Rotational Conservation

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

Conservation  With no external torque, angular momentum is constant. dL / dt = 0, L = constantdL / dt = 0, L = constant  r  I = mr 2 m r/2 I = mr 2 /4

Faster Ride  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. I d = 320 kg m 2 I = I d + I c = 390 kg m 2  The angular momentum comes from the period. L = I  = I(2  /T)  Since L is conserved we can find the final period. I(2  /T) = I d (2  /T f ) T f = I d T / I = 12 s m M r

Internal Angular Momentum  A system may have more than one rotating axis.  The total angular momentum is the sum of separate vectors. L total = L s + L w = L w  LwLw L s = 0

Internal Movement  Internal torques cancel out.  Conservation requires that the sum stay constant. L total = L s + (- L w ) = L w L s = 2 L w  -L w L s = 2 L w

Larger System  A child jumping on a merry go round adds angular momentum. Initial momentum in a straight lineInitial momentum in a straight line Not at axis – contributes angular momentumNot at axis – contributes angular momentum  L  L+rpsin  p

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

Gyroscope  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.  mg L  r

Boomerang  Boomerangs move due to gravitational torque. Aerodynamic lift is the forceAerodynamic lift is the force The lever arm is the length of each armThe lever arm is the length of each arm next  L r  F