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# Rotational Conservation

Rotational Conservation. Angular Momentum Conserved. With no net external torque, angular momentum is constant. The angular momentum of an isolated system is conserved. Conservation. With no external torque, angular momentum is constant. dL / dt = 0 , L = constant. 4w. w. m. r /2. r.

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## Rotational Conservation

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1. Rotational Conservation

2. Angular Momentum Conserved • With no net external torque, angular momentum is constant. • The angular momentum of an isolated system is conserved

3. Conservation • With no external torque, angular momentum is constant. • dL/dt= 0, L = constant 4w w m r/2 r I = mr2 I = mr2/4

4. A child of 180 N sits at the edge of a merry-go-round with radius 2.0 m and mass 160 kg. The ride is spinning with a period of 15 s. If the child moves to the center, what is the new period of rotation? The moments of inertia for the disk and combination were found before. Id = 320 kg m2 I = Id + Ic =390 kg m2 The angular momentum comes from the period. L = Iw = I(2p/T) Since L is conserved we can find the final period. I(2p/T) = Id(2p/Tf) Tf= Id T / I = 12 s Faster Ride m M r

5. A system may have more than one rotating axis. The total angular momentum is the sum of separate vectors. Ltotal = Ls + Lw = Lw Internal Angular Momentum Lw w Ls = 0

6. Internal torques cancel out. Conservation requires that the sum stay constant. Ltotal = Ls + (-Lw) = Lw Ls = 2Lw Internal Movement Ls = 2 Lw -w -Lw

7. Larger System • A child jumping on a merry go round adds angular momentum. • Initial momentum in a straight line • Not at axis – contributes angular momentum L+rpsinq L w w p

8. Tops use torque. Gravity supplies the torque. The lever arm is the axis of rotation. Gravity is directed down. The torque is at right angles to the lever arm and horizontal. The top will precess in a circle. Gravitational Torque L t w r mg

9. A gyroscope acts like a top, and precesses if its axis is at an angle. If the gyroscope axis is vertical the torque from gravity is zero. If the base moves, the gyroscope stays vertical. Gyroscope L w t = 0 mg r

10. Boomerang • Boomerangs move due to gravitational torque. • Aerodynamic lift is the force • The lever arm is the length of each arm F t r L w next

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