Angular Momentum Angular momentum of rigid bodies Newton’s 2nd Law for rotational motion Torques and angular momentum in 3-D Text sections 11.1 - 11.6 Physics 1D03 - Lecture 30

“Angular momentum” is the rotational analogue of linear momentum. Recall linear momentum: for a particle, p = mv . Newton’s 2nd Law: The net external force on a particle is equal to the rate of change of its momentum. To get the corresponding angular relations for a rigid body, replace: m I v w F t p L (“angular momentum”) Physics 1D03 - Lecture 30

Angular momentum of a rotating rigid body: Angular momentum, L, is the product of the moment of inertia and the angular velocity. L = Iw Units: kg m2/s (no special name). Note similarity to: p=mv Newton’s 2nd Law for rotation: the torque due to external forces is equal to the rate of change of L. For a rigid body (constant I ), So, sometimes (but not always). Physics 1D03 - Lecture 30

Conservation of Angular momentum There are three great conservation laws in classical mechanics: Conservation of Energy Conservation of linear momentum and now, Conservation of Angular momentum: In an isolated system (no external torques), the total angular momentum is constant. Physics 1D03 - Lecture 30

Angular Momentum Vector w For a symmetrical, rotating, rigid body, the vector L will be along the axis of rotation, parallel to the vector w, and L = I w. L (In general L is not parallel to w, but Iw is still equal to the component of L along the rotation axis.) Physics 1D03 - Lecture 30

Angular momentum of a particle z f r v L O x y m This is the real definition of L. L is a vector. Like torque, it depends on the choice of origin (or “pivot”). If the particle motion is all in the x-y plane, L is parallel to the z axis.. Physics 1D03 - Lecture 30

Angular momentum of a particle (2-D): |L| = mrvt = mvr sin θ r v m For a particle travelling in a circle (constant |r|, θ=90), vt = rw, so: L = mrvt = mr2w = Iw Physics 1D03 - Lecture 30

Quiz As a car travels forwards, the angular momentum vector L of one of its wheels points: forwards backwards C) up D) down E) left F) right Physics 1D03 - Lecture 30

Quiz A physicist is spinning at the center of a frictionless turntable, holding a heavy physics book in each hand with his arms outstretched. As he brings his arms in, what happens to the angular momentum? increases decreases remains constant What happens to the angular velocity? Physics 1D03 - Lecture 30

Example: A student sits on a rotating chair, holding two weights each of mass 3.0kg. When his arms are extended to 1.0m from the axis of rotation his angular speed is 0.75 rad/s. The students then pulls the weights horizontally inward to 0.3m from the axis of rotation. Given that I = 3.0 kg m2 for the student and chair, what is the new angular speed of the student ? Physics 1D03 - Lecture 30

Example Angular momentum provides a neat approach to Atwood’s Machine. We will find the accelerations of the masses using “external torque = rate of change of L”. m1 m2 v w O R Physics 1D03 - Lecture 30

w m2 v m1 p1 Atwoods Machine, frictionless (at pivot), massive pulley For m1 : L1 = |r1 x p1|= Rp1 m1 m2 v w R O p1 so L1 = m1vR L2 = m2vR Lpulley= Iw = Iv/R Thus L = (m1 + m2 + I/R2)v R so dL/dt = (m1 + m2 + I/R2)a R p1 r R Torque, t = m1gR - m2gR = (m1 - m2 )gR Write t = dL/dt, and complete the calculation to solve for a. Note that we only consider the external torques on the entire system. Physics 1D03 - Lecture 30

Solution Physics 1D03 - Lecture 30

Summary Particle: Any collection of particles: L = I w. Rotating rigid body: L = I w. Newton’s 2nd Law for rotation: Angular momentum is conserved if there is no external torque. Physics 1D03 - Lecture 30