Chapter 11 General Rotation

Vector cross product b a  where Right-hand rule

Expressed by cross product The torque vector r O P Expressed by cross product Torque vector about point O Example1: A particle is at position , Calculate the torque about origin if . Solution: We use the determinant form

Angular momentum of a particle Angular momentum about point O magnitude Angular momentum theorem Rotational equivalent of Newton’s second law 4

Angular momentum in circle Example2: Determine the angular momentum of a particle in uniform circular motion (m, v, r). Solution: It depends on the choice of point O! First calculate it about the center of circle Agreed with the rigid body case O l r O’ Direction? What about another O’ ? Component along axis OO’ 5

Conservation of angular momentum The angular momentum of a particle remains constant if there is no net torque acting on it. Typical case: Acted by a central force Kepler’s 2nd law: A line from the sun to a planet sweeps out equal area in equal time. Area: 6

Solution: No torque, L about o is conserved Move in a spiral line Example3: A mass m connected by a rope moves on frictionless table circularly with uniform  and r. Then one pulls the rope slowly through the center, determine the work done when r changes to r/2. Solution: No torque, L about o is conserved Work F  r o m 7

Homework A massless spring (l0=0.2m, k=100N/m) connects mass m=1kg to point o on a horizontal frictionless table. Mass m moves with v0=5m/s ⊥ the spring, and the length of spring becomes l=0.5m after rotating 90°, determine the final velocity v’ and θ. 90° 90° 8

Angular quantities for a system Consider a system of particles (rigid body or not) The total angular momentum: The net torque: Angular momentum theorem for a system: (about the same origin O) Valid in a inertial frame or frame of the CM (about the CM of system) 9

For a rigid body rotates about a fixed axis Rigid body & fixed axis For a rigid body rotates about a fixed axis Consider the component along axis Total angular momentum Angular momentum theorem for a rigid body: (rotational theorem) 1) Vector & Component 2) 1-dimensional case 10

Conservation of L for system The total angular momentum of a system remains constant if the net external torque is zero. Example4: A uniform thin rod (m, l) rotates about fixed axis o with  on frictionless horizontal table, and collides elastically with a resting mass m at its end. Determine the final angular velocity. o m, l m . Solution: Elastic collision: 11

Rotating about varying axis Rigid body rotates about a fixed axis For a body rotating about a symmetry axis (symmetry axis, through CM) Angular momentum theorem: where may have different directions! How angular momentum changes? 1) No initial rotation 2) With initial rotation 12

Motion of a rapidly spinning top, or a gyroscope *The spinning top Motion of a rapidly spinning top, or a gyroscope External torques are acted by a pair of forces: We always have L Only change the direction of W N It is called precession Precession of Earth Bullet, bicycle, … 13

Noninertial reference frame Newton’s first law does not hold in such frames S’ S To use Newton’s laws, we have to use a trick Inertial force: a type of fictitious force 14

Dynamics in noninertial frame With considering inertial forces, N-2 is still valid 1) It is not a real force: no object exerts it 2) For rotating frame, also called centrifugal force (opposite to the centripetal force) Weight loss 15

Solution: Centrifugal force → virtual gravity Ring form spaceship Example5: A ring-form spaceship with radius r is rotating to obtain a virtual gravity of g, determine the period. Solution: Centrifugal force → virtual gravity r=50m O 16

If a body is moving relative to a rotating frame *The Coriolis effect If a body is moving relative to a rotating frame There is another inertial force: Coriolis force where is the relative velocity Coriolis effect River, wind, whirlpool Falling objects Foucault pendulum 17

*Foucault pendulum  Earth 1 1 2 2 3 18