8) B.Example Calculation: Merry-go-round I M = 800 kgm 2, r = 2 m, 40 kg child at edge I T ?  required to cause  = 0.05 rad/s 2 ?

II.Conservation of Angular Momentum A.What happens to  when radius changes? 1)As skater pulls her arms in,  increases 2)Why does this happen? B.Angular Momentum 1)Linear momentum: p = mv 2)Angular momentum = L = I  3)Should Angular Momentum be conserved? C.Conservation of Momentum 1)Linear momentum is conserved: If F = 0,  p = 0 2)Angular momentum is conserved: If  = 0,  L =0 3)I = mr 2 a)Mass (arms) farther from body, larger I 1 b)Mass (arms) closer to body, smaller I 2 c)L 1 = L 2 for conservation of momentum If I large,  is small Rotate slowly If I small,  is large Rotate quickly

4)Example Calculation: I 1 (arms out) = 1.2 kgm 2, I 2 (arms in) = 0.5 kgm 2 w 1 = 1 rev/s w 2 ? 5)Demo: student on a rotating chair 6)Other Examples of Conservation of Angular Momentum Kepler’s 2 nd Law: equal areas in equal times L = mvr L 1 = L 2 mv 1 r 1 = mv 2 r 2

III.Everyday Applications A.Direction of  and L 1)p is a vector with the same direction as v 2)L is a vector with the direction determined by  3)Right-hand rule establishes direction of  and L B.Riding a Bike  is applied to wheel to make it turn 2)L is horizontal for the motion of the turning wheel 3)To tip over the bike, L 1 must be changed by another torque 4)That torque will be gravity working on C.O.M. of the rider/bike a)Axis of rotation is the line on the ground of bike’s path b)When upright the force is in line with the axis, so  = 0 (l = 0) c)Conservation of momentum keeps the bike balanced d)When tilted, L 2 is forward or back along the road axis i.Standing still, bike falls over ii.Moving, L 1 + L 2 =  L  bike changes direction iii.Turn the wheel slightly to compensate and you stay up e)Angular momentum of a moving bike makes it easier to balance

L i = L W L i = L f = L S – L W L S = 2 L W