Chapt. 10: Angular Momentum Angular momentum conservation And applications 4/27/2017 Phys 201, 2011

Angular Momentum Conservation where and In the absence of external torques Ii ωi = If ωf Total angular momentum is conserved. (demos) 4/27/2017 Phys 201, 2011

Because the torque exerted by the ice is Spinning skater: Because the torque exerted by the ice is small, the angular momentum of the skater is approximately constant. When she reduces her moment of inertia by drawing in her arms, her angular speed increases. 4/27/2017 Phys 201, 2011

A student sitting on a stool that rests on a turntable with frictionless bearings is holding a rapidly spinning bicycle Wheel (a). The rotation axis of the wheel is initially horizontal, and the magnitude of the spin-angular-momentum vector of the spinning wheel is What will happen if the student suddenly tips the axle of the wheel (b) so that after the rotation the spin axis of the wheel is vertical and the wheel is spinning counterclockwise (when viewed from above)? Answer: The turntable, stool, and student will be rotating clockwise with an angular momentum about the vertical axis of the turntable of magnitude 4/27/2017 Phys 201, 2011

Example: Two Disks A disk of mass M and radius R rotates around the z axis with angular velocity i. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity . What is the relation ship between and ? analogous to an inelastic collision: p1 + p2(=0)  p3 z z 4/27/2017 Phys 201, 2011

Example: Two Disks First realize that there are no external torques acting on the two-disk system of combined mass M. Angular momentum will be conserved! Initially, the total angular momentum is due only to the disk on the bottom: z z 2 1 4/27/2017 Phys 201, 2011

Example: Two Disks Since Li = Lf An inelastic collision, since E is not conserved (friction)! z z Li Lf 4/27/2017 Phys 201, 2011

Example: bullet hitting a stick A uniform stick of mass m and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. Knowing that the initial speed of the bullet is v1, and the final speed is v2. What is the angular speed ωf of the stick immediately after the collision? (Ignore gravity) 4/27/2017 Phys 201, 2011

Example: bullet hitting a stick Initial angular momentum: Final angular momentum: where Conservation of angular momentum around pivot axis: 4/27/2017 Phys 201, 2011

Example: throw ball from stool 4/27/2017 Phys 201, 2011

Angular momentum 4/27/2017 Phys 201, 2011

Angular momentum 4/27/2017 Phys 201, 2011

A 25-kg child in a playground runs with an initial speed of 2.5 m/s along a path tangent to the rim of a merry-go-round, whose radius is 2.0 m. The merry-go-round, which is initially at rest, has a moment of inertia of 500 kg · m2 The child then jumps on to it. Find the final angular velocity of the child and the merry-go-round together. 4/27/2017 Phys 201, 2011

Summary of Dynamics Dynamics (cause – effect) Force – Linear acceleration Change in linear momentum, Fnet = dp/dt Torque – angular acceleration Change in angular momentum, τ = dL/dt Conservation of momentum When net external force = zero Conservation of angular momentum When net external torque = zero Conservation of Energy Mechanical energy = kinetic + potential Non-conservative forces e.g. friction Energy lost to the environment, e.g., heat 4/27/2017 Phys 201, 2011

Dynamical Reprise: Linear The change of motion of an object is described by Newton as F=ma where F is the force, m is the mass, and a is the acceleration. For a set of discrete point particles, all forces act on the center of mass The center of mass is a calculable property of the object. If F=0, then there is no change of motion -- if at rest, the object will remain at rest. 4/27/2017 Phys 201, 2011

Dynamical Reprise: Rotation About a fixed rotation axis, you can always write where is the torque, I is the moment of inertia, and is the angular acceleration. For a set of discrete point particles, The parallel axis theorem lets you calculate the moment of inertia about an axis parallel to an axis through the CM if you know ICM : IPARALLEL = ICM + MD2 L D M x CM ICM IPARALLEL 4/27/2017 Phys 201, 2011