Welcome back to Physics 215 Today’s agenda: • Moment of Inertia • Rotational Dynamics • Angular Momentum Physics 215 – Fall 2014 Lecture 11-1 1 Current homework assignment • HW8: – Knight Textbook Ch.12: 8, 26, 54, 58, 60, 62, 64 – Due Monday, Nov. 5th in recitation Physics 215 – Fall 2014 Lecture 11-1 2 Discussion Dw/Dt) S miri2 = tnet a - angular acceleration Moment of inertia, I Ia = tnet compare this with Newton’s 2nd law Ma = F Physics 215 – Fall 2014 Lecture 11-1 3 Moment of Inertia N I = m r + m2 r2 + … + m N rN = å mi ri 2 11 2 2 i=1 Physics 215 – Fall 2014 2 * I must be defined with respect to a particular axis Lecture 11-1 4 Moment of Inertia of Continuous Body Dm a 0 å N Þ I = å mi ri i=1 Physics 215 – Fall 2014 2 ò r Dm Þ I = ò r dm 2 Lecture 11-1 5 Tabulated Results for Moments of Inertia of some rigid, uniform objects Physics 215 – Fall 2014 Lecture 11-1 (from p.299 of University Physics, Young & Freedman) 6 Parallel-Axis Theorem D I = ICM + M D CM 2 D *Smallest I will always be along axis passing through CM Physics 215 – Fall 2014 Lecture 11-1 7 Practical Comments on Calculation of Moment of Inertia for Complex Object 1. To find I for a complex object, split it into simple geometrical shapes that can be found in Table 9.2 2. Use Table 9.2 to get ICM for each part about the axis parallel to the axis of rotation and going through the center-of-mass 3. If needed use parallel-axis theorem to get I for each part about the axis of rotation 4. Add up moments of inertia of all parts Physics 215 – Fall 2014 Lecture 11-1 8 Beam resting on pivot N r CM of beam r rm x M=? Vertical equilibrium? Mb = 2m SF = Rotational equilibrium? Physics 215 – Fall 2014 m St = M= N= Lecture 11-1 9 Suppose M replaced by M/2 ? • vertical equilibrium? • rotational dynamics? • net torque? SF = St = • which way rotates? • initial angular acceleration? Physics 215 – Fall 2014 Lecture 11-1 10 Moment of Inertia? I = Smiri2 * depends on pivot position! I= * Hence a = tI = Physics 215 – Fall 2014 Lecture 11-1 11 Rotational Kinetic Energy K = Si(12mivi2 = (1/2)w2Simiri2 • Hence K = (1/2)Iw2 • This is the energy that a rigid body possesses by virtue of rotation Physics 215 – Fall 2014 Lecture 11-1 12 Spinning a cylinder 2R F Cable wrapped around cylinder. Pull off with constant force F. Suppose unwind a distance d of cable • What is final angular speed of cylinder? • Use work-KE theorem W = Fd = Kf = (1/2)Iw2 • Mom. of inertia of cyl.? -- from table: (1/2)mR2 – from table: (1/2)mR2 w = [2Fd/(mR2/2)]1/2 = [4Fd/(mR2)]1/2 Physics 215 – Fall 2014 Lecture 11-1 13 cylinder+cable problem -constant acceleration method N F extended free body diagram * no torque due to N or FW * why direction of N ? FW radius R * torque due to t = FR * hence a = FR/[(1/2)MR2] = 2F/(MR) D = (1/2)at2 = d/R; t = [(MR/F)(d/R)]1/2 w = at = [4Fd/(MR2)] 1/2 Physics 215 – Fall 2014 Lecture 11-1 14 Angular Momentum • can define rotational analog of linear momentum called angular momentum • in absence of external torque it will be conserved in time • True even in situations where Newton’s laws fail …. Physics 215 – Fall 2014 Lecture 11-1 15 Definition of Angular Momentum * Back to slide on rotational dynamics: miri2Dw/Dt = ti * Rewrite, using li = Dli/Dt = ti miri2w w pivot ri Fi mi * Summing over all particles in body: DL/Dt = text L = angular momentum = Iw Physics 215 – Fall 2014 Lecture 11-1 16 An ice skater spins about a vertical axis through her body with her arms held out. As she draws her arms in, her angular velocity • • • • 1. increases 2. decreases 3. remains the same 4. need more information Physics 215 – Fall 2014 Lecture 11-1 17 Angular Momentum 1. r p O Point particle: |L| = |r||p|sin() = m|r||v| sin() vector form L = r x p – direction of L given by right hand rule (into paper here) L = mvr if v is at 900 to r for single particle Physics 215 – Fall 2014 Lecture 11-1 18 Angular Momentum 2. w o rigid body: * |L| = I w fixed axis of rotation) * direction – along axis – into paper here Physics 215 – Fall 2014 Lecture 11-1 19 Rotational Dynamics t = Ia DL/Dt = t These are equivalent statements If no net external torque: t = 0 * L is constant in time * Conservation of Angular Momentum * Internal forces/torques do not contribute to external torque. Physics 215 – Fall 2014 Lecture 11-1 20 Bicycle wheel demo • Spin wheel, then step onto platform • Apply force to tilt axle of wheel Physics 215 – Fall 2014 Lecture 11-1 21 Linear and rotational motion • Force • Torque • Acceleration • Angular acceleration Fnet = å F = ma • Momentum • Kinetic energy K = mv Physics 215 – Fall 2014 • Angular momentum L = Iw p = mv 1 2 t net = åt = Ia • Kinetic energy 2 Lecture 11-1 22 A hammer is held horizontally and then released. Which way will it fall? Physics 215 – Fall 2014 Lecture 11-1 23 General motion of extended objects • Net force acceleration of CM • Net torque about CM angular acceleration (rotation) about CM • Resultant motion is superposition of these two motions • Total kinetic energy K = KCM + Krot Physics 215 – Fall 2014 Lecture 11-1 24 Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown. After the force has stopped acting on each block, which block will spin the fastest? 1. 2. 3. 4. A. B. C. A and C. Physics 215 – Fall 2014 Top-view diagram Lecture 11-1 25 Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown. After each force has stopped acting, which block’s center of mass will have the greatest speed? 1. 2. 3. 4. A. B. C. Top-view diagram A, B, and C have the same C.O.M. speed. Physics 215 – Fall 2014 Lecture 11-1 26 Reading assignment • Rotational dynamics, rolling • Remainder of Ch. 12 in textbook Physics 215 – Fall 2014 Lecture 11-1 27