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Chapter 8 Lecture 2 Rotational Inertia

Chapter 8 Lecture 2 Rotational Inertia. Rotational Inertia Newton’s 2 nd law: F = ma Rewrite for Rotational Motion : t = m a But, should we really use m (mass)? In Linear motion, Mass = measure of inertia Mass = resistance to change in motion

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Chapter 8 Lecture 2 Rotational Inertia

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  1. Chapter 8 Lecture 2 Rotational Inertia • Rotational Inertia • Newton’s 2nd law: F = ma • Rewrite for Rotational Motion: t = ma • But, should we really use m (mass)? • In Linear motion, • Mass = measure of inertia • Mass = resistance to change in motion • In Rotational motion, mass and how far from center (r) determine inertia • Rotational Inertia = Moment of Inertia = resistance to change in rotational motion = I • I = mr2 (Units = kg x m2) for a point mass rotating around center

  2. I depends on shape for solid objects • I replaces m for rotational motion • t = Ia is Newton’s 2nd Law for Rotational Motion 8) • Example Calculation: Merry-go-round IM = 800 kgm2, r = 2 m, 40 kg child at edge IT? t required to cause a = 0.05 rad/s2?

  3. Conservation of Angular Momentum • What happens to w when radius changes? • As skater pulls her arms in, w increases • Why does this happen? • Angular Momentum • Linear momentum: p = mv • Angular momentum = L = Iw • Should Angular Momentum be conserved? • Conservation of Momentum • Linear momentum is conserved: If F = 0, Dp = 0 • Angular momentum is conserved: If t = 0, DL =0 • I = mr2 • Mass (arms) farther from body, larger I1 • Mass (arms) closer to body, smaller I2 • L1 = L2 for conservation of momentum If I large, w is small Rotate slowly If I small, w is large Rotate quickly

  4. Example Calculation: I1 (arms out) = 1.2 kgm2, I2 (arms in) = 0.5 kgm2 w1 = 1 rev/s w2? • Demo: student on a rotating chair • Other Examples of Conservation of Angular Momentum Kepler’s 2nd Law: equal areas in equal times L = mvr L1 = L2 mv1r1 = mv2r2

  5. Everyday Applications • Direction of w and L • p is a vector with the same direction as v • L is a vector with the direction determined by w • Right-hand rule establishes direction of w and L • Riding a Bike • tis applied to wheel to make it turn • L is horizontal for the motion of the turning wheel • To tip over the bike, L1 must be changed by another torque • That torque will be gravity working on C.O.M. of the rider/bike • Axis of rotation is the line on the ground of bike’s path • When upright the force is in line with the axis, so t = 0 (l = 0) • Conservation of momentum keeps the bike balanced • When tilted, L2 is forward or back along the road axis • Standing still, bike falls over • Moving, L1 + L2 = DL  bike changes direction • Turn the wheel slightly to compensate and you stay up • Angular momentum of a moving bike makes it easier to balance

  6. Li = LW Li = Lf = LS – LW LS = 2 LW

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