8:30 9:45 A.M. on 2/18, Friday (arrive by 8:15am) FMH 307 Exam covers Week 1 Week 3 in syllabus (Section in text book)

Transcription

1 Common exam 8:30 9:45 A.M. on /8, Friday (arrie by 8:5am) FMH 307 Exam coers Week Week 3 in syllabus (Section in text book) Bring scientific calculators To combat cheating, while taking the exams ) students ts must show their ID upon entering the classroom, ) no cell phone use, 3) if a student leaes the room during test time, e.g.men s/ladies room, he/she forfeits finishing the exam. hysics 06 Week 4 Work, Energy, Rolling, SJ 7 th Ed.: Chap 0.8 to 9, Chap. Work and rotational kinetic energy Today Rolling Kinetic energy of rolling Examples of Second Law applied to rolling

2 hys05 Reiew: Work/energy for linear motion Work-Kinetic Energy Theorem ΔK Kf Ki Δ W... work done by external forces Work : dw Fids F ds cosθ Kinetic energy K m point mass only, no rota : tion (including potentials) Mechanical Energy K(kinetic energy) + U(potential energy) ΔE ΔK + ΔU mech W non-conseratie only U otential energy, e.g., graity, U Δ mgh, and spring, U(/)kx, potential. Non-conseratie forces in 05/06 : all forces except graity and spring forces dw ds power F F dt dt Work done by a pure rotational motion Apply force F to mass at point r, causing rotation-only about axis. Displacement is only along θ (transerse) direction Find the work done by force applied to the object at as it rotates through an infinitesimal distance ds r dθr dθ d W F ds as for F cos(90 - F rsin( φ)dθ φ )ds Only the transerse component of the force (along the displacement) does work the same component that contributes to the torque: τ F rsin( φ) F translation dw τdθ The radial component of the force does no work because it is perpendicular to the displacement

3 Work-Kinetic Energy Theorem for pure rotation As object rotates from θ i to θ f work is done by the torque. Integrate ΔW θ f dw τdθ dw θ θ i θ i f τdθ Apply the Second Law dw ΔW ( τδθ τ I α d Iα dθ I dθ I d dt f I i if τ does not depend on θ d I is constant for rigid bodies f ΔW W I I K K Δ K i f,rot i,rot rot ) Instantaneous ower: Diide both sides of dw τdθ by dt dw dt dθ τ τ dt Example: The power output of a certain car is adertised to be 00 hp at 6000 rpm. What is the corresponding torque? hp 746 Watts 3

4 Example: An electric motor attached to a grindstone exerts a constant torque of τ 0 N.m. The moment of inertia of the grindstone is I.0 kg.m. The system starts from rest. a) Find the kinetic energy after 8 seconds. b) Find the work done by the motor during this time (Hint: Wtorque x angle change for a constant torque). c) Find the aerage power deliered by the motor. d) iclicker Q: Find instantaneous power at 0 second e) Find instantaneous power at 8 second. For motion with translation and rotation about center of mass Example: Rolling Ktotal Krot + K K rot I E K + U mech tot K U M graity Mgh 4

5 Energy Conseration: for translation + rotation about mass center () ΔK ΔE tot mech ΔK OR rot ΔK + ΔK tot ΔW + ΔU ΔW nc ΔW includes both conseratie and non-conseratie forces, treated as external to system ΔW nc includes only non-conseratie forces, ΔU contains the conseratie forces Comparison: pure translation ersus pure rotation ure Translation (Fixed Direction) ure Rotation (Fixed Axis) osition coordinate x Angular position θ Velocity dx/dy Angular elocity dθ/dt Acceleration a d/dt Mass m Newton's second law F net ma Angular acceleration α d/dt Rotational inertia I Newton's second law τ net Iα Work dw F dx Work dw τ dθ Kinetic energy K (m/) Kinetic energy K rot (I/) ower (constant force) F. ower (constant torque) τ Work KE theorem ΔW ΔK Same, include K rot ΔW ΔK A wheel rolling without slipping on a table The green line aboe is the path of the mass center of a wheel. The red cure shows the path (called a cycloid) swept out by a point on the rim of the wheel. When there is no slipping, there are simple relationships between the translational (mass center) and rotational motion. s Rθ R a αr 5

6 Rolling without slipping What does it mean for a wheel to roll rather than slide? No slipping contact point is stationary, o/w it would be sliding friction at is static friction fs μsn distance coered s arc swept out s as wheel rotates by Δθ Mass center moes along while wheel rotates around an axle (axis, ) Friction at the point of contact matches the rotation rate to the mass center speed, proiding torque. time t later time t r + s s rδθ Δθ + Connection between constant mass center elocity and the angular elocity of the wheel from the translational motion: s com Δt from the rotation: s r Δ θ r Δt NO SLIING IMLIES: s_s_ r tang Rolling pure rotation around CM + pure translation of CM Combination motions actual rolling motion the portion of the wheel at the bottom (at point ) is stationary the point at the top (at point T) is moing at speed com, (fastest) a) ure rotation b) ure translation c) Rolling motion Someone moing with As seen from the lab with the CM of the wheel rotation turned off, all sees tangential speed points on the wheel would tang r com moe at speed com for no slipping at center : lab 0 + com at bottom : lab com + com 0 at top : lab com + com com total elocity for a point lab tang + com 6

7 Rolling as pure rotation about contact point Complementary iews a snapshot in time Contact point is constantly changing A R cos( ϕ) R tang A φ φ R R tang 0 Stationary obserer sees rotation about the instantaneous center of rotation with angular elocity w_p R α p α Angular elocity and acceleration are the same about contact point or about CM. Kinetic energy for rolling, using contact point Show that: K tot K rot + K The total kinetic energy of a rolling object is the sum of the translational energy of its mass center of mass plus the rotational kinetic energy about its center of mass Stationary obserer ure rotation about (snapshot) K Ip Apply parallel axis theorem I I + MR K I + M R R rolling condition r KE of rotation about mass center axis KE of oerall mass center motion as a particle with mass M 7