Page 1 of 5 3/25/2009 © M.Dubson, University of Colorado at Boulder Conservation of Angular Momentum = "Spin" We can assign a direction to the angular velocity: direction of = direction of axis + right hand rule (with right hand, curl fingers in direction of rotation, thumb points in direction of .) Similarly, we can assign a direction to angular acceleration : 2121ddt t t t Direction of is direction of , not the direction of (like direction of a is direction of v, not the direction of v) Can also assign direction to torque with cross product. The cross-product of two vectors A and B is a third vector A B C defined like this: The magnitude is| A B| ABsin . The direction of C A Bis the direction perpendicular to the plane defined by the vectors A and B , plus right-hand-rule. (Curl fingers from first vector A to second vector B, thumb points in direction of AB ˆ ˆ ˆ ˆ ˆi j k i i 0ˆ ˆ ˆ ˆ ˆj k i j j 0 etc. Vector torque defined as ˆrF AB A B spinning, and slowing x y z i ^ j ^ k ^Page 2 of 5 3/25/2009 © M.Dubson, University of Colorado at Boulder The vector torque is defined with respect to an origin (which is usually, but not always, the axis of rotation). So, if you change the origin, you change the torque (since changing the origin changes the position vector r). With these definitions of vector angular acceleration and vector torque, the fixed-axis equation I becomes dIIdt ( like dvF ma mdt) Now, a new concept: angular momentum L = "spin". Angular momentum, a vector, is the rotational analogue of linear momentum. So, based on our analogy between translation and rotation, we expect LI ( like p mv) . Note that this equation implies that the direction of L is the direction of . Definition of angular momentum of a particle with momentum p = mv at position r relative to an origin is L r p Like torque , the angular momentum L is defined w.r.t. an origin, often the axis of rotation. r origin F point of application of force (out of page) = vector out of page = vector into page L r origin p particle L (out of page) Page 3 of 5 3/25/2009 © M.Dubson, University of Colorado at Boulder We now show that the total angular momentum of a object spinning about a fixed axis is LI. Consider a object spinning about an axis pointing along the +z direction. We place the orgin at the axis. tot i i i i i ii i i2i i i i iiitotL L r p r (m v )ˆ ˆ ˆz r m v z m r z ILI If something has a big moment of inertia I and is spinning fast (big ), then it has a big "spin", big angular momentum. Angular momentum is a very useful concept, because angular momentum is conserved. Important fact: the angular momentum of a object spinning about an axis that passes through the center of mass is given by L = Icmindependent of the location of the origin; that is, even if the origin is chosen to be outside the spinning object, the angular momentum has the same value as if the origin was chosen to be at the axis. (Proof not given here). Conservation of Angular Momentum: If a system is isolated from external torques, then its total angular momentum L is constant. ext = 0 Ltot = I = constant ( like Fext = 0 ptot = constant ) Here is a plausibility argument for conservation of angular momentum (proof is a bit too messy): First, we argue that netdLdt ( like netdpFdt ) : net(I ) LI I (assuming I const) =t t t ( This is not a proof, because we assumed that I = constant. The equation netLt turns out to be true even if I constant. ) ri mi vi x y z vi = ri axisPage 4 of 5 3/25/2009 © M.Dubson, University of Colorado at Boulder So now we have, netLt if net = 0 , then L0 L constantt . Done. It turns out that only 4 things are conserved: - Energy - Linear momentum p - Angular momentum L - Charge q Conservation of Angular Momentum is very useful for analyzing the motion of spinning objects isolated from external torques — like a skater or a spinning star. If ext = 0 , then L = I = constant. If I decreases, must increase, to keep L = constant. Example: spinning skater. Example: rotation of collapsing star. A star shines by converting hydrogen (H) into helium (He) in a nuclear reaction. When the H is used up, the nuclear fire stops, and gravity causes the star to collapse inward. As the star collapses (pulls its arms in), the star rotates faster and faster. Star's radius can get much smaller: Ri 1 million miles Rf 30 miles Ii i = If f ( I big, small ) ( I small, big ) nuclear gravity slow fast!!Page 5 of 5 3/25/2009 © M.Dubson, University of Colorado at Boulder 22i i f f52222i i f f5522i i f f2i f i2f i fI I (Sphere I = M R )M R M RRRRT2( using = 2 f )R T T If Ri >> Rf, then Ti >>>Tf . The sun rotates once every 27 days. "Neutron stars" with diameter of about 30 miles typically rotates 100 time per second. Let's review the correspondence between translational and rotational motion Translation Rotation x xvt = t vat = t F = r F M I = m r2 Fnet = M a net = I KEtrans = (1/2)M v2 KErot = (1/2 ) I 2 p = m v L = I Fnet = p / t net = L / t If Fext = 0, ptot = constant If ext = 0, Ltot =
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